Reinforcement Learning

In the forward reinforcement-learning problem, the reward is fixed and known; the learner is asked to find a good policy or value function. Here we turn the question around. Given offline data generated by an expert, can we recover the reward the expert was optimizing? This is the inverse reinforcement learning problem, and remarkably, two communities, structural econometricians studying dynamic discrete choice (DDC) and machine learners studying entropy-regularized IRL, have been working on exactly the same probabilistic model under different names. We begin by proving their equivalence. We then develop the classical identification result of Magnac and Thesmar and the classical computational paradigms that grew out of it: Rust's nested fixed-point algorithm, the conditional-choice-probability approach of Hotz and Miller, and the two temporal-difference approaches of Adusumilli and Eckardt: linear semi-gradient TD and approximate value iteration. Each route has its limits: dimensionality, transition-kernel estimation, the deadly triad, or projected fixed-point bias. We then walk through the modern ML/IRL strand: adversarial IRL, occupancy matching, IQ-Learn, and offline ML-IRL, deriving each method's actual objective and stating precisely what it does and does not identify. We close with the empirical-risk-minimization framework of Kang et al., which yields a gradient-based estimator for offline IRL/DDC.

We identify a new dimension for enhancing rollout diversity in Group Relative Policy Optimization (GRPO) for LLMs. While GRPO relies on diverse rollouts, prevailing strategies primarily increase diversity by injecting more token-level randomness, which may introduce step-wise noise and lead to incoherent trajectories. We uncover that smaller models within the same model family inherently exhibit higher policy-level diversity, indicated by their superior pass@k relative to larger counterparts as sample counts increase. Unlike token-level noise, this diversity is temporally correlated, preserves logical consistency, and provides structured exploration signals for gradient estimation. We thus propose S2L-PO (Small-to-Large Policy Optimization), a framework that leverages fixed small models as natural explorers to train larger models. To balance exploration and exploitation, we design a progressive annealing strategy that transitions from offline small-model rollouts to the large learner's own sampling. This shift elegantly avoids mid-training performance drops caused by the small model's capacity limits, achieving faster convergence and unlocking a higher performance ceiling. S2L-PO improves accuracy on diverse mathematical reasoning benchmarks (e.g., +8.8% on AIME 24 using a 1.7B explorer to guide the 8B model) while reducing rollout compute.

Offline-to-Online Reinforcement Learning (O2O-RL) leverages an offline, pre-trained policy to minimize costly online interactions. Although data-efficient, O2O-RL is susceptible to shifts between offline and online distributions. Existing work aims to mitigate the harm of this shift by finetuning the policy on trajectory data sampled from a diffusion model. Inspired by this line of work, we propose DUAL: an efficient \textbf{D}iffusion \textbf{U}ncertainty-\textbf{A}ware framework for offline-to-online reinforcement \textbf{L}earning. DUAL utilizes the prior knowledge of the diffusion model to distill a fast-sampling diffusion actor policy and transition model in the offline phase. DUAL also employs a Laplace approximation and distance transition-state-shift detection, thereby using uncertainty quantification to improve exploration versus exploitation in the online phase. We formally show that our actor loss with the Laplace approximation provides a proxy for a principled estimate of epistemic uncertainty. Empirically, DUAL improves the online expected return over O2O-RL baselines across multiple settings and environments.

In many practical reinforcement learning environments, observations are far higher-dimensional than the variables that matter for control. In this work, we ask: can we learn representations that capture only control-relevant features of the environment? We study this question through the empowerment objective, which maximizes an agent's influence over the environment and is widely used for unsupervised skill learning. We show that empowerment agents induce two distinct representations -- forward and backward -- that capture complementary aspects of the state, and both of which are invariant to control-irrelevant features. Thus, empowerment maximization leads agents to learn an implicit, control-centric model of the world. Our analysis highlights the importance of learning representations through interaction rather than from passive datasets: interaction aimed at maximizing control is essential for learning useful invariance properties, a perspective that aligns closely with the causal learning literature.

Continuous control policies trained with off-policy reinforcement learning frequently exhibit high-frequency action jitter, impractical for direct deployment on physical actuators. Post-hoc filtering attenuates jitter but adds phase lag; embedding smoothness penalties in the actor's loss couples them with the RL gradient and conflates reward regression with over-aggressive smoothing. We present ZAPS-DA, which reduces action jitter at deployment with negligible phase lag and no post-processing. ZAPS-DA pairs an unmodified main actor (trained by the base RL loss) with a separate decoupled actor trained via supervised imitation of zero-phase filtered targets stored in the replay buffer. The deployed policy is the decoupled actor: a feed-forward map from observation to smooth action, with no inference-time filter and no action-history input -- causal distillation of a non-causal filter. A magnitude-matched MSE loss gives zero-hyperparameter portability across optimizer classes. Validated with Soft Actor-Critic and a Savitzky-Golay filter in two driving simulators (paired n=150): on MetaDrive (anchor protocol), ZAPS-DA cuts steering jitter 14-21x and throttle jitter 3-5x (all $p<10^{-4}$, Bonferroni) while matching task-completion (p=0.28 success, p=0.31 crash) at 6.3% reward cost; on a custom Webots adaptive cruise control task, the same configuration yields a Pareto improvement -- reward parity (p=0.121), 8-45x steering-jitter reduction, task-failure rate 2.0% to 0.7%. Against CAPS, the standard penalty-based baseline -- at both its auto-entropy and native fixed-entropy operating points, with penalty weight, spatial noise, and entropy coefficient re-tuned per environment -- ZAPS-DA reaches 14.7x steering-jitter reduction versus CAPS's best 3.2x at matched seeds, a ~4.6x gap, with no per-environment tuning of the smoothness signal and post-hoc applicability to trained policies.

Exploration in reinforcement learning for autonomous driving is inherently unsafe: agents must experience novel behaviors to learn, yet exploration can lead to collisions or off-road driving. We propose an uncertainty-aware framework that leverages expert advice to guide exploration while avoiding long-term dependence. Advice is triggered when epistemic or aleatoric uncertainty exceeds adaptive thresholds derived from rolling buffers, ensuring advice evolves with the agent's confidence. A commitment-cooldown strategy with a stochastic early-stop heuristic regulates the duration and frequency of guidance, exposing the agent to coherent maneuvers without exhausting the advice budget. Expert and agent experiences are combined in a shared replay buffer within an off-policy implicit quantile network (IQN) backbone, enabling efficient reuse of expert trajectories. Experiments in CARLA show that our method outperforms the IQN baseline, improving success by 5-7% and reducing failures, demonstrating that risk-sensitive uncertainty coupled with regulated expert integration enables safer and more efficient exploration for sensor-based RL policy learning in unsignalized intersection navigation.

Learning to reach arbitrary goals from sparse feedback requires agents to infer a rich notion of reachability across state--goal pairs. Goal-conditioned reinforcement learning (GCRL) tackles this challenge by learning policies that generalize across goals, but this generalization becomes increasingly difficult as the underlying dynamics become high-dimensional, hybrid, or contact-dependent. To address this issue, physics-informed GCRL (Pi-GCRL) introduces optimal-control-inspired inductive biases into goal-conditioned value learning. While Pi-GCRL methods have proven effective in navigation and object-free goal-reaching domains, their reliability in contact-rich tasks remains unclear, where contact interactions induce hybrid dynamics, mode-dependent controllability, and nonsmooth value landscapes. In this work, we show that these structural properties can cause existing Pi-GCRL methods to degrade when applied naively to contact-rich manipulation. Motivated by this analysis, we introduce contact-aware and hierarchical formulations that apply physics-informed inductive biases selectively across the manipulation problem. Our results provide a principled step toward extending Pi-GCRL to contact-rich manipulation.

We present Stable-Layers, a reinforcement learning framework that eliminates the need for paired supervision by fine-tuning a pretrained layer decomposition model using only feedback from a vision-language model (VLM). Starting from Qwen-Image-Layered, we apply Flow-GRPO with LoRA adaptation, sampling multiple candidate decompositions per image, scoring them with a VLM, and optimising the policy from group-relative advantages. The key challenge lies in designing a reliable reward signal: VLMs scoring samples in isolation tend to compress their judgements into a narrow band, leaving GRPO with little within-group variance to learn from. We address this with a two-stage evaluation pipeline that pairs structured per-sample scoring across five edit-centric criteria with a grid-based calibration step in which the VLM re-scores all candidates side-by-side. Stable-Layers produces decompositions with stronger layer separation, fewer blank or artifact-heavy layers, and lower per-layer reconstruction error on the Crello dataset compared to the base model.

While Reinforcement Learning with Verifiable Rewards (RLVR) is effective for deterministically checkable tasks, many vision-language tasks are partially verifiable, demanding multi-criteria supervision (e.g., perceptual details, reasoning steps, and constraints). Rubrics provide a natural interface for this fine-grained supervision, but their effectiveness depends on the execution accuracy during online RL. We propose Reinforcement Learning with Robust Rubric Rewards ($\text{RLR}^3$), extending RLVR from task-level verification to criterion-level verification. $\text{RLR}^3$ routes instance-specific rubrics through two execution paths: an LLM-as-an-extractor paired with a deterministic verifier, or an LLM-as-a-Judge for non-verifiable criteria. To ensure faithful scoring, $\text{RLR}^3$ introduce a minimal exposure strategy that masks ground truths from extractors and images from judges. Furthermore, $\text{RLR}^3$ employs hierarchical aggregation to prioritize essential criteria over additional criteria, and mitigates score saturation within rollout groups. Evaluated on Qwen3-VL-30B-A3B across 15 benchmarks, $\text{RLR}^3$ consistently outperforms RLVR, yielding a 4.7-point improvement over the base model and exceeding the official instruct-to-thinking model gap. Controlled audits confirm our deterministic verification and minimal exposure significantly reduce exploitable false positives.

We introduce TriSearch, a reinforcement learning framework for optimizing objectives over triangulations of a polytope via bistellar flips. The key idea is a circuit-supported subtriangulation action representation: feasible flips are encoded by their supporting circuit and realized local subtriangulation, enabling a learned policy to rank them using local geometric and combinatorial features. This yields a dimension-agnostic interface and enables efficient traversal of the flip graph without explicit enumeration of the full triangulation space. Instantiated in 3D and 4D, TriSearch generalizes zero-shot from small training instances to larger polytopes with exponentially larger search spaces. It achieves top performance on metric objectives in 3D and, in 4D, discovers more distinct Fine, Regular, Star triangulations of reflexive polytopes, corresponding to Calabi-Yau threefolds, than existing samplers under a fixed budget.

Chaotic dynamical systems pose a fundamental challenge for Reinforcement Learning (RL): exponential sensitivity to initial conditions induces high-variance bootstrap targets and poorly conditioned gradient updates. Chaotic dynamics arise across scientific and engineering domains, from fluid flows and climate systems to multi-agent systems, where reliable learning is highly desirable. Standard RL methods optimise expected returns through scalar value functions, implicitly averaging over diverging trajectories and entangling trajectory level instability with the learning objective. We show that under mild statistical stability assumptions, the return distribution evolves more regularly than individual trajectories when measured under the $1$-Wasserstein metric, yielding a smoother distributional Bellman objective. By aligning optimisation with this measure level structure, distributional RL provides better conditioned learning. We offer a principled explanation for the advantages of distributional methods in chaotic systems and the geometries of RL objectives under chaos.

Correctness-based Reinforcement Learning with Verifiable Rewards (RLVR) trains language models from binary feedback on sampled outputs, but the objective optimized in expectation and the stochastic update geometry induced by finite rollout groups are often conflated. This paper develops RL2ML, a family of finite-rollout surrogate objectives with a closed-form, exactly unbiased gradient estimator. The family continuously connects standard reinforcement learning, maximum-likelihood-like training, and beyond-maximum-likelihood objectives while preserving estimator-objective alignment under a fixed rollout budget. We introduce the group-level update scale to characterize how a rollout group is reweighted after its empirical success count is observed, revealing a subcritical-supercritical update-scale transition that is hidden by population-level objective notation alone. Building on this distinction, calibrated metric-gain analysis and exact variance decomposition show that the best choice of surrogate objective is determined neither by proximity to maximum likelihood nor by the population-level weight alone. Instead, it depends jointly on the evaluation metric, local sensitivity, and estimator variance. The remaining degree of freedom in the surrogate objective family can therefore be formulated as a one-dimensional optimization problem rather than treated as an unconstrained hyperparameter.

Multi-step robot manipulation requires acting under uncertainty about how the scene will evolve, making exploration and policy adaptation challenging. We study whether short-horizon, task-consistent future videos can provide useful structured priors for control and reinforcement-learning fine-tuning. We formalize this idea through Future-Experience Conditioning (FEC), a simple interface that conditions closed-loop policies on a latent representation of a short future video. In our simulation setup, future clips are generated in three stages, an LLM reasoner operating over a task ontology initialized from the current scene state, a robot-free digital-twin rollout of the intended object motion, and a mask-free video diffusion model that synthesizes a robot-consistent future clip without requiring segmentation at inference. We instantiate this future-conditioning interface primarily with BC and BC+RL, and compare against a future-conditioned Streaming Flow Policy (SFP) baseline on RoboCasa and CALVIN under NoFuture, GTFuture, GenFuture, and WrongFuture. Generated futures improve performance over no-future conditioning, while mismatched futures degrade it, and our BC+RL instantiation achieves the strongest overall results. An average BC+RL learning-curve analysis across 8 CALVIN tasks further shows that GTFuture improves fastest, GenFuture improves earlier and to a higher level than NoFuture, and WrongFuture remains at zero throughout training. These results suggest that short-horizon future videos can serve as useful structured priors for exploration and policy adaptation under imperfect future predictions. https://enact2026.github.io/

When a large language model under reinforcement learning commits a wrong reasoning step early in a trajectory, standard algorithms force it to keep generating until the maximum horizon, spending compute on tokens that never receive positive reward and polluting advantage estimates with post-failure noise. We propose ESPO (Early-Stopping Proximal Policy Optimization), which detects trajectory failure on-the-fly and terminates rollouts early. At each generation step, ESPO computes a surrogate regret using only the logits already computed during sampling, and terminates when the smoothed cumulative regret significantly exceeds its estimated values. Truncated trajectories are treated as absorbing failure states with a terminal reward, concentrating negative temporal-difference (TD) errors near the detected failure step without any additional reward model or human annotation. On DeepSeek-R1-Distill-Qwen-7B trained for mathematical reasoning, ESPO surpasses PPO on AIME2024 (46.28% vs. 45.25%), AMC2023 (85.83% vs. 82.94%), and MATH-500 (87.42% vs. 85.43%), while saving more than 20% rollout tokens cumulatively.

Reinforcement Learning (RL) has advanced Large Language Models (LLMs) in verifiable domains, while open-ended generation remains challenging due to the absence of definitive rewards. Rubric-based RL provides explicit evaluation criteria, but learning to construct these criteria remains challenging when final-answer correctness is not verifiable. We propose EvoRubric, a co-evolutionary RL framework that combines criterion-validity feedback, response discrimination, and peer agreement to learn rubrics for open-ended generation. A shared policy acts as both a Reasoner and a Rubric Generator, using its current responses and historical rubrics to discover new evaluation dimensions. To combine adaptive rubric discovery with a stable validity check, a frozen copy of the initial policy serves as the Meta-Verifier, while a frozen Grader scores responses against the retained criteria. Discriminative feedback, Leave-One-Out peer consensus, and a persistent memory pool transform this feedback into complementary rewards that jointly optimize both policy roles, closing the loop between response improvement and rubric discovery. EvoRubric improves over matched static and external evolving-rubric baselines across five benchmarks in Medical, Writing, and Science. Across three training seeds, it achieves five-benchmark averages of 56.28 at 8B and 61.13 at 14B, exceeding the strongest matched baselines by 3.09 and 2.06 points, respectively. Human audits assess criterion validity and response quality, and experiments with expert-initialized rubrics demonstrate compatibility with human priors.

Urban traffic congestion is a growing global issue contributing significantly to long commute times and environmental pollution. Traditional traffic signal control systems often fail to adapt to dynamic traffic conditions. Adaptive traffic signal control can improve urban traffic without changing road infrastructure. Deep Reinforcement Learning (DRL) has shown strong performance for this task, but existing delay and queue-based rewards often produce short-sighted or unstable policies. This paper proposes a Momentum-Based Reward Function (MBRF) that encourages vehicles to keep moving rather than penalizing congestion alone. The method is evaluated in SUMO (Simulation of Urban MObility) using standard traffic metrics such as waiting time, queue length, throughput, and CO2 emissions. Results show that the proposed reward produces better throughput-emission trade-offs and more stable learning behavior than delay or queue-based rewards, as well as classical controllers such as Max Pressure and LQF.

Tool-Integrated Reasoning (TIR) extends LLM capabilities by leveraging external environments. However, existing methods lack the deliberation during sequential tool invocation required for strategic planning and self-correction. While RL mitigates this, conventional approaches for Tool-Integrated Reasoning are hindered by sparse outcome-based rewards, failing to supervise intermediate reasoning steps and tool invocations. To address this, we propose DeepTool, a novel framework that scales deliberate thinking within the interleaved process of thinking, action, and observation at each turn. In DeepTool, we first introduce a synthesis pipeline that evolves extended thinking into interleaved trajectories, integrating adversarial perturbations to ensure robustness and self-correction. Secondly, we devise Process-Supervised Reinforcement Learning based on GRPO, which utilizes an Action-Centric Process Reward to reinforce intermediate interleaved thinking and enforce precise tool invocation at every turn. Extensive experiments demonstrate that DeepTool achieves superior performance, boosting Qwen2.5-7B significantly across six benchmarks (e.g., AIME24: 3.2% -> 40.4% and HMMT25: 0.0% -> 28.6%). Furthermore, the token cost-effectiveness analysis confirms the utility of interleaved thinking, demonstrating DeepTool's optimal balance between performance and token efficiency.

Open-ended post-training benefits from rewards that make prompt-specific success conditions explicit, rather than relying only on post-hoc scalar scores. In instruction following, writing, and decision-support tasks, response quality depends on local requirements, holistic preferences, and explicit constraints, but existing reward methods often leave these criteria implicit or cover only narrowly verifiable cases. We propose a prompt-level reward specification framework that separates reward specification from reward computation. Given only prompts, our framework constructs reusable task-adaptive rubrics and executable hard-constraint checkers offline, making reward criteria explicit before training and reusable across rollouts. At scoring time, artifact-anchored rubric and code scores are combined with an independent global score for residual holistic quality, yielding a normalized hybrid reward over requirement satisfaction, holistic quality, and deterministic constraints. The framework requires no human preference annotations, reference answers, or a separately trained reward model. Experiments show that the resulting reward improves offline RM-style response ranking and supports online reinforcement learning across multiple open-ended benchmarks. Ablations further show that rubrics, global scoring, and executable verification provide complementary supervision.

Computer use agents (CUAs) have shown strong potential for automating complex digital workflows, yet their training remains constrained by costly live environment interaction and limited high-quality supervision. Existing filtered behavior cloning pipelines suffer from imitation bottlenecks, including distribution shift from the expert demonstration and the absence of negative learning signals. Meanwhile, standard trajectory-level reinforcement learning struggles with sparse rewards, ambiguous credit assignment, and high infrastructure costs for long-horizon GUI interaction. In this work, we propose PRO-CUA, a process-reward optimization framework for training CUAs with iterative step-level reinforcement learning. PRO-CUA decouples on-policy environment interaction from policy optimization: the current policy collects states through live rollouts, generates diverse candidate actions for each state, receives step-level feedback from a process reward model (PRM), and is optimized with group-relative advantages. This design enables dense and flexible credit assignment without relying on golden answers or offline expert trajectories, while reducing distribution shift by training on the agent's own execution states. Experiments on live web benchmarks demonstrate the effectiveness of PRO-CUA and the reliability of PRM-guided step-level training.

Event-driven scheduling policies are increasingly deployed in industrial environments, where decisions are made under asynchronous and partially observed system states. As a result, decision states are not temporally consistent, action admissibility is not explicitly defined, and the origin of execution errors remains ambiguous. These issues limit both reliability and interpretability. To address this gap, a policy-neutral execution and measurement layer is proposed to mediate between scheduling policies and the industrial execution environment. The layer constructs decision-valid snapshots from asynchronous event streams, defines a standardized execution contract with explicit action admissibility, and records outcomes as divergences between policy intent, transactional outcomes, physical execution, and human intervention. This enables a separation between decision semantics and execution behavior and makes deployment mismatch observable and structurally attributable. The proposed framework is evaluated using a discrete-event simulation. The results show analytical benefits across all observation lag regimes, as undifferentiated execution failures are transformed into structured, typed outcomes with full attribution coverage. Operational benefits are strongest under low observation lag, where avoidable execution errors can be prevented before commitment. Overall, the layer turns execution uncertainty into supervisory data for evaluation and policy refinement.

Conditioned Sequence Models (CSMs) learn policies by treating return-to-go (RTG) as a control signal. However, existing CSMs often treat the RTGs as simple numerical inputs rather than aligning them with the performance of their policies. In this paper, we propose Q-ALIGN DT, a framework that enforces this alignment by ensuring the $Q$-value of the output policy is consistent with the input RTG. By leveraging a $Q$ function to provide dense guidance to CSMs and further fine-tuning it using an RTG-perturbation technique with the CSM, our method ensures that higher RTGs are consistently mapped to trajectories with higher expected returns. Theoretically, we show that Q-ALIGN DT can efficiently learn the desired policy and output a near-optimal one when the RTG is sufficiently high. Empirically, we demonstrate through extensive experiments that Q-ALIGN DT achieves superior controllability and performance across the D4RL benchmark. Remarkably, our model effectively learns a structured family of policies that maintains precise alignment and generalizes to tasks like velocity-tracking where prior methods fail.

Post-training large language models with reinforcement learning is bottlenecked by the reward signal. Existing approaches require either ground-truth verifiable rewards, restricting training to domains with automatic correctness checks (e.g., mathematics, code execution), or human preference labels, which are expensive to collect and prone to reward hacking. Recent label-free methods replace ground-truth verifiers with self-referential signals like majority voting or token entropy over a model's own outputs, but risk reinforcing a model's own errors. In this work we propose Cross-Model Entropy (CME), the mean log-likelihood of a generator's response under a separate verifier model, as a label-free reward signal for RL post-training. CME is continuous, training-free, and grounded in the principle that responses a verifier finds unsurprising are likely correct or high quality. Because the verifier is independent of the generator, the signal cannot be gamed through self-consistency. We integrate CME into GRPO with no other changes to the training loop, extending label-free RL to open-ended instruction following -- a regime where self-referential signals are inapplicable or poorly suited. On open-ended instruction following (UltraFeedback prompts, evaluated on AlpacaEval 2.0), CME rewards beat the untrained base in head-to-head LLM-as-Judge comparisons across four model families (Qwen, Llama, Gemma, OLMo) and three training regimes (pretrained, SFT, and instruction-tuned), with tie-adjusted win rates ranging from 52.5% to 71.4%. Code will be released upon publication.

Data acquisition efficiency is a central challenge in deploying reinforcement learning in business and healthcare operations, where interactions are costly, slow, and often involve humans in the loop. This paper develops a unified large deviations framework for data acquisition in infinite-horizon reinforcement learning. We introduce the exponential decay rate of the policy-selection error probability as a principled efficiency metric and derive a variational characterization of this rate via large deviations theory for Markov chains, yielding a nested optimization problem. Based on this characterization, we formalize two complementary notions of optimality in terms of the optimal solution of the nested problem. Because the resulting program is implicit and generally intractable, we propose a tractable convex relaxation with explicit constraints. We then develop a lazy one-step projected subgradient method to solve the relaxed problem and use its iterates to construct an adaptive data acquisition policy. We prove that the resulting reinforcement learning algorithm is near-robustly optimal under our optimality criterion, up to a constant factor. Finally, we extend the framework to linear function approximation to improve scalability, and numerical experiments support the effectiveness of the proposed approach.

Reinforcement Learning from Verifiable Rewards (RLVR) has improved language models in domains such as mathematics and code, where correctness can be checked automatically. However, many important tasks are only partially verifiable: prompts contain multiple requirements, responses may satisfy some but not all of them, or no single reference answer might exist. We introduce Soft-RLVR, a framework for reinforcement learning from decomposed, learned verification signals. Soft-RLVR converts each prompt into a checklist of atomic requirements, scores candidate responses item by item with an LLM verifier, and trains on the resulting soft reward. Checklist-based rewards turn sparse pass/fail supervision into a denser partial-credit signal, but they also introduce a tradeoff: averaging item-level judgments can reduce verifier noise, while partial credit can reward incomplete responses. We formalize this tradeoff and identify conditions under which checklist-based verification gives a more reliable RL training signal than holistic verification. We further introduce Soft-SVeRL, a self-verifying variant of Soft-RLVR in which the policy also acts as the verifier. We show that self-verification is prone to reward inflation from overly permissive self-judgments, and that explicit stabilization is needed to prevent this collapse. In a controlled instruction-following setting with rule-based ground-truth evaluation, checklist-based Soft-RLVR improves IFEval by up to 11.1 points using only learned verifier rewards. Our experiments further show that verifier quality and checklist quality both affect downstream RL outcomes, and that explicit stabilization is essential for effective self-verification.

Reinforcement Learning with Verifiable Reward (RLVR) is empirically shown to notably enhance the reasoning performance of large language models (LLMs), particularly in mathematics and programming. However, the mechanistic role of Sample Difficulty in RLVR remains poorly understood. In this paper, we investigate RLVR through the lens of difficulty-wise and one-sample analysis. We find that sample difficulty has a non-monotonic effect on RLVR: easy and medium-difficulty problems yield the strongest and most stable reasoning improvements, whereas overly hard problems often provide weak learning signals, induce degenerate behaviors such as answer repetition or skipping necessary computation, and can ultimately degrade the model's pre-existing capabilities. Beyond the obverse of response, we further analyze the model's internal feature dynamics using Temporal Sparse Autoencoders (T-SAE). Easy problems mainly reinforce direct-answer and basic-computation features while suppressing deliberative-reasoning features; hard problems activate reasoning-related features but become useful only when successful trajectories are sampled; medium-difficulty problems provide a more balanced signal, strengthening both computation and multi-step reasoning features. Motivated by these findings, we propose difficulty-adaptive strategies for hard-sample utilization, using backward-reasoning reformulation and T-SAE-guided training signals to improve reward density and credit assignment during RLVR. Overall, our results identify sample difficulty as a key factor governing both the optimization dynamics and representation evolution of RLVR.

Reinforcement learning with multinomial logistic (MNL) function approximation has become an important framework due to its flexibility and broad applicability. While existing studies have established regret guarantees under worst-case analysis, they do not capture how performance depends on the variability of the interaction between the learner and the environment. In this paper, we develop a new theoretical analysis for MNL-based Markov decision processes that yields explicit variance-adaptive regret bounds. Our algorithm is computationally efficient and achieves the instance-wise optimal rate of regret, narrowing the gap between upper and lower bounds. Our numerical experiments validate that our method learns optimal policies more efficiently than conventional approaches.

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.

Reinforcement learning with verifiable rewards (RLVR) has become a key technique for en- hancing LLM reasoning, yet its data ineffi- ciency remains a major bottleneck. Existing methods address this problem only partially, each missing at least one of subset-level cov- erage, verifier signal use, or interpretability. To address this gap, we present IRDS (Inter- pretable RLVR Data Selection), which selects RLVR training instances on a sparse autoen- coder (SAE) cluster basis so the selection itself is auditable on recognizable problem motifs. To select instances the model both fails on and can still learn from, we introduce a verifier- coupled coverage objective on the SAE basis and solve it by greedy log-determinant max- imization. Experiments on three instruction- tuned models and six math reasoning bench- marks show that IRDS achieves the highest overall accuracy, exceeding the strongest base- line by +3.9/+4.0 pp on the two Qwen models and by +0.5 pp on Llama-3.1-8B, while run- ning an order of magnitude cheaper than the trajectory-based baseline.

Large scale reinforcement learning has become a central tool for improving reasoning in large language models. At this scale, generation is often lagged or asynchronous, so updates are performed on data collected by older policies. This makes learning inherently off-policy. Most existing approaches nevertheless remain rooted in PPO-style trust-region objectives, treating training as approximately on-policy and using importance weights to correct distribution mismatch. These corrections can introduce high variance, destabilize optimization, and accelerate entropy collapse. Recent work suggests an alternative: rather than correcting the mismatch, one can embrace off-policy data and remove importance weights, often yielding stronger algorithms. In this paper, we provide an intuitive construction of off-policy objectives that include successful off-policy objectives and show that their effectiveness can be understood through implicit pessimism: they optimize toward target policies that are more conservative than their nominal objectives suggest. This perspective explains why some particular implementation choices improve stability: they implicitly control the effective target distribution. We then propose a principled modification that stabilize this induced distribution and improve off-policy learning.

Agentic large language models are increasingly used to solve real-world tasks by reasoning over goals, invoking tools, and interacting with external environments. Reinforcement learning provides a natural framework for improving these behaviors, and recent agent RL methods have achieved strong results across domains. However, the training dynamics of agent RL remain poorly understood, limiting our ability to diagnose instabilities and design more effective training algorithms. In this work, we identify a previously underexplored phenomenon in agent RL, which we term cyclical entropy eruption. Unlike single-turn reasoning RL, where entropy typically collapses and stays low, agent RL training exhibits unique recurring cycles of sharp entropy eruption and gradual subsidence. We decompose this dynamic into three phases and provide theoretical and empirical analyses of each, explaining the mechanisms underlying its cyclical oscillation. We further show that degenerate patterns such as sentence duplication and hallucination, once acquired during eruption, can persist and accumulate across cycles. Motivated by these findings, we propose SEAL (Separation-Enhanced Agent Learning), a lightweight auxiliary loss that separates correct and incorrect trajectories in representation space, directly targeting the root cause of entropy eruption. Experiments across multiple benchmarks, models, and RL algorithms demonstrate that SEAL stabilizes training and yields stronger downstream agent performance.