ICLR 2026 Papers

Combinatorial online learning is a fundamental task for selecting the optimal action (or super arm) as a combination of base arms in sequential interactions with systems providing stochastic rewards. It is applicable to diverse domains such as robotics, social advertising, network routing, and recommendation systems. In many real-world scenarios, we often encounter rising rewards, where playing a base arm not only provides an instantaneous reward but also contributes to the enhancement of future rewards, e.g., robots improving through practice and social influence strengthening in the history of successful recommendations. Crucially, these enhancements may propagate to multiple super arms that share the same base arms, introducing dependencies beyond the scope of existing bandit models. To address this gap, we introduce the Combinatorial Rising Bandit (CRB) framework and propose a provably efficient and empirically effective algorithm, Combinatorial Rising Upper Confidence Bound (CRUCB). We empirically demonstrate the effectiveness of CRUCB in realistic deep reinforcement learning environments and synthetic settings, while our theoretical analysis establishes tight regret bounds. Together, they underscore the practical impact and theoretical rigor of our approach.

Reinforcement learning (RL) promises to expand the capabilities of language models, but it is unclear if current RL techniques promote the discovery of novel behaviors, or simply sharpen those already present in the base model. In this paper, we investigate the value of deliberate exploration---explicitly incentivizing the model to discover novel and diverse behaviors---and aim to understand how the knowledge in pre-trained models can guide this search. Our main finding is that exploration with a simple, principled, representation-based bonus derived from the pre-trained language model's hidden states significantly improves diversity and pass@k rates---both for post-training, and in a novel inference-time scaling setting we introduce. (1) For inference-time, exploration with representation-based diversity improves efficiency, consistently improving pass@k rates across a variety of models and reasoning tasks. For example, for Qwen-2.5-14b-Instruct we obtain over 50\% improvement in verifier efficiency on almost all considered tasks. (2) For post-training, we show that integrating this exploration strategy into an RL pipeline improves reasoning performance over that of the initial model and over standard RL post-training. For example, on AIME 2024, our post-trained Qwen-2.5-7b-Instruct's pass@80 matches the pass@256 of GRPO on the same model, demonstrating a 3x improvement in test-time sample efficiency. Overall, our findings suggest that deliberate exploration---with the right notion of diversity---is a practical path toward discovery of new behaviors beyond sharpening.

The Schrödinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms allow to infer such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schrödinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schrödinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method.

Code evaluation and reinforcement learning rely critically on test cases. However, collecting golden test cases is hard and expensive, motivating the use of LLMs for automatic test case generation. This, in turn, raises a pivotal challenge: how can we rigorously evaluate the quality of the generated test cases? Existing benchmarks often evaluate the exclusion ratio on large, unstructured collections of wrong codes, leading to high computational costs and severe score inflation. Furthermore, they inadvertently reward generators that detect common, trivial bugs, while failing to penalize their inability to identify rare yet critical faults. In this work, we connect two fundamental questions: (1) What is the minimal set of wrong codes sufficient to represent the entire error space? and (2) What is the minimal set of test cases needed to distinguish them? We introduce a novel framework that formalizes benchmark construction as finding an optimal diagnostic basis in a binary code-test matrix, where rows represent wrong codes and columns represent test case results. The rank of this matrix plays a dual role. It specifies the minimal number of independent error patterns, which determines the size of wrong codes. It also provides a tight upper bound on the number of test cases required for complete fault coverage. Our objective is to identify a basis of size equal to the matrix rank that maximizes internal diversity, which is defined as the average pairwise Jaccard similarity of the codes' failure signatures (i.e., the matrix rows). To tackle this NP-hard problem, we propose WrongSelect, an efficient approximation algorithm combining pre-filtering and random-restart local search to select maximally diverse wrong codes. Applying this framework to millions of competitive programming submissions, we construct TC-Bench, a compact, diverse, and inflation-resistant benchmark. Extensive experiments show that even the most advanced test case generation methods achieve only ~60\% exclusion rates on TC-Bench, exposing a significant gap in their diagnostic power and highlighting substantial room for future improvement.

Someone who learns to walk shortest paths in New York can, upon receiving a map of Paris, immediately apply the same rule to navigate, despite never practicing there. This ability to recombine known rules to solve novel problems exemplifies compositional generalization (CG), a hallmark of human cognition. Yet our understanding of what drives the success or failure of such extrapolative problem solving, particularly the roles of training data properties and optimization paradigms, remains limited. In this work, we introduce a controlled map-navigation testbed that cleanly separates two dimensions of CG: ***spatial transfer*** (systematicity across environments) and ***length scaling*** (productivity along problem difficulty). Through quantitative experiments, we show that transfer is enabled by sufficient distinct questions with high coverage and modest diversity, while scaling critically depends on exposure to neighboring-but-longer examples. Finally, we find that reinforcement learning (RL) stabilizes optimization but does not surpass the ceiling set by supervised fine-tuning (SFT). Together, these results provide principled insights into how data properties and training paradigms shape extrapolative problem solving.

A common and effective human strategy to improve a poor outcome is to first identify prior experiences most relevant to the outcome and then focus on learning from those experiences. This paper investigates whether this human strategy can improve generalization of meta-reinforcement learning (MRL). MRL learns a meta-prior from a set of training tasks such that the meta-prior can adapt to new tasks in a distribution. However, the meta-prior usually has imbalanced generalization, i.e., it adapts well to some tasks but adapts poorly to others. We propose a two-stage approach to improve generalization. The first stage identifies "critical" training tasks that are most relevant to achieve good performance on the poorly adapted tasks. The second stage improves generalization by encouraging the meta-prior to pay more attention to the critical tasks. We use conditional mutual information to mathematically formalize the notion of "paying more attention". We formulate a bilevel optimization problem to maximize the conditional mutual information by augmenting the critical tasks and propose an algorithm to solve the bilevel optimization problem. We theoretically guarantee that (1) the algorithm converges at the rate of $O(1/\sqrt{K})$ and (2) the generalization improves after the task augmentation. We use two real-world experiments, two MuJoCo experiments, and a Meta-World experiment to validate the algorithm.

Traditional on-policy Reinforcement Learning with Verifiable Rewards (RLVR) frameworks suffer from experience waste and reward homogeneity, which directly hinders learning efficiency on difficult samples during large language models post-training. In this paper, we introduce Batch Adaptation Policy Optimization (BAPO), an off-policy RLVR framework to improve the data efficiency in large language models post-training. It dynamically selects training batches by re-evaluating historically difficult samples and reusing high-quality ones, while holding a lower bound guarantee for policy improvement. Extensive experiments further demonstrate that BAPO achieves an average 12.5\% improvement over GRPO across mathematics, planning, and visual reasoning tasks. Crucially, BAPO successfully resolves 40.7\% of problems that base models consistently fail to solve.

Recent advancements in reinforcement learning from verifiable rewards (RLVR), particularly through Group Relative Policy Optimization (GRPO), have significantly improved the capabilities of large language models (LLMs) for interactive coding agents. However, these methods overlook process-verifiable environment feedback (e.g., code execution failures), leading to inaccurate advantage estimation at each reasoning step and insufficient learning. To address this issue, we propose Group Verification-based Policy Optimization (GVPO), a novel RL algorithm that introduces an advantage shaping framework integrating both outcome-verifiable and process-verifiable signals. While outcome-verifiable rewards ensure alignment with long-term task objectives, process-verifiable feedback derived from intermediate execution traces (e.g., syntax errors, runtime exceptions) serves as corrective shaping terms at the step level. By jointly leveraging these two forms of verifiability, GVPO achieves more accurate credit assignment, balancing short-term process guidance with long-term outcome alignment. This unified formulation yields more stable optimization, faster convergence, and stronger generalization in complex interactive environments. A 32B-parameter agent trained with GVPO in the AppWorld environment outperforms OpenAI’s o1 agent by 12.6\%s on the more challenging Test-C split and surpasses the strongest 32B RL-trained state-of-the-art baseline by 3.6\%.

In policy gradient reinforcement learning, access to a differentiable model enables 1st-order gradient estimation that accelerates learning compared to relying solely on derivative-free 0th-order estimators. However, discontinuous dynamics cause bias and undermine the effectiveness of 1st-order estimators. Prior work addressed this bias by constructing a confidence interval around the REINFORCE 0th-order gradient estimator and using these bounds to detect discontinuities. However, the REINFORCE estimator is notoriously noisy, and we find that this method requires task-specific hyperparameter tuning and has low sample efficiency. This paper asks whether such bias is the primary obstacle and what minimal fixes suffice. First, we re-examine standard discontinuous settings from prior work and introduce DDCG, a lightweight test that switches estimators in nonsmooth regions; with a single hyperparameter, DDCG achieves robust performance and remains reliable with small samples. Second, on differentiable robotics control tasks, we present IVW-H, a per-step inverse-variance implementation that stabilizes variance without explicit discontinuity detection and yields strong results. Together, these findings indicate that while estimator switching improves robustness in controlled studies, careful variance control often dominates in practical deployments.

We introduce Music Flamingo, a novel large audio–language model, designed to advance music (including song) understanding in foundational audio models. While audio–language research has progressed rapidly, music remains challenging due to its dynamic, layered, and information-dense nature. Progress has been further limited by the difficulty of scaling open audio understanding models, primarily because of the scarcity of high-quality music data and annotations. As a result, prior models are restricted to producing short, high-level captions, answering only surface-level questions, and showing limited generalization across diverse musical cultures. To address these challenges, we curate MF-Skills, a large-scale dataset labeled through a multi-stage pipeline that yields rich captions and question–answer pairs covering harmony, structure, timbre, lyrics, and cultural context. We fine-tune an enhanced Audio Flamingo 3 backbone on MF-Skills and further strengthen multiple skills relevant to music understanding. To improve the model's reasoning abilities, we introduce a post-training recipe: we first cold-start with MF-Think, a novel chain-of-thought dataset grounded in music theory, followed by GRPO-based reinforcement learning with custom rewards. Music Flamingo achieves state-of-the-art results across 10+ benchmarks for music understanding and reasoning, establishing itself as a generalist and musically intelligent audio–language model. Beyond strong empirical results, Music Flamingo sets a new standard for advanced music understanding by demonstrating how models can move from surface-level recognition toward layered, human-like perception of songs. We believe this work provides both a benchmark and a foundation for the community to build the next generation of models that engage with music as richly and meaningfully as humans do. Demo: https://musicflamingo.github.io

Large language models (LLMs) are increasingly adapted to downstream tasks via reinforcement learning (RL) methods like Group Relative Policy Optimization (GRPO), which often require thousands of rollouts to learn new tasks. We argue that the interpretable nature of language often provides a much richer learning medium for LLMs, compared to policy gradients derived from sparse, scalar rewards. To test this, we introduce GEPA (Genetic-Pareto), a prompt optimizer that thoroughly incorporates natural language reflection to learn high-level rules from trial and error. Given any AI system containing one or more LLM prompts, GEPA samples trajectories (e.g., reasoning, tool calls, and tool outputs) and reflects on them in natural language to diagnose problems, propose and test prompt updates, and combine complementary lessons from the Pareto frontier of its own attempts. As a result of GEPA's design, it can often turn even just a few rollouts into a large quality gain. Across four tasks, GEPA outperforms GRPO by 6% on average and by up to 20%, while using up to 35x fewer rollouts. GEPA also outperforms the leading prompt optimizer, MIPROv2, by over 10% (e.g., +10% accuracy on AIME-2025).

With the advent of DeepSeek-R1, a new wave of reinforcement learning (RL) methods has emerged that seem to unlock stronger mathematical reasoning. However, a closer look at the open-source ecosystem reveals a critical limitation: with sufficiently many draws (e.g., $\texttt{pass@1024}$), existing base models already solve nearly all questions on widely used math benchmarks such as MATH-500 and AIME 2024. This suggests that the RL fine-tuning methods prevalent in the LLM reasoning literature largely sharpen existing solution modes rather than discovering entirely new ones. Such sharpening stands in contrast to the broader promise of RL: to foster exploration and to acquire new skills. To move beyond this plateau, we introduce MATH-Beyond (MATH-B), a benchmark deliberately constructed to defeat common open-source models of up to 8B parameters even under large sampling budgets. Improving performance on our benchmark via RL requires methods that learn to reason in ways that go beyond base model capabilities in repeated sampling. Since the problems are drawn from subsets of DAPO-Math-17K and DeepScaleR datasets, they remain topically equivalent to standard high-school math. Validating our premise, RL fine-tuned models such as Nemotron-Research-Reasoning-Qwen-1.5B and DeepScaleR-1.5B-Preview perform poorly on MATH-B at $\texttt{pass@1024}$, showing how existing approaches fall short on tackling harder instances. We hope MATH-B will catalyze exploration-driven RL approaches that elicit deeper reasoning capabilities.

Reinforcement Learning with Verifiable Reward (RLVR) effectively solves complex tasks but demands extremely long context lengths during training, leading to substantial computational costs. While multi-stage training can partially mitigate this, starting with overly short contexts often causes irreversible performance degradation, ultimately failing to reduce overall training compute significantly. In this paper, we introduce **T**hinking-**F**ree **P**olicy **I**nitialization (**TFPI**), a simple yet effective adaptation to RLVR that bridges long Chain-of-Thought (CoT) distillation and standard RLVR. TFPI employs a simple *ThinkFree* operation, explicitly discarding the thinking content via a direct *</think>* append, to reduce token usage during inference. Training with *ThinkFree*-adapted inputs improves performance and lowers token consumption, even in the original slow-thinking mode. Extensive experiments across various benchmarks have shown that {\method} accelerates RL convergence, achieves a higher performance ceiling, and yields more token-efficient reasoning models without specialized rewards or complex training designs. With TFPI only, we train a 4B model to reach 89.0% accuracy on AIME24 and 65.5% on LiveCodeBench using less than 4K H20 hours.

Diffusion-based policies have gained growing popularity in solving a wide range of decision-making tasks due to their superior expressiveness and controllable generation during inference. However, effectively training large diffusion policies using reinforcement learning (RL) remains challenging. Existing methods either suffer from unstable training due to directly maximizing value objectives, or face computational issues due to relying on crude Gaussian likelihood approximation, which requires a large amount of sufficiently small denoising steps. In this work, we propose DIPOLE (Dichotomous diffusion Policy improvement), a novel RL algorithm designed for stable and controllable diffusion policy optimization. We begin by revisiting the KL-regularized objective in RL, which offers a desirable weighted regression objective for diffusion policy extraction, but often struggles to balance greediness and stability. We then formulate a greedified policy regularization scheme, which naturally enables decomposing the optimal policy into a pair of stably learned dichotomous policies: one aims at reward maximization, and the other focuses on reward minimization. Under such a design, optimized actions can be generated by linearly combining the scores of dichotomous policies during inference, thereby enabling flexible control over the level of greediness.Evaluations in offline and offline-to-online RL settings on ExORL and OGBench demonstrate the effectiveness of our approach. We also use DIPOLE to train a large vision-language-action (VLA) model for end-to-end autonomous driving (AD) and evaluate it on the large-scale real-world AD benchmark NAVSIM, highlighting its potential for complex real-world applications.

The Transformer architecture, despite its widespread success, struggles with long-context scenarios due to quadratic computation and linear memory growth. While various linear attention variants mitigate these efficiency constraints by compressing context into fixed-size states, they often degrade performance in tasks such as in-context retrieval and reasoning. To address this limitation and achieve more effective context compression, we propose two key innovations. First, we introduce a row-sparse update formulation for linear attention by conceptualizing state updating as information classification. This enables sparse state updates via softmax-based top-$k$ hard classification, thereby extending receptive fields and reducing inter-class interference. Second, we present Sparse State Expansion (SSE) within the sparse framework, which expands the contextual state into multiple partitions, effectively decoupling parameter size from state capacity while maintaining the sparse classification paradigm. Supported by efficient parallelized implementations, our design achieves effective classification and highly discriminative state representations. We extensively validate SSE in both pure linear and hybrid (SSE-H) architectures across language modeling, in-context retrieval, and mathematical reasoning benchmarks. SSE demonstrates strong retrieval performance and scales favorably with state size. Moreover, after reinforcement learning (RL) training, our 2B SSE-H model achieves state-of-the-art mathematical reasoning performance among small reasoning models, scoring 64.5 on AIME24 and 50.2 on AIME25, significantly outperforming similarly sized open-source Transformers. These results highlight SSE as a promising and efficient architecture for long-context modeling.

Large reasoning models (LRMs) often consume excessive tokens, inflating computational cost and latency. We challenge the assumption that longer responses improve accuracy. By penalizing the reasoning tokens using a discounted reinforcement-learning setup (interpretable as a small per-token cost) and analyzing Blackwell optimality in restricted policy classes, we encourage concise yet accurate reasoning; in practice we discount only the environment (correctness) reward. Experiments confirm our theoretical results that this approach shortens chains of thought while preserving accuracy.

Multi-domain reinforcement learning (RL) for large language models (LLMs) involves highly intricate reward surfaces, posing significant challenges in finding parameters that excel across all domains. Recent empirical studies have further highlighted conflicts among domains, where gains in one capability often come at the expense of another. However, approaches to mitigate such conflicts and enhance multi-domain reasoning remain largely underexplored. To address this challenge, we propose **C**urvature-**G**uided **P**olicy **O**ptimization (**CGPO**), a principled and scalable training framework to advance the multi-domain reasoning of LLMs. Inspired by Newton's method, CGPO exploits the geometric structure in the reward surface, while sidestepping the prohibitive cost of Hessian computation. At each update, CGPO processes domains in random order, preconditioning their gradients with curvature information from other domains to foster richer cross-domain interactions. This mechanism further promotes implicit gradient alignment by maximizing inter-domain inner products in expectation, steering the parameters toward regions that jointly enhance multi-domain performance. Extensive experiments on a mixed dataset covering math, coding, science, and creative writing, evaluated across seven widely-used benchmarks, show that CGPO significantly outperforms all baselines in terms of faster reward improvement and stronger multi-domain capability.

Reinforcement learning (RL) 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 applying RL techniques and a fragmented understanding of their underlying mechanisms. In addition, 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 show that a minimalist combination of two techniques can unlock the learning capability of critic-free policies with a vanilla PPO loss. The results demonstrate that our simple combination consistently improves performance, surpassing strategies such as GRPO and DAPO.

Weather forecasting is a fundamental task in spatiotemporal data analysis, with broad applications across a wide range of domains. Existing data-driven forecasting methods typically model atmospheric dynamics over a fixed short time interval, e.g., 6 hours, and rely on naive autoregression-based rollout for long-term forecastsing, e.g., 5 days. However, this paradigm suffers from two key limitations: (1) it often inadequately models the spatial and multi-scale temporal dependencies inherent in global weather systems, and (2) the rollout strategy struggles to balance error accumulation with the capture of fine-grained atmospheric variations. In this study, we propose ARROW, an Adaptive-Rollout Multi-scale temporal Routing method for Global Weather Forecasting. To contend with the first limitation, we construct a multi-interval forecasting model that forecasts weather across different time intervals. Within the model, the Shared-Private Mixture-of-Experts captures both shared patterns and specific characteristics of atmospheric dynamics across different time scales, while Ring Positional Encoding accurately encodes the circular latitude structure of the Earth when representing spatial information. For the second limitation, we develop an adaptive rollout scheduler based on reinforcement learning, which selects the most suitable time interval to forecast according to the current weather state. Experimental results demonstrate that ARROW achieves state-of-the-art performance in global weather forecasting, establishing a promising paradigm in this field.