Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.

Reinforcement Learning with Verifiable Rewards (RLVR) has achieved substantial gains in single-attempt accuracy (Pass@1) on reasoning tasks, yet often suffers from reduced multi-sample coverage (Pass@K), indicating diversity collapse. We identify a structural cause for this degradation: common RLVR objectives, such as GRPO, are indifferent to how probability mass is distributed among correct solutions. Combined with stochastic training dynamics, this indifference induces a self-reinforcing collapse, in which probability mass concentrates on a narrow subset of correct outputs while alternative valid solutions are suppressed. We formalize this collapse mechanism and further characterize the optimal policy structure under two complementary criteria: robustness and entropy-regularized optimality, which identify the Uniform-Correct Policy as uniquely optimal. Motivated by this analysis, we propose Uniform-Correct Policy Optimization (UCPO), a modification to GRPO that adds a conditional uniformity penalty on the policy's distribution over correct solutions. The penalty redistributes gradient signal toward underrepresented correct responses, encouraging uniform allocation of probability mass within the correct set. Across three models (1.5B-7B parameters) and five mathematical reasoning benchmarks, UCPO improves Pass@K and diversity while maintaining competitive Pass@1, achieving up to +10\% absolute improvement on AIME24 at Pass@64 and up to 45\% higher equation-level diversity within the correct set. The code is available at https://github.com/AnamikaLochab/UCPO.

Adapting large language models (LLMs) for specific tasks usually involves fine-tuning through reinforcement learning with human feedback (RLHF) on preference data. While these data often come from diverse labelers' groups (e.g., different demographics, ethnicities, company teams, etc.), traditional RLHF approaches adopt a one-size-fits-all approach, i.e., they indiscriminately assume and optimize a single preference model, thus not being robust to unique characteristics and needs of the various groups. To address this limitation, we propose a novel Group Robust Preference Optimization (GRPO) method to align LLMs to individual groups' preferences robustly. Our approach builds upon reward-free direct preference optimization methods, but unlike previous approaches, it seeks a robust policy which maximizes the worst-case group performance. To achieve this, GRPO adaptively and sequentially weights the importance of different groups, prioritizing groups with worse cumulative loss. We theoretically study the feasibility of GRPO and analyze its convergence for the log-linear policy class. By fine-tuning LLMs with GRPO using diverse group-based global opinion data, we significantly improved performance for the worst-performing groups, reduced loss imbalances across groups, and improved probability accuracies compared to non-robust baselines.

Recent research shows that Preference Alignment (PA) objectives act as divergence estimators between aligned (chosen) and unaligned (rejected) response distributions. In this work, we extend this divergence-based perspective to general alignment settings, such as reinforcement learning with verifiable rewards (RLVR), where only environmental rewards are available. Within this unified framework, we propose f-Group Relative Policy Optimization (f-GRPO), a class of on-policy reinforcement learning, and f-Hybrid Alignment Loss (f-HAL), a hybrid on/off policy objectives, for general LLM alignment based on variational representation of f-divergences. We provide theoretical guarantees that these classes of objectives improve the average reward after alignment. Empirically, we validate our framework on both RLVR (Math Reasoning) and PA tasks (Safety Alignment), demonstrating superior performance and flexibility compared to current methods.

Group Relative Policy Optimization (GRPO) has recently emerged as a practical recipe for aligning large language models with verifiable objectives. However, under sparse terminal rewards, GRPO often stalls because rollouts within a group frequently receive identical rewards, causing relative advantages to collapse and updates to vanish. We propose self-hint aligned GRPO with privileged supervision (SAGE), an on-policy reinforcement learning framework that injects privileged hints during training to reshape the rollout distribution under the same terminal verifier reward. For each prompt $x$, the model samples a compact hint $h$ (e.g., a plan or decomposition) and then generates a solution $τ$ conditioned on $(x,h)$. Crucially, the task reward $R(x,τ)$ is unchanged; hints only increase within-group outcome diversity under finite sampling, preventing GRPO advantages from collapsing under sparse rewards. At test time, we set $h=\varnothing$ and deploy the no-hint policy without any privileged information. Moreover, sampling diverse self-hints serves as an adaptive curriculum that tracks the learner's bottlenecks more effectively than fixed hints from an initial policy or a stronger external model. Experiments over 6 benchmarks with 3 LLMs show that SAGE consistently outperforms GRPO, on average +2.0 on Llama-3.2-3B-Instruct, +1.2 on Qwen2.5-7B-Instruct and +1.3 on Qwen3-4B-Instruct. The code is available at https://github.com/BaohaoLiao/SAGE.

It is widely recognized that reinforcement learning (RL) fine-tuning of large language models often leads to diversity collapse, where outputs lack variety. Prior work has proposed a range of heuristics to counteract this effect, but these methods are ad hoc: they frequently trade off correctness for diversity, their effectiveness varies across tasks, and in some cases they even contradict one another. In this work, we place these observations on a rigorous foundation. We first provide a formal proof of why RL fine-tuning exhibits diversity collapse via a selection and reinforcement bias. Next, we make a key observation that any reward modification to address diversity collapse only needs to be applied on the correct trajectories. Building directly on this analysis, we introduce a principled method -- differential smoothing -- that provably improves both correctness and diversity, outperforming vanilla RL as well as widely used entropy-based heuristics. Our theory precisely characterizes when existing heuristics help and why they fail, while showing that differential smoothing is universally superior. Extensive experiments with models from 1B to 7B parameters, across domains including CountDown and real-world mathematical reasoning, demonstrate consistent gains. Differential smoothing improves both Pass@1 and Pass@k, with up to 6.7% improvements on AIME24 dataset.

Group Relative Policy Optimization (GRPO) is a promising policy-based approach for Large Language Model alignment, yet its performance is often limited by training instability and suboptimal convergence. In this paper, we identify and analyze two main GRPO issues: (i) the token-level penalization, where valuable tokens shared across different responses receive contradictory feedback signals, leading to conflicting gradient updates that can reduce their likelihood; and (ii) the policy collapse, where negatively rewarded completions may penalize confident responses and shift model decisions toward unlikely tokens, destabilizing training process. To address these issues we introduce GTPO (Group-relative Trajectory-based Policy Optimization), which prevents conflicting gradients on valuable tokens by skipping negative updates while amplifying positive ones and filters out completions whose entropy exceeds a provable threshold, to prevent policy collapse. Unlike GRPO, GTPO does not rely on KL-divergence regularization, eliminating the need for a reference model during training, while still ensuring greater training stability and improved performance, as validated through multiple experiments on GSM8K, MA TH, AIME 2024, AIME 2025 and AMC 2023.

Vision-language reinforcement learning (RL) has primarily focused on narrow domains (e.g. geometry or chart reasoning). This leaves broader training scenarios and resources underexplored, limiting the exploration and learning of Vision Language Models (VLMs) through RL. We find video games inherently provide rich visual elements and mechanics that are easy to verify. To fully use the multimodal and verifiable reward in video games, we propose Game-RL, constructing diverse game tasks for RL training to boost VLMs general reasoning ability. To obtain training data, we propose Code2Logic, a novel approach that adapts game code to synthesize game reasoning task data, thus obtaining the GameQA dataset of 30 games and 158 tasks with controllable difficulty gradation. Unexpectedly, RL training solely on GameQA enables multiple VLMs to achieve performance improvements across 7 diverse vision-language benchmarks, demonstrating the value of Game-RL for enhancing VLMs' general reasoning. Furthermore, this suggests that video games may serve as valuable scenarios and resources to boost general reasoning abilities. Our code, dataset and models are available at the GitHub repository.

This study presents a systematic comparison of three Reinforcement Learning (RL) algorithms (PPO, GRPO, and DAPO) for improving complex reasoning in large language models (LLMs). Our main contribution is a controlled transfer-learning evaluation: models are first fine-tuned on the specialized Countdown Game and then assessed on a suite of general-purpose reasoning benchmarks. Across all tasks, RL-trained models outperform their corresponding base models, although the degree of improvement differs by benchmark. Our parametric analysis offers practical guidance for RL-based LLM training. Increasing the group size in GRPO and DAPO leads to more stable training dynamics and higher accuracy, while the impact of the KL-penalty coefficient is non-monotonic. Additionally, we find that the Dynamic Sampling (DS) component in DAPO does not improve performance; in fact, the best overall results are achieved with DAPO when DS is disabled.

We study what actually works and what doesn't for training large language models as agents via multi-turn reinforcement learning. Despite rapid progress, existing frameworks and definitions are fragmented, and there is no systematic formulation or analysis of which design choices matter across tasks. We address this gap by first breaking down the design space into three inter-related pillars--environment, reward, and policy--and empirically derive a recipe for training LLM agents in situated textual domains. In particular, we test TextWorld and ALFWorld, popular domains for testing situated embodied reasoning, as well as SWE-Gym for more software engineering style tasks. Training LLMs as autonomous agents to navigate open-ended environments presents unique challenges: planning across extended horizons, making multi-turn sequential decisions, and optimizing for multi-turn rewards. The transition from static single-turn problem-solving to dynamic multi-step reasoning is essential for agentic benchmarks such as interactive text and embodied simulations (TextWorld (C ˆ ot e et al., 2018), ALFWorld (Shridhar et al., 2021), etc.), real-world software programming (OSWorld (Xie et al., 2024), SWE-gym (Pan et al., 2025), etc.), and abstract reasoning in novel situations (ARC-AGI (Chollet et al., 2025)). However, existing multi-turn RL implementations vary widely: some refer to tool-augmented single queries as multi-turn (Zeng et al., 2025), while many rely on model-based assumptions (Wang et al., 2025). This fragmentation has led to incomparable results across papers and confusion about what constitutes true multi-turn learning versus pseudo-multi-turn adaptations of single-turn methods. This paper aims to facilitate research efforts on the open research question: What factors are practically important in making multi-turn RL for LLM agent learning work. Motivated by the lack of standardization of multi-turn RL approaches, we systematically decompose the design space into three interdependent pillars--environment, reward, and policy--and empirically derive a recipe for training LLM agents in situated textual domains (Figure 1). We evaluate our approach on TextWorld and ALFWorld for embodied reasoning, and SWE-gym for real-world programming, revealing critical insights for each pillar.