This work establishes the first non-vacuous generalization bounds for parameter-efficient RLVR fine-tuning at the billion-parameter scale, and proposes the Progressive RLVR framework, which integrates RLVR with on-policy distillation, TinyLoRA, and model quantization.
Abstract
While reinforcement learning with verifiable rewards (RLVR) is widely used to improve the reasoning capabilities of large language models (LLMs), the generalizability of the resulting models remains poorly understood. In this work, we establish the first non-vacuous generalization bounds for parameter-efficient RLVR fine-tuning at the billion-parameter scale. Our approach adapts PAC-Bayes compression bounds to this setting, and addresses the inherent stochasticity of token generation by applying the Gumbel-max reparameterization trick. To operationalize these bounds, we propose the Progressive RLVR framework, which integrates RLVR with on-policy distillation, TinyLoRA, and model quantization. Progressive RLVR empirically retains 84-97% performance of standard LoRA fine-tuning while producing models that are 14,796x more compressible. We show that this framework yields non-vacuous generalization bounds in four domains: mathematical problem-solving, programming, general-knowledge reasoning, and Text-to-SQL. Our bounds exceed the accuracy of the base model by 9-51% and lie within 6-11% of the accuracy of the fine-tuned models.
Reinforcement learning with verifiable rewards (RLVR) has been effective on tasks with structured solutions like math and coding, but its reliance on simple, rule-based verifiers creates a fundamental bottleneck. We find their applicability is surprisingly narrow even in structured domains, a limitation that is compounded at scale: rule-based systems can paradoxically degrade in performance as multi-domain, free-form training data increases. To overcome these challenges, we propose a new RLVR framework that uses a generative verifier to provide soft, probabilistic rewards. Our key insight is that powerful LLMs show high agreement with human evaluators when judging answer correctness given a ground-truth reference, allowing us to automate reward generation without costly human annotation. Our experiments demonstrate the effectiveness of this approach. We show that a compact 7B generative reward model can guide a 7B policy model to decisively outperform models up to 10x its size, including the 72B Qwen2.5-Instruct (by a margin of +8.6%). This effectiveness is robust, holding true across diverse training datasets with answers sourced from experts, web users, and other LLMs, and generalizes strongly to seven out-of-distribution benchmarks. Our work provides a scalable and effective framework for extending RLVR beyond the limitations of pattern-based verification to complex, noisy, real-world domains.
Yi Su, Dian Yu, Linfeng Song et al.· Annual Meeting of the Associ...· 1 citation
This project investigates scaling test-time compute through a Generative Verifier (GV) on the Countdown mathematical reasoning task using a computationally constrained 0.5B parameter regime, hypothesizing that the “verification gap” will widen at higher values of N due to the model’s limited semantic capacity.
Reinforcement Learning with Verifiable Rewards (RLVR) and on-policy distillation (OPD) have become two widely adopted paradigms for post-training large language models. However, RLVR suffers from sparse task-level feedback, while OPD provides dense token-level guidance but ignores trajectory correctness, limiting its performance to that of the teacher. Combining them is a promising direction: OPD supplies dense supervisory signals, while RLVR provides task-level correctness. Nevertheless, existing integrations often rely on weighted combination or heuristic switching, introducing extra hyperparameters and trade-offs. We propose On-policy Distillation with Verifiable Reward (OPDVR), a simple yet effective method that seamlessly combines OPD and RLVR without adding any hyperparameters. We first reformulate the implicit reward of sampled-token OPD based on trajectory correctness, then apply a ReLU gating mechanism to ensure that correct trajectories receive non-negative rewards and incorrect ones receive non-positive rewards---thereby aligning the distillation signal with task success while preserving the teacher's distributional guidance. Furthermore, our modification transforms sampled-token OPD into a proper RLVR method, making it readily combinable with any policy gradient algorithm, such as GRPO. Experiments on six reasoning benchmarks show that OPDVR consistently outperforms standard OPD. Our code is available at https://github.com/LeapLabTHU/OPDVR.
Wenze Lin, Jiale Zhao, Xitai Jiang et al.· 0 citations
Although reinforcement learning with verifiable rewards (RLVR) has improved the performance of large language models (LLMs) across a variety of reasoning tasks, there is significant debate as to whether RLVR expands the reasoning capability boundary, or just improves sampling efficiency. In this paper, we investigate the nature of test-time exploration in RLVR-trained LLMs by employing controlled maze-solving experiments and extracting a tree structure from mathematical reasoning traces (BODHI-Trees) based on semantic equivalence. This helps us delineate between entropy arising from stylistic variations and genuine inferential branching. Our findings demonstrate that the policy entropy collapse observed in RLVR models is not merely syntactic, and is accompanied by a significant reduction in semantic branching entropy. While RLVR improves adherence to environmental constraints and backtracking capabilities, it constricts the space of continuations; we provide evidence suggesting that this might be responsible for the sample efficiency gains of RLVR, albeit at the cost of genuine rollout diversity.
Soumadeep Saha, Krish Sharma, Akshay Chaturvedi et al.· 0 citations