Free Energy-Driven Reinforcement Learning with Adaptive Advantage Shaping for Unsupervised Reasoning in LLMs

Reinforcement Learning with Verifiable Rewards (RLVR) is an essential paradigm that enhances the reasoning capabilities of Large Language Models (LLMs). However, existing methods typically rely on static policy optimization schemes that misalign with the model's evolving reasoning capabilities. To address this issue, we propose Adaptive Power-Mean Policy Optimization (APMPO), which comprises two main innovations: Power-Mean Policy Optimization (PMPO) and Feedback-Adaptive Clipping (FAC). Specifically, PMPO introduces a generalized power-mean objective. This enables the model to adaptively transition from the signal-amplifying behavior of the arithmetic mean to the consistency-enforcing behavior of the geometric mean. FAC adaptively adjusts clipping bounds based on real-time reward statistics to overcome the limitations of static mechanisms. Capitalizing on these innovations, APMPO improves learning dynamics and reasoning performance. Extensive experiments on nine datasets across three reasoning tasks showcase the superiority of APMPO over state-of-the-art RLVR-based baselines. For instance, APMPO boosts the average Pass@1 score on mathematical reasoning benchmarks by 3.0 points compared to GRPO when using Qwen2.5-3B-Instruct.

Although outcome-based reinforcement learning (RL) significantly advances the mathematical reasoning capabilities of Large Language Models (LLMs), its reliance on computationally expensive ground-truth annotations imposes a severe scalability bottleneck. Unsupervised RL guided by intrinsic rewards offers a scalable alternative, yet it suffers from opaque training dynamics and catastrophic instability, such as policy collapse and reward hacking. In this paper, we first design and evaluate a suite of intrinsic rewards that explicitly enforce concise and certain generation. Second, to discover the boundaries of this approach, we test base models across a spectrum of intrinsic reasoning capabilities, revealing how a model's foundational logical prior dictates its success or failure. Finally, to demystify why certain configurations stabilize while others collapse, we introduce a novel geometric diagnostic lens, showing that successful cases are enveloped by manifolds. Ultimately, our work goes beyond merely demonstrating that enforcing concise and certain responses successfully boosts mathematical reasoning; we reveal when this unsupervised approach breaks down and geometrically diagnose why.

Reinforcement learning has become a central paradigm for advancing reasoning in large language models, yet most existing methods still depend on stronger teacher models or heavily curated difficult datasets, limiting scalable capability improvement. In this paper, we introduce DenoiseRL, a reinforcement learning framework that substitutes external supervision with recovery-oriented optimization over failures from weak models. Instead of relying on stronger supervision or carefully engineered data, DenoiseRL learns directly from incorrect reasoning traces by converting them into opportunities for improvement, making training more scalable and less dependent on external resources. This yields a richer and more diverse learning signal, improving exploration efficiency from imperfect model behavior. As a result, DenoiseRL improves reasoning performance and overall training efficiency while reducing the need for expensive data curation or stronger teacher models. Empirically, DenoiseRL consistently outperforms strong on-policy RL baselines across competitive mathematical and general reasoning benchmarks and promotes stronger self-corrective behavior as training difficulty increases, highlighting an effective and scalable alternative pathway for improving reasoning in large language models.