When and Why Does Unsupervised RL Succeed in Mathematical Reasoning? A Manifold Envelopment Perspective
This work addresses the scalability bottleneck in RL for mathematical reasoning by exploring unsupervised alternatives, though it is incremental as it builds on existing intrinsic reward methods to analyze and diagnose their limitations.
The paper tackled the problem of unsupervised reinforcement learning (RL) for mathematical reasoning in Large Language Models (LLMs), which suffers from instability and opaque dynamics, and found that enforcing concise and certain generation via intrinsic rewards can boost performance, but it breaks down depending on the model's foundational logical prior, with successful cases geometrically diagnosed as enveloped by manifolds.
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.