When and Why Does Unsupervised RL Succeed in Mathematical Reasoning? A Manifold Envelopment Perspective

• URL: http://arxiv.org/abs/2603.16578v1
• Date: Tue, 17 Mar 2026 14:29:38 GMT
• Title: When and Why Does Unsupervised RL Succeed in Mathematical Reasoning? A Manifold Envelopment Perspective
• Abstract summary: Outcome-based reinforcement learning (RL) significantly advances the mathematical reasoning capabilities of Large Language Models (LLMs)<n>Its reliance on computationally expensive ground-truth annotations imposes a severe scalability bottleneck.<n>Unsupervised RL guided by intrinsic rewards offers a scalable alternative.
• Score: 23.239530219575936
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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