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

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.

Figures (16)

• Figure 1: Overview of Research Pipeline.(a) Design Rewards: enforce concise and certain. (b) Find Boundary Conditions: evaluate across models with different reasoning abilities. (c) Find Diagnostic Lens: DTW clustering to 3D phase space.
• Figure 2: Visualization of 3D Semantic Manifold Boundaries. The three axes define the phase space corresponding to the mean entropy of the Thinking, Logic, and Execution semantic clusters, as established in Section \ref{['sec:dtw_analysis']}. (a-c) Exploration Manifolds: The translucent polygons denote the 3D convex hulls encompassing all aggregated trajectory points for specific training methods. The legend identifies the training method and provides the calculated volume of its corresponding hull. DS-Distill denotes DeepSeek-R1-Distill-Llama-8B.
• Figure 3: Token Entropy Trajectories on Qwen3-1.7B (Part 1 of 2). Clustering results evaluated on the AIME 24 dataset across six different reward formulations. The consistent emergence of Execution, Logic, and Thinking clusters is observed despite the smaller parameter scale. The text boxes within each subplot display the top frequency tokens corresponding to that cluster.
• Figure 4: Token Entropy Trajectories on Qwen3-1.7B (Part 2 of 2). (Continued from previous page.)
• Figure 5: Token Entropy Trajectories on Qwen3-8B (Standard Setup) (Part 1 of 2). Visualizations of the optimization dynamics under standard context length on our primary testbed model. The text boxes within each subplot display the top frequency tokens corresponding to that cluster.