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Learning to Steer, Steering to See: Unveiling the Geometry of RLVR in Large Language Models via Trainable Vectors

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Briefings

This study investigates reinforcement learning with verifiable rewards (RLVR) in large language models by identifying a low-dimensional effective manifold in activation space. Key findings include two geometric properties: effective manifold capacity is small but not infinitely compressible, and control directions lie in the low-variance complement of the principal subspace. Experiments on five LLMs and six verifiable-reward tasks validate these geometric insights, with Alpha-Stabler proposed as a plug-and-play framework to stabilize training.

Key points

• Effective manifold capacity required to reproduce RL gains is small but not infinitely compressible, with intervention dimensionality and input-dependent expressiveness becoming key constraints at extremely low capacity.
• Control directions for RLVR lie mainly in the low-variance complement of the activation principal subspace, and learned geometry remains largely consistent across training configurations and tasks.

Context

Reinforcement learning with verifiable rewards (RLVR) is a key paradigm for enhancing large language model reasoning, yet high dimensionality of parameter updates makes training dynamics hard to analyze. This research provides geometric insights into RLVR dynamics.

Limitations

Findings are based on experiments on five LLMs and six verifiable-reward tasks. The study does not claim causal claims or forecast future performance; results remain consistent with observed training dynamics.

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