Student Guides Teacher: Weak-to-Strong Inference via Spectral Orthogonal Exploration

TL;DR

This paper introduces Spectral Orthogonal Exploration (SOE), a geometric inference method that enhances reasoning diversity in Large Language Models, significantly improving accuracy and exploration efficiency on mathematical, logic, and code generation tasks.

Contribution

The paper proposes SOE, a novel geometric inference framework that uses orthogonal probes to diversify reasoning trajectories and mitigate reasoning collapse in LLMs.

Findings

SOE improves average accuracy by 62.4% on mathematical benchmarks.

SOE increases sampling efficiency by 113.7% over baseline methods.

Preliminary results show SOE's effectiveness on logic and code generation tasks.

Abstract

Large Language Models (LLMs) often suffer from ''Reasoning Collapse'' on challenging mathematical reasoning tasks, where stochastic sampling produces lexical variations of the same erroneous logic rather than genuine semantic exploration. We observe that failed reasoning traces are often associated with a low-rank bias manifold in the model's hidden-state geometry, which reduces exploration toward corrective solution directions. To address this, we propose Spectral Orthogonal Exploration (SOE), a geometric inference framework under a ''Student Guides Teacher'' paradigm. Instead of using a weak auxiliary agent for imitation, SOE uses it as an orthogonal probe to introduce semantically heterogeneous reasoning signals into the teacher's orthogonal complement of its dominant subspace. This intervention steers the teacher toward more diverse reasoning trajectories and improves exploration beyond standard sampling. Experiments on mathematical benchmarks show that SOE improves average accuracy by 62.4\% and average sampling efficiency by 113.7\% over baseline methods, suggesting that geometric interventions can be effective for mitigating reasoning collapse in mathematical reasoning. We further provide preliminary evidence that SOE is also effective on logic and code generation benchmarks.