HyperGuide: Hyperbolic Guidance for Efficient Multi-Step Reasoning in Large Language Models

2026-05-22

AI Summary

This work addresses the trade-off between efficiency and accuracy in multi-step reasoning with large language models: while single-pass generation is fast but error-prone, tree search methods are accurate yet computationally expensive. The paper introduces hyperbolic geometry into reasoning state modeling—a novel approach that leverages the exponential volume growth of hyperbolic space to encode solution proximity via distance from the origin and distinguish reasoning branches through angular separation, thereby enabling lightweight guidance of reasoning paths. By integrating hyperbolic embeddings, a lightweight projection head, low-rank adapter fine-tuning, and self-guided training, the proposed method consistently enhances performance across multiple benchmarks, with gains becoming more pronounced as reasoning chains deepen.

Abstract

Multi-step reasoning remains a central challenge for large language models: single-pass generation is efficient but lacks accuracy; tree-search methods explore multiple paths but are computation-heavy. We address this gap by distilling reasoning progress into a hyperbolic geometric signal that guides step-by-step generation. Our approach is motivated by a structural observation: in combinatorial reasoning trees, solution-bearing states are few while dead ends are exponentially numerous. The hyperbolic space matches this asymmetry, with compact volume near the origin and exponentially expanding capacity toward the boundary, so that distance-to-origin naturally encodes solution proximity while angular separation distinguishes branches requiring different next operations. We train a lightweight head to project LLM hidden states into this space, then fine-tune a low-rank adapter interactively on its own reasoning attempts to act on the injected signal. Across multiple benchmarks, the geometric signal yields consistent gains, with larger improvements on deeper reasoning chains. Our code is publicly available at https://github.com/yuyuliu11037/HyperGuide.

Problem

• multi-step reasoning
• large language models
• efficiency-accuracy trade-off
• combinatorial reasoning
• reasoning paths

Innovation

• hyperbolic geometry
• multi-step reasoning
• large language models
• geometric guidance
• low-rank adaptation