SAGE: Mitigating Long-Horizon Reasoning Biases via Topological Guidance

2026-09-24

AI Summary

This study addresses the challenges of exploration and compounding bias encountered by large language models during long-horizon reasoning under sparse rewards. To this end, we propose SAGE, a novel framework that introduces symbolic closure analysis theory for the first time. By modeling reasoning states through algebraic subspace projection and hyperbolic negative-curvature space embeddings, SAGE effectively integrates algebraic sparsification with structure-guided reasoning to mitigate long-horizon reasoning biases and enhance model stability. Experimental results demonstrate that SAGE comprehensively outperforms existing baseline methods across twelve benchmarks, achieving an eightfold performance improvement on the Andrews-Curtis problem.

Abstract

Long-horizon reasoning remains a central challenge for large language models (LLMs) under sparse-reward regimes. We argue that this brittleness arises from two biases induced by complex reasoning spaces: an exploration bias, where models are drawn toward locally plausible but structurally unstable branches, and a compounding bias, where small local deviations accumulate across depth and suppress rare rewards. We introduce Symbolic Closure Analysis (SCA) as a theoretical lens characterizing how branching structures and sparse rewards induce these biases in long-horizon reasoning with local admissibility, and as a design principle for structural priors in less formal reasoning tasks. Motivated by this analysis, we propose SAGE (Structural Admissibility-Guided Exploration), a unified framework that injects structural guidance to alleviate exploration bias and compounding bias in long-horizon reasoning. SAGE combines two complementary structural guidance: algebraic sparsification, which projects locally admissible candidates onto operator-indexed algebraic subspaces to suppress spurious branching and mitigate exploration bias, and hyperbolic structural guidance, which embeds reasoning states into a negatively curved space to provide dense depth-wise signals and mitigate compounding bias. Across 12 benchmarks and 7 model families, SAGE outperforms competitive baselines. In particular, SAGE achieves up to an 8-fold improvement on the Andrews-Curtis problem, an open real-world long-horizon task. Code is available at: https://github.com/Susan571/SAGE-NeurIPS2026.

Problem

• long-horizon reasoning
• sparse rewards
• exploration bias
• compounding bias
• large language models

Innovation

• Long-Horizon Reasoning
• Symbolic Closure Analysis
• Algebraic Sparsification
• Hyperbolic Embedding
• Structural Guidance