HS-STaR: Hierarchical Sampling for Self-Taught Reasoners via Difficulty Estimation and Budget Reallocation

TL;DR

This paper introduces HS-STaR, a hierarchical sampling method that improves self-taught reasoning in large language models by focusing on problems near the reasoning boundary, leading to better training data efficiency and performance.

Contribution

The paper proposes a novel hierarchical sampling framework that dynamically reallocates sampling budget based on difficulty estimation, enhancing self-taught reasoning in LLMs.

Findings

HS-STaR outperforms baseline methods across multiple benchmarks.

Focusing on boundary-level problems yields higher learning utility.

Dynamic budget reallocation improves training efficiency.

Abstract

Self-taught reasoners (STaRs) enhance the mathematical reasoning abilities of large language models (LLMs) by leveraging self-generated responses for self-training. Recent studies have incorporated reward models to guide response selection or decoding, aiming to obtain higher-quality data. However, they typically allocate a uniform sampling budget across all problems, overlooking the varying utility of problems at different difficulty levels. In this work, we conduct an empirical study and find that problems near the boundary of the LLM's reasoning capability offer significantly greater learning utility than both easy and overly difficult ones. To identify and exploit such problems, we propose HS-STaR, a Hierarchical Sampling framework for Self-Taught Reasoners. Given a fixed sampling budget, HS-STaR first performs lightweight pre-sampling with a reward-guided difficulty estimation strategy to efficiently identify boundary-level problems. Subsequently, it dynamically reallocates the remaining budget toward these high-utility problems during a re-sampling phase, maximizing the generation of valuable training data. Extensive experiments across multiple reasoning benchmarks and backbone LLMs demonstrate that HS-STaR significantly outperforms other baselines without requiring additional sampling budget.