SalaMAnder: Shapley-based Mathematical Expression Attribution and Metric for Chain-of-Thought Reasoning

Abstract

Chain-of-Thought (CoT) prompting enhances the math reasoning capability of large language models (LLMs) to a large margin. However, the mechanism underlying such improvements remains unexplored. In this paper, we present \textbf{SalaMAnder} (\textbf{S}h\textbf{a}p\textbf{l}ey-b\textbf{a}sed \textbf{M}athematical Expression \textbf{A}ttribution a\textbf{nd} M\textbf{e}t\textbf{r}ic), a theoretically grounded methodology as well as a mathematically rigorous evaluation metric for quantifying component-level contributions in few-shot CoT reasoning. Concretely, we leverage the Shapley value for mathematical expression attribution and develop an efficient stratified sampling algorithm that significantly reduces the computational complexity. Besides, we develop the \textbf{CoSP} (\textbf{C}ardinality \textbf{o}f \textbf{S}hapley \textbf{P}ositives) metric through covariance analysis. Comprehensive validation across popular LLM models and diverse mathematical benchmarks demonstrates that the CoSP metric within our SalaMAnder framework exhibits a robust monotonic correlation with model performance, not only providing theoretical explanations for the empirical success of existing few-shot CoT but also establishing mathematically rigorous principles for prompt construction optimization. Furthermore, we verify the reliability of the explanation, based on which we unify the insights of previous work.