Effective Reasoning Chains Reduce Intrinsic Dimensionality

TL;DR

This paper introduces intrinsic dimensionality as a metric to evaluate reasoning strategies in language models, showing that effective reasoning chains reduce this dimensionality and improve generalization.

Contribution

It establishes intrinsic dimensionality as a new quantitative measure linking reasoning strategies to model generalization performance.

Findings

Effective reasoning strategies lower intrinsic dimensionality.

Lower dimensionality correlates with better generalization on in- and out-of-distribution data.

Intrinsic dimensionality serves as a useful metric for analyzing reasoning effectiveness.

Abstract

Chain-of-thought (CoT) reasoning and its variants have substantially improved the performance of language models on complex reasoning tasks, yet the precise mechanisms by which different strategies facilitate generalization remain poorly understood. While current explanations often point to increased test-time computation or structural guidance, establishing a consistent, quantifiable link between these factors and generalization remains challenging. In this work, we identify intrinsic dimensionality as a quantitative measure for characterizing the effectiveness of reasoning chains. Intrinsic dimensionality quantifies the minimum number of model dimensions needed to reach a given accuracy threshold on a given task. By keeping the model architecture fixed and varying the task formulation through different reasoning strategies, we demonstrate that effective reasoning strategies consistently reduce the intrinsic dimensionality of the task. Validating this on GSM8K with Gemma-3 1B and 4B, we observe a strong inverse correlation between the intrinsic dimensionality of a reasoning strategy and its generalization performance on both in-distribution and out-of-distribution data. Our findings suggest that effective reasoning chains facilitate learning by better compressing the task using fewer parameters, offering a new quantitative metric for analyzing reasoning processes.