CoT Information: Improved Sample Complexity under Chain-of-Thought Supervision

TL;DR

This paper provides a theoretical analysis showing that chain-of-thought supervision can significantly reduce the sample complexity needed for learning complex functions, by quantifying the additional information gained from reasoning steps.

Contribution

It introduces the CoT information measure and links it to sample complexity bounds, offering the first statistical theory for learning with CoT supervision.

Findings

CoT supervision leads to faster learning rates compared to end-to-end supervision.

Sample complexity scales with the inverse of the CoT information measure.

Theoretical lower bounds are established based on CoT information.

Abstract

Learning complex functions that involve multi-step reasoning poses a significant challenge for standard supervised learning from input-output examples. Chain-of-thought (CoT) supervision, which provides intermediate reasoning steps together with the final output, has emerged as a powerful empirical technique, underpinning much of the recent progress in the reasoning capabilities of large language models. This paper develops a statistical theory of learning under CoT supervision. A key characteristic of the CoT setting, in contrast to standard supervision, is the mismatch between the training objective (CoT risk) and the test objective (end-to-end risk). A central part of our analysis, distinguished from prior work, is explicitly linking those two types of risk to achieve sharper sample complexity bounds. This is achieved via the *CoT information measure* $\mathcal{I}_{\mathcal{D}, h_\star}^{\mathrm{CoT}}(\epsilon; \calH)$, which quantifies the additional discriminative power gained from observing the reasoning process. The main theoretical results demonstrate how CoT supervision can yield significantly faster learning rates compared to standard E2E supervision. Specifically, it is shown that the sample complexity required to achieve a target E2E error $\epsilon$ scales as $d/\mathcal{I}_{\mathcal{D}, h_\star}^{\mathrm{CoT}}(\epsilon; \calH)$, where $d$ is a measure of hypothesis class complexity, which can be much faster than standard $d/\epsilon$ rates. Information-theoretic lower bounds in terms of the CoT information are also obtained. Together, these results suggest that CoT information is a fundamental measure of statistical complexity for learning under chain-of-thought supervision.