Training Nonlinear Transformers for Chain-of-Thought Inference: A Theoretical Generalization Analysis

TL;DR

This paper provides the first theoretical analysis of training nonlinear Transformers for Chain-of-Thought reasoning, quantifying sample complexity, convergence, and generalization, including robustness to noisy examples.

Contribution

It offers a novel theoretical framework for understanding how to train Transformers for CoT reasoning and their ability to generalize to unseen tasks.

Findings

Quantifies training sample and iteration requirements.

Proves CoT generalization on distribution-shifted data.

Characterizes conditions for accurate reasoning with noisy examples.

Abstract

Chain-of-Thought (CoT) is an efficient prompting method that enables the reasoning ability of large language models by augmenting the query using multiple examples with multiple intermediate steps. Despite the empirical success, the theoretical understanding of how to train a Transformer to achieve the CoT ability remains less explored. This is primarily due to the technical challenges involved in analyzing the nonconvex optimization on nonlinear attention models. To the best of our knowledge, this work provides the first theoretical study of training Transformers with nonlinear attention to obtain the CoT generalization capability so that the resulting model can inference on unseen tasks when the input is augmented by examples of the new task. We first quantify the required training samples and iterations to train a Transformer model towards CoT ability. We then prove the success of its CoT generalization on unseen tasks with distribution-shifted testing data. Moreover, we theoretically characterize the conditions for an accurate reasoning output by CoT even when the provided reasoning examples contain noises and are not always accurate. In contrast, in-context learning (ICL), which can be viewed as one-step CoT without intermediate steps, may fail to provide an accurate output when CoT does. These theoretical findings are justified through experiments.