Transformers Provably Solve Parity Efficiently with Chain of Thought

TL;DR

This paper provides a theoretical analysis of how transformers can efficiently learn to solve parity problems through chain-of-thought reasoning, highlighting the importance of intermediate supervision and data augmentation.

Contribution

It introduces the first theoretical framework showing transformers can learn parity efficiently with chain-of-thought, especially when using intermediate supervision and data augmentation.

Findings

Transformers require many iterations without intermediate supervision.

With teacher forcing, parity can be learned in one gradient step.

Data augmentation enables end-to-end learning without teacher forcing.

Abstract

This work provides the first theoretical analysis of training transformers to solve complex problems by recursively generating intermediate states, analogous to fine-tuning for chain-of-thought (CoT) reasoning. We consider training a one-layer transformer to solve the fundamental $k$-parity problem, extending the work on RNNs by Wies et al. (2023). We establish three key results: (1) any finite-precision gradient-based algorithm, without intermediate supervision, requires substantial iterations to solve parity with finite samples. (2) In contrast, when intermediate parities are incorporated into the loss function, our model can learn parity in one gradient update when aided by \emph{teacher forcing}, where ground-truth labels of the reasoning chain are provided at each generation step. (3) Even without teacher forcing, where the model must generate CoT chains end-to-end, parity can be learned efficiently if augmented data is employed to internally verify the soundness of intermediate steps. Our findings, supported by numerical experiments, show that task decomposition and stepwise reasoning naturally arise from optimizing transformers with CoT; moreover, self-consistency checking can improve multi-step reasoning ability, aligning with empirical studies of CoT.