Multi-head Transformers Provably Learn Symbolic Multi-step Reasoning via Gradient Descent

TL;DR

This paper provides a theoretical analysis demonstrating that trained one-layer multi-head transformers can provably learn symbolic multi-step reasoning tasks, with guarantees of generalization and insights into the emergence of reasoning abilities.

Contribution

It offers the first provable guarantees for how shallow transformers learn multi-step symbolic reasoning via gradient descent, explaining the emergence of reasoning capabilities.

Findings

Trained one-layer transformers can solve chain-of-thought reasoning tasks with generalization guarantees.

Attention heads learn to specialize and coordinate to perform complex reasoning steps.

Shallow multi-head transformers can implement multi-step reasoning typically associated with deeper models.

Abstract

Transformers have demonstrated remarkable capabilities in multi-step reasoning tasks. However, understandings of the underlying mechanisms by which they acquire these abilities through training remain limited, particularly from a theoretical standpoint. This work investigates how transformers learn to solve symbolic multi-step reasoning problems through chain-of-thought processes, focusing on path-finding in trees. We analyze two intertwined tasks: a backward reasoning task, where the model outputs a path from a goal node to the root, and a more complex forward reasoning task, where the model implements two-stage reasoning by first identifying the goal-to-root path and then reversing it to produce the root-to-goal path. Our theoretical analysis, grounded in the dynamics of gradient descent, shows that trained one-layer transformers can provably solve both tasks with generalization guarantees to unseen trees. In particular, our multi-phase training dynamics for forward reasoning elucidate how different attention heads learn to specialize and coordinate autonomously to solve the two subtasks in a single autoregressive path. These results provide a mechanistic explanation of how trained transformers can implement sequential algorithmic procedures. Moreover, they offer insights into the emergence of reasoning abilities, suggesting that when tasks are structured to take intermediate chain-of-thought steps, even shallow multi-head transformers can effectively solve problems that would otherwise require deeper architectures.