A Theoretical Understanding of Shallow Vision Transformers: Learning, Generalization, and Sample Complexity

TL;DR

This paper provides the first theoretical analysis of shallow Vision Transformers, characterizing their sample complexity, attention sparsity, and how token sparsification can improve performance, supported by empirical validation.

Contribution

It offers a novel theoretical framework for understanding shallow ViTs, including sample complexity bounds and the impact of token sparsification on generalization.

Findings

Sample complexity depends on label-relevant token fraction and noise level.

SGD training leads to sparse attention maps, verifying intuition.

Token sparsification improves test performance by removing irrelevant tokens.

Abstract

Vision Transformers (ViTs) with self-attention modules have recently achieved great empirical success in many vision tasks. Due to non-convex interactions across layers, however, theoretical learning and generalization analysis is mostly elusive. Based on a data model characterizing both label-relevant and label-irrelevant tokens, this paper provides the first theoretical analysis of training a shallow ViT, i.e., one self-attention layer followed by a two-layer perceptron, for a classification task. We characterize the sample complexity to achieve a zero generalization error. Our sample complexity bound is positively correlated with the inverse of the fraction of label-relevant tokens, the token noise level, and the initial model error. We also prove that a training process using stochastic gradient descent (SGD) leads to a sparse attention map, which is a formal verification of the general intuition about the success of attention. Moreover, this paper indicates that a proper token sparsification can improve the test performance by removing label-irrelevant and/or noisy tokens, including spurious correlations. Empirical experiments on synthetic data and CIFAR-10 dataset justify our theoretical results and generalize to deeper ViTs.