Patch-level Routing in Mixture-of-Experts is Provably Sample-efficient for Convolutional Neural Networks

In deep learning, mixture-of-experts (MoE) activates one or few experts (sub-networks) on a per-sample or per-token basis, resulting in significant computation reduction. The recently proposed \underline{p}atch-level routing in \underline{MoE} (pMoE) divides each input into $n$ patches (or tokens) and sends $l$ patches ($l\ll n$) to each expert through prioritized routing. pMoE has demonstrated great empirical success in reducing training and inference costs while maintaining test accuracy. However, the theoretical explanation of pMoE and the general MoE remains elusive. Focusing on a supervised classification task using a mixture of two-layer convolutional neural networks (CNNs), we show for the first time that pMoE provably reduces the required number of training samples to achieve desirable generalization (referred to as the sample complexity) by a factor in the polynomial order of $n/l$, and outperforms its single-expert counterpart of the same or even larger capacity. The advantage results from the discriminative routing property, which is justified in both theory and practice that pMoE routers can filter label-irrelevant patches and route similar class-discriminative patches to the same expert. Our experimental results on MNIST, CIFAR-10, and CelebA support our theoretical findings on pMoE's generalization and show that pMoE can avoid learning spurious correlations.