  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.
