  Convolutional residual neural networks (ConvResNets), though
overparameterized, can achieve remarkable prediction performance in practice,
which cannot be well explained by conventional wisdom. To bridge this gap, we
study the performance of ConvResNeXts, which cover ConvResNets as a special
case, trained with weight decay from the perspective of nonparametric
classification. Our analysis allows for infinitely many building blocks in
ConvResNeXts, and shows that weight decay implicitly enforces sparsity on these
blocks. Specifically, we consider a smooth target function supported on a
low-dimensional manifold, then prove that ConvResNeXts can adapt to the
function smoothness and low-dimensional structures and efficiently learn the
function without suffering from the curse of dimensionality. Our findings
partially justify the advantage of overparameterized ConvResNeXts over
conventional machine learning models.
