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

Nonparametric Classification on Low Dimensional Manifolds using Overparameterized Convolutional Residual Networks