Nonparametric Classification on Low Dimensional Manifolds using Overparameterized Convolutional Residual Networks

Zixuan Zhang, Kaiqi Zhang, Minshuo Chen, Yuma Takeda, Mengdi Wang, Tuo Zhao, Yu-Xiang Wang

Keywords: Nonparametric Classification; Low Dimensional Manifolds; Overparameterized ResNets; Function Approximation.

TL;DR: We prove that an overparameterized ResNeXts achieves close-to-minimax rate in learning Besov funtions on a low dimension manifold under weight decay.

Abstract: 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.

Submission Number: 66