Nonparametric Classification on Low Dimensional Manifolds using Overparameterized Convolutional Residual Networks

TL;DR

This paper analyzes overparameterized convolutional residual networks (ConvResNeXts) trained with weight decay, showing they adapt to low-dimensional manifolds and efficiently learn smooth functions, explaining their strong practical performance.

Contribution

It provides a nonparametric theoretical framework for ConvResNeXts, demonstrating how weight decay induces sparsity and enables adaptation to low-dimensional structures.

Findings

ConvResNeXts can adapt to function smoothness and low-dimensional manifolds.

Weight decay implicitly enforces sparsity on network blocks.

Overparameterized ConvResNeXts outperform traditional models in low-dimensional settings.

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.