Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraint

TL;DR

This paper proves that overparameterized convolutional residual networks can approximate functions accurately while maintaining smoothness, explaining their effectiveness in practice, especially for robust image classification.

Contribution

It establishes that large ConvResNets can approximate functions with both accuracy and smoothness, extending theory to manifold-supported functions.

Findings

Large ConvResNets achieve function approximation with smoothness.

Theoretical support for using deep and wide networks in practice.

Numerical experiments confirm the theory on robust image classification.

Abstract

Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation theories suggest that with sufficiently many parameters, neural networks can well approximate certain classes of functions in terms of the function value. The neural network themselves, however, can be highly nonsmooth. To bridge this gap, we take convolutional residual networks (ConvResNets) as an example, and prove that large ConvResNets can not only approximate a target function in terms of function value, but also exhibit sufficient first-order smoothness. Moreover, we extend our theory to approximating functions supported on a low-dimensional manifold. Our theory partially justifies the benefits of using deep and wide networks in practice. Numerical experiments on adversarial robust image classification are provided to support our theory.