Convergence of Deep Convolutional Neural Networks

TL;DR

This paper investigates the convergence behavior of deep ReLU neural networks with increasing widths, including convolutional architectures, by analyzing infinite matrix products and establishing conditions for convergence.

Contribution

It extends previous convergence analysis to networks with increasing widths, specifically applying to deep convolutional neural networks, and introduces new conditions for their convergence.

Findings

Convergence reduces to infinite products of matrices with increasing sizes.

Established sufficient conditions for convergence of these matrix products.

Provided conditions for piecewise and pointwise convergence of deep ReLU networks.

Abstract

Convergence of deep neural networks as the depth of the networks tends to infinity is fundamental in building the mathematical foundation for deep learning. In a previous study, we investigated this question for deep ReLU networks with a fixed width. This does not cover the important convolutional neural networks where the widths are increasing from layer to layer. For this reason, we first study convergence of general ReLU networks with increasing widths and then apply the results obtained to deep convolutional neural networks. It turns out the convergence reduces to convergence of infinite products of matrices with increasing sizes, which has not been considered in the literature. We establish sufficient conditions for convergence of such infinite products of matrices. Based on the conditions, we present sufficient conditions for piecewise convergence of general deep ReLU networks with increasing widths, and as well as pointwise convergence of deep ReLU convolutional neural networks.