On the growth of the parameters of approximating ReLU neural networks

TL;DR

This paper analyzes how the number of parameters in ReLU neural networks grows as they approximate smooth functions, showing polynomial growth rates that are better than existing results, especially in high dimensions.

Contribution

It establishes that for near-optimal approximation, ReLU networks' parameters grow at most polynomially, improving upon previous bounds especially in high-dimensional settings.

Findings

Parameters grow at most polynomially with network size.

Polynomial growth rate is superior to previous results in high dimensions.

Results are relevant for error analysis and training consistency.

Abstract

This work focuses on the analysis of fully connected feed forward ReLU neural networks as they approximate a given, smooth function. In contrast to conventionally studied universal approximation properties under increasing architectures, e.g., in terms of width or depth of the networks, we are concerned with the asymptotic growth of the parameters of approximating networks. Such results are of interest, e.g., for error analysis or consistency results for neural network training. The main result of our work is that, for a ReLU architecture with state of the art approximation error, the realizing parameters grow at most polynomially. The obtained rate with respect to a normalized network size is compared to existing results and is shown to be superior in most cases, in particular for high dimensional input.