Normal approximation of Random Gaussian Neural Networks

TL;DR

This paper derives explicit bounds on how closely the outputs of random Gaussian neural networks approximate Gaussian distributions, considering various architectures and activation functions, using Stein's method.

Contribution

It introduces a novel application of Stein's method to quantify Gaussian approximation in deep and shallow Gaussian neural networks with general activation functions.

Findings

Bounds depend on layer widths and activation functions.

Results apply to total variation and convex distances.

Provides estimates for the probability of outputs in specific regions.

Abstract

In this paper we provide explicit upper bounds on some distances between the (law of the) output of a random Gaussian NN and (the law of) a random Gaussian vector. Our results concern both shallow random Gaussian neural networks with univariate output and fully connected and deep random Gaussian neural networks, with a rather general activation function. The upper bounds show how the widths of the layers, the activation functions and other architecture parameters affect the Gaussian approximation of the ouput. Our techniques, relying on Stein's method and integration by parts formulas for the Gaussian law, yield estimates on distances which are indeed integral probability metrics, and include the total variation and the convex distances. These latter metrics are defined by testing against indicator functions of suitable measurable sets, and so allow for accurate estimates of the probability that the output is localized in some region of the space. Such estimates have a significant interest both from a practitioner's and a theorist's perspective.