Gaussian random field approximation via Stein's method with applications to wide random neural networks

TL;DR

This paper develops new bounds on how closely wide random neural networks approximate Gaussian random fields, using Stein's method and Gaussian smoothing, with explicit error bounds based on network width and activation function smoothness.

Contribution

It introduces a novel Gaussian smoothing technique and provides the first bounds for Gaussian approximation of wide neural networks at the random field level.

Findings

Bounds depend explicitly on network widths and weight moments.

Tighter bounds achieved with thrice differentiable activation functions.

Extends Gaussian approximation results beyond one-dimensional index sets.

Abstract

We derive upper bounds on the Wasserstein distance ($W_1$), with respect to $\sup$-norm, between any continuous $\mathbb{R}^d$ valued random field indexed by the $n$-sphere and the Gaussian, based on Stein's method. We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the $W_1$ distance. The smoothing is based on covariance functions constructed using powers of Laplacian operators, designed so that the associated Gaussian process has a tractable Cameron-Martin or Reproducing Kernel Hilbert Space. This feature enables us to move beyond one dimensional interval-based index sets that were previously considered in the literature. Specializing our general result, we obtain the first bounds on the Gaussian random field approximation of wide random neural networks of any depth and Lipschitz activation functions at the random field level. Our bounds are explicitly expressed in terms of the widths of the network and moments of the random weights. We also obtain tighter bounds when the activation function has three bounded derivatives.