Lifting Finite-Sample Gaussian Approximation of Wide Neural Networks to Functional Convergence

Abstract: Randomly initialised neural networks of width $m$ converge to Gaussian processes as $m \to \infty$, with two types of quantitative bounds characterising this convergence. Finite-sample bounds compare the network with its Gaussian limit at a fixed finite sample set, on an arbitrary domain, in the quadratic Wasserstein distance $\mathcal W_2$, achieving a convergence rate of $m<sup>{-1/2}$ with explicit constants. However, these constants grow with the number of samples, so the bound becomes uninformative as the sample set is refined. Functional bounds remove that dependence by comparing the two as random functions, at the price of spherical input, metrics weaker than $\mathcal W_2$, or slower rates. In this paper, we therefore lift a finite-sample bound to a functional one. The lifting refines the sample set until it is dense in the domain, so the bound must first be made independent of the number of samples. We achieve this by measuring the error per sample, which makes the constant uniform. For a polynomial activation and any compact input set, this gives a rate of $m<sup>{-1/2}$ in the supremum norm, with an explicit constant.