Quantitative CLTs in Deep Neural Networks

TL;DR

This paper provides quantitative bounds on how closely deep neural networks with Gaussian weights approximate Gaussian processes as width increases, with improved and optimal convergence rates.

Contribution

It introduces new, stronger bounds on the finite-width approximation of neural networks to Gaussian processes, valid at large but finite widths and fixed depths.

Findings

Bounds scale as n^{-mma} for the discrepancy between network and Gaussian process

Bounds are strictly stronger than previous results in the literature

In one-dimensional case, bounds are proven to be optimal with matching lower bounds

Abstract

We study the distribution of a fully connected neural network with random Gaussian weights and biases in which the hidden layer widths are proportional to a large constant $n$. Under mild assumptions on the non-linearity, we obtain quantitative bounds on normal approximations valid at large but finite $n$ and any fixed network depth. Our theorems show both for the finite-dimensional distributions and the entire process, that the distance between a random fully connected network (and its derivatives) to the corresponding infinite width Gaussian process scales like $n^{-\gamma}$ for $\gamma>0$, with the exponent depending on the metric used to measure discrepancy. Our bounds are strictly stronger in terms of their dependence on network width than any previously available in the literature; in the one-dimensional case, we also prove that they are optimal, i.e., we establish matching lower bounds.