Gaussian Process Behaviour in Wide Deep Neural Networks

TL;DR

This paper demonstrates that deep, wide neural networks tend to behave like Gaussian processes, extending Neal's 1996 results to multiple layers and providing empirical convergence analysis and comparisons.

Contribution

It extends the theoretical understanding of deep neural networks by formalizing their convergence to Gaussian processes as width increases, with empirical validation.

Findings

Deep networks converge to Gaussian processes as width increases.

Empirical convergence rates are evaluated using maximum mean discrepancy.

Finite Bayesian networks can closely match Gaussian process predictions.

Abstract

Whilst deep neural networks have shown great empirical success, there is still much work to be done to understand their theoretical properties. In this paper, we study the relationship between random, wide, fully connected, feedforward networks with more than one hidden layer and Gaussian processes with a recursive kernel definition. We show that, under broad conditions, as we make the architecture increasingly wide, the implied random function converges in distribution to a Gaussian process, formalising and extending existing results by Neal (1996) to deep networks. To evaluate convergence rates empirically, we use maximum mean discrepancy. We then compare finite Bayesian deep networks from the literature to Gaussian processes in terms of the key predictive quantities of interest, finding that in some cases the agreement can be very close. We discuss the desirability of Gaussian process behaviour and review non-Gaussian alternative models from the literature.