Bayesian Interpolation with Deep Linear Networks

TL;DR

This paper provides a complete theoretical analysis of Bayesian linear networks, revealing how depth, width, and data size influence model quality, and demonstrating the benefits of increased depth for model selection and prediction.

Contribution

It offers exact non-asymptotic expressions for Bayesian posteriors and evidence in linear networks, introducing the concept of effective depth and analyzing the impact of depth and width.

Findings

Infinite depth networks have optimal predictions with data-agnostic priors.

Bayesian evidence is maximized at infinite depth in wide networks.

Effective depth influences the structure of the posterior in large-data regimes.

Abstract

Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find non-asymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through novel asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is a novel emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit.