Understanding Priors in Bayesian Neural Networks at the Unit Level

TL;DR

This paper explores how Gaussian priors in deep Bayesian neural networks influence the distribution of unit activations, revealing increasingly heavy-tailed behaviors with depth, and providing new theoretical insights supported by simulations.

Contribution

It characterizes the evolving distributional properties of unit activations in deep Bayesian neural networks with Gaussian priors, revealing a progression from Gaussian to sub-Weibull distributions.

Findings

First layer units are Gaussian.

Second layer units are sub-exponential.

Deeper layer units are sub-Weibull.

Abstract

We investigate deep Bayesian neural networks with Gaussian weight priors and a class of ReLU-like nonlinearities. Bayesian neural networks with Gaussian priors are well known to induce an L2, "weight decay", regularization. Our results characterize a more intricate regularization effect at the level of the unit activations. Our main result establishes that the induced prior distribution on the units before and after activation becomes increasingly heavy-tailed with the depth of the layer. We show that first layer units are Gaussian, second layer units are sub-exponential, and units in deeper layers are characterized by sub-Weibull distributions. Our results provide new theoretical insight on deep Bayesian neural networks, which we corroborate with simulation experiments.