Trace-class Gaussian priors for Bayesian learning of neural networks with MCMC

TL;DR

This paper proposes a new Gaussian neural network prior that is scalable in high dimensions and compatible with Hilbert space MCMC, enabling efficient Bayesian inference for functions in high-dimensional spaces.

Contribution

It introduces a trace-class Gaussian prior for neural networks that remains well-defined in the infinite-width limit and demonstrates its effectiveness with MCMC in high-dimensional Bayesian learning.

Findings

The prior is scalable and well-defined in the infinite-width limit.

MCMC sampling remains stable under mesh refinement with this prior.

Numerical experiments show advantages over traditional priors and applications in Bayesian Reinforcement Learning.

Abstract

This paper introduces a new neural network based prior for real valued functions on $\mathbb R^d$ which, by construction, is more easily and cheaply scaled up in the domain dimension $d$ compared to the usual Karhunen-Lo\`eve function space prior. The new prior is a Gaussian neural network prior, where each weight and bias has an independent Gaussian prior, but with the key difference that the variances decrease in the width of the network in such a way that the resulting function is \emph{almost surely} well defined in the limit of an infinite width network. We show that in a Bayesian treatment of inferring unknown functions, the induced posterior over functions is amenable to Monte Carlo sampling using Hilbert space Markov chain Monte Carlo (MCMC) methods. This type of MCMC is popular, e.g. in the Bayesian Inverse Problems literature, because it is stable under \emph{mesh refinement}, i.e. the acceptance probability does not shrink to $0$ as more parameters of the function's prior are introduced, even \emph{ad infinitum}. In numerical examples we demonstrate these stated competitive advantages over other function space priors. We also implement examples in Bayesian Reinforcement Learning to automate tasks from data and demonstrate, for the first time, stability of MCMC to mesh refinement for these type of problems.