The Ridgelet Prior: A Covariance Function Approach to Prior Specification for Bayesian Neural Networks

TL;DR

This paper introduces the ridgelet prior, a novel covariance function-based approach for specifying meaningful priors in Bayesian neural networks, enabling better approximation of Gaussian processes with finite-sample guarantees.

Contribution

It proposes a non-asymptotic ridgelet prior that approximates Gaussian process covariance functions, establishing universality and providing finite-sample error bounds.

Findings

Ridgelet prior can outperform unstructured priors in regression tasks.

The approach offers finite-sample error bounds for neural network approximation.

Demonstrates the potential for Gaussian process-inspired priors in Bayesian neural networks.

Abstract

Bayesian neural networks attempt to combine the strong predictive performance of neural networks with formal quantification of uncertainty associated with the predictive output in the Bayesian framework. However, it remains unclear how to endow the parameters of the network with a prior distribution that is meaningful when lifted into the output space of the network. A possible solution is proposed that enables the user to posit an appropriate Gaussian process covariance function for the task at hand. Our approach constructs a prior distribution for the parameters of the network, called a ridgelet prior, that approximates the posited Gaussian process in the output space of the network. In contrast to existing work on the connection between neural networks and Gaussian processes, our analysis is non-asymptotic, with finite sample-size error bounds provided. This establishes the universality property that a Bayesian neural network can approximate any Gaussian process whose covariance function is sufficiently regular. Our experimental assessment is limited to a proof-of-concept, where we demonstrate that the ridgelet prior can out-perform an unstructured prior on regression problems for which a suitable Gaussian process prior can be provided.