All You Need is a Good Functional Prior for Bayesian Deep Learning

TL;DR

This paper introduces a method to align neural network priors with functional priors using Wasserstein distance, significantly improving Bayesian deep learning performance and making fully Bayesian neural networks more feasible.

Contribution

It proposes a novel framework to match neural network priors with functional priors via Wasserstein distance, enhancing Bayesian deep learning methods.

Findings

Large performance improvements over existing methods

Effective coupling of neural priors with Gaussian process priors

Scalable MCMC sampling enhances Bayesian neural network training

Abstract

The Bayesian treatment of neural networks dictates that a prior distribution is specified over their weight and bias parameters. This poses a challenge because modern neural networks are characterized by a large number of parameters, and the choice of these priors has an uncontrolled effect on the induced functional prior, which is the distribution of the functions obtained by sampling the parameters from their prior distribution. We argue that this is a hugely limiting aspect of Bayesian deep learning, and this work tackles this limitation in a practical and effective way. Our proposal is to reason in terms of functional priors, which are easier to elicit, and to "tune" the priors of neural network parameters in a way that they reflect such functional priors. Gaussian processes offer a rigorous framework to define prior distributions over functions, and we propose a novel and robust framework to match their prior with the functional prior of neural networks based on the minimization of their Wasserstein distance. We provide vast experimental evidence that coupling these priors with scalable Markov chain Monte Carlo sampling offers systematically large performance improvements over alternative choices of priors and state-of-the-art approximate Bayesian deep learning approaches. We consider this work a considerable step in the direction of making the long-standing challenge of carrying out a fully Bayesian treatment of neural networks, including convolutional neural networks, a concrete possibility.