Functional Variational Bayesian Neural Networks

TL;DR

This paper introduces functional variational Bayesian neural networks (fBNNs) that optimize an ELBO over functions, enabling flexible priors, better uncertainty, and scalability in Bayesian neural network modeling.

Contribution

The paper proposes a novel functional ELBO for BNNs, allowing direct optimization over distributions of functions and enabling rich prior specifications.

Findings

fBNNs extrapolate well with structured priors

fBNNs provide reliable uncertainty estimates

fBNNs scale to large datasets

Abstract

Variational Bayesian neural networks (BNNs) perform variational inference over weights, but it is difficult to specify meaningful priors and approximate posteriors in a high-dimensional weight space. We introduce functional variational Bayesian neural networks (fBNNs), which maximize an Evidence Lower BOund (ELBO) defined directly on stochastic processes, i.e. distributions over functions. We prove that the KL divergence between stochastic processes equals the supremum of marginal KL divergences over all finite sets of inputs. Based on this, we introduce a practical training objective which approximates the functional ELBO using finite measurement sets and the spectral Stein gradient estimator. With fBNNs, we can specify priors entailing rich structures, including Gaussian processes and implicit stochastic processes. Empirically, we find fBNNs extrapolate well using various structured priors, provide reliable uncertainty estimates, and scale to large datasets.