Variational Bayesian Neural Networks via Resolution of Singularities

TL;DR

This paper emphasizes the importance of singular learning theory in understanding and improving variational inference in Bayesian neural networks, proposing a novel normalizing flow-based variational family that enhances performance.

Contribution

It introduces a SLT-informed variational family using normalizing flows with a generalized gamma base, improving inference in Bayesian neural networks.

Findings

Improved variational free energy with the proposed method

Enhanced variational generalization error

Clarified the link between SLT and variational objectives

Abstract

In this work, we advocate for the importance of singular learning theory (SLT) as it pertains to the theory and practice of variational inference in Bayesian neural networks (BNNs). To begin, using SLT, we lay to rest some of the confusion surrounding discrepancies between downstream predictive performance measured via e.g., the test log predictive density, and the variational objective. Next, we use the SLT-corrected asymptotic form for singular posterior distributions to inform the design of the variational family itself. Specifically, we build upon the idealized variational family introduced in \citet{bhattacharya_evidence_2020} which is theoretically appealing but practically intractable. Our proposal takes shape as a normalizing flow where the base distribution is a carefully-initialized generalized gamma. We conduct experiments comparing this to the canonical Gaussian base distribution and show improvements in terms of variational free energy and variational generalization error.