Variational Bayesian Neural Networks via Resolution of Singularities

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, we lay to rest some of the confusion surrounding discrepancies between downstream predictive performance measured via 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 Bhattacharya, Pati, and Plummer 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. Supplemental appendices and code for the article are available online.