Variational Bayesian Neural Networks via Resolution of Singularities
This work addresses theoretical and practical challenges in variational inference for Bayesian neural networks, offering an incremental improvement for researchers in machine learning and statistics.
The authors tackled the discrepancy between predictive performance and variational objectives in Bayesian neural networks by applying singular learning theory to design a new variational family, resulting in improved variational free energy and generalization error compared to Gaussian-based methods.
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.