On the Expressiveness of Approximate Inference in Bayesian Neural Networks

TL;DR

This paper investigates the limitations of common variational inference methods in Bayesian neural networks, revealing fundamental issues in uncertainty estimation especially in shallow networks, and highlighting the importance of exact inference.

Contribution

It provides theoretical proofs of limitations in variational methods for shallow BNNs and empirical evidence of their shortcomings compared to exact inference, emphasizing careful use of approximations.

Findings

Variational methods have fundamental limitations in shallow BNNs.

Exact inference does not exhibit the same uncertainty pathologies.

Deep networks can have more flexible uncertainty estimates despite similar issues.

Abstract

While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood. We study the quality of common variational methods in approximating the Bayesian predictive distribution. For single-hidden layer ReLU BNNs, we prove a fundamental limitation in function-space of two of the most commonly used distributions defined in weight-space: mean-field Gaussian and Monte Carlo dropout. We find there are simple cases where neither method can have substantially increased uncertainty in between well-separated regions of low uncertainty. We provide strong empirical evidence that exact inference does not have this pathology, hence it is due to the approximation and not the model. In contrast, for deep networks, we prove a universality result showing that there exist approximate posteriors in the above classes which provide flexible uncertainty estimates. However, we find empirically that pathologies of a similar form as in the single-hidden layer case can persist when performing variational inference in deeper networks. Our results motivate careful consideration of the implications of approximate inference methods in BNNs.