On the Expressiveness of Approximate Inference in Bayesian Neural Networks
This work addresses the problem of unreliable uncertainty estimation in Bayesian neural networks for practitioners, highlighting incremental insights into the limitations of existing approximation methods.
The study reveals that common variational inference methods in Bayesian neural networks, such as mean-field Gaussian and Monte Carlo dropout, have fundamental limitations in approximating uncertainty for single-hidden layer networks, failing to increase uncertainty between separated low-uncertainty regions, while exact inference does not show this issue; for deep networks, theoretical universality exists but similar pathologies persist empirically.
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.