On the Convergence of Locally Adaptive and Scalable Diffusion-Based Sampling Methods for Deep Bayesian Neural Network Posteriors

Achieving robust uncertainty quantification for deep neural networks represents an important requirement in many real-world applications of deep learning such as medical imaging where it is necessary to assess the reliability of a neural network's prediction.Bayesian neural networks are a promising approach for modeling uncertainties in deep neural networks. Unfortunately, generating samples from the posterior distribution of neural networks is a major challenge. One significant advance in that direction would be the incorporation of adaptive step sizes, similar to modern neural network optimizers, into Monte Carlo Markov chain sampling algorithms without significantly increasing computational demand.Over the past years, several papers have introduced sampling algorithms with corresponding theorems stating that they achieve this property. In this paper, we demonstrate that these methods can have a substantial bias in the distribution they sample, even in the limit of vanishing step sizes and at full batch size. Furthermore, for most of the algorithms, we show that convergence to the correct distribution can be restored with a simple fix at the cost of increasing computational demand.