Function-Space MCMC for Bayesian Wide Neural Networks

Bayesian Neural Networks represent a fascinating confluence of deep learning and probabilistic reasoning, offering a compelling framework for understanding uncertainty in complex predictive models. In this paper, we investigate the use of the preconditioned Crank-Nicolson algorithm and its Langevin version to sample from the reparametrised posterior distribution of the weights as the widths of Bayesian Neural Networks grow larger. In addition to being robust in the infinite-dimensional setting, we prove that the acceptance probabilities of the proposed methods approach 1 as the width of the network increases, independently of any stepsize tuning. Moreover, we examine and compare how the mixing speeds of the underdamped Langevin Monte Carlo, the preconditioned Crank-Nicolson and the preconditioned Crank-Nicolson Langevin samplers are influenced by changes in the network width in some real-world cases. Our findings suggest that, in wide Bayesian Neural Networks configurations, the preconditioned Crank-Nicolson method allows for more efficient sampling of the reparametrised posterior distribution, as evidenced by a higher effective sample size and improved diagnostic results compared with the other analysed algorithms.