What Are Bayesian Neural Network Posteriors Really Like?

TL;DR

This paper uses full-batch Hamiltonian Monte Carlo to analyze Bayesian neural network posteriors, revealing insights into their performance, robustness, and differences from approximate methods, challenging some recent assumptions.

Contribution

It demonstrates the effectiveness of HMC in capturing BNN posteriors, compares it with other methods, and investigates properties like posterior tempering and generalization under domain shift.

Findings

BNNs with HMC outperform standard training and deep ensembles.

A single long HMC chain suffices for posterior approximation.

Posterior tempering is unnecessary for near-optimal performance.

Abstract

The posterior over Bayesian neural network (BNN) parameters is extremely high-dimensional and non-convex. For computational reasons, researchers approximate this posterior using inexpensive mini-batch methods such as mean-field variational inference or stochastic-gradient Markov chain Monte Carlo (SGMCMC). To investigate foundational questions in Bayesian deep learning, we instead use full-batch Hamiltonian Monte Carlo (HMC) on modern architectures. We show that (1) BNNs can achieve significant performance gains over standard training and deep ensembles; (2) a single long HMC chain can provide a comparable representation of the posterior to multiple shorter chains; (3) in contrast to recent studies, we find posterior tempering is not needed for near-optimal performance, with little evidence for a "cold posterior" effect, which we show is largely an artifact of data augmentation; (4) BMA performance is robust to the choice of prior scale, and relatively similar for diagonal Gaussian, mixture of Gaussian, and logistic priors; (5) Bayesian neural networks show surprisingly poor generalization under domain shift; (6) while cheaper alternatives such as deep ensembles and SGMCMC methods can provide good generalization, they provide distinct predictive distributions from HMC. Notably, deep ensemble predictive distributions are similarly close to HMC as standard SGLD, and closer than standard variational inference.