  The goal of Bayesian deep learning is to provide uncertainty quantification
via the posterior distribution. However, exact inference over the weight space
is computationally intractable due to the ultra-high dimensions of the neural
network. Variational inference (VI) is a promising approach, but naive
application on weight space does not scale well and often underperform on
predictive accuracy. In this paper, we propose a new adaptive variational
Bayesian algorithm to train neural networks on weight space that achieves high
predictive accuracy. By showing that there is an equivalence to Stochastic
Gradient Hamiltonian Monte Carlo(SGHMC) with preconditioning matrix, we then
propose an MCMC within EM algorithm, which incorporates the spike-and-slab
prior to capture the sparsity of the neural network. The EM-MCMC algorithm
allows us to perform optimization and model pruning within one-shot. We
evaluate our methods on CIFAR-10, CIFAR-100 and ImageNet datasets, and
demonstrate that our dense model can reach the state-of-the-art performance and
our sparse model perform very well compared to previously proposed pruning
schemes.
