Structured and Efficient Variational Deep Learning with Matrix Gaussian Posteriors

We introduce a variational Bayesian neural network where the parameters are governed via a probability distribution on random matrices. Specifically, we employ a matrix variate Gaussian (Gupta & Nagar ’99) parameter posterior distribution where we explicitly model the covariance among the input and output dimensions of each layer. Furthermore, with approximate covariance matrices we can achieve a more efficient way to represent those correlations that is also cheaper than fully factorized parameter posteriors. We further show that with the “local reprarametrization trick" (Kingma & Welling ’15) on this posterior distribution we arrive at a Gaussian Process (Rasmussen ’06) interpretation of the hidden units in each layer and we, similarly with (Gal & Ghahramani ’15), provide connections with deep Gaussian processes. We continue in taking advantage of this duality and incorporate “pseudo-data” (Snelson & Ghahramani ’05) in our model, which in turn allows for more efficient posterior sampling while maintaining the properties of the original model. The validity of the proposed approach is verified through extensive experiments

Multiplicative Normalizing Flows for Variational Bayesian Neural Networks

We reinterpret multiplicative noise in neural networks as auxiliary random variables that augment the approximate posterior in a variational setting for Bayesian neural networks. We show that through this interpretation it is both efficient and straightforward to improve the approximation by employing normalizing flows while still allowing for local reparametrizations and a tractable lower bound. In experiments we show that with this new approximation we can significantly improve upon classical mean field for Bayesian neural networks on both predictive accuracy as well as predictive uncertainty