We establish the first mathematically rigorous link between Bayesian, variational Bayesian, and ensemble methods. A key step towards this it to reformulate the non-convex optimisation problem typically encountered in deep learning as a convex optimisation in the space of probability measures. On a technical level, our contribution amounts to studying generalised variational inference through the lense of Wasserstein gradient flows. The result is a unified theory of various seemingly disconnected approaches that are commonly used for uncertainty quantification in deep learning---including deep ensembles and (variational) Bayesian methods. This offers a fresh perspective on the reasons behind the success of deep ensembles over procedures based on parameterised variational inference, and allows the derivation of new ensembling schemes with convergence guarantees. We showcase this by proposing a family of interacting deep ensembles with direct parallels to the interactions of particle systems in thermodynamics, and use our theory to prove the convergence of these algorithms to a well-defined global minimiser on the space of probability measures.

Variational Gaussian process (GP) approximations have become a standard tool in fast GP inference. This technique requires a user to select variational features to increase efficiency. So far the common choices in the literature are disparate and lacking generality. We propose to view the GP as lying in a Banach space which then facilitates a unified perspective. This is used to understand the relationship between existing features and to draw a connection between kernel ridge regression and variational GP approximations.

We develop a framework for generalized variational inference in infinite-dimensional function spaces and use it to construct a method termed Gaussian Wasserstein inference (GWI). GWI leverages the Wasserstein distance between Gaussian measures on the Hilbert space of square-integrable functions in order to determine a variational posterior using a tractable optimization criterion. It avoids pathologies arising in standard variational function space inference. An exciting application of GWI is the ability to use deep neural networks in the variational parametrization of GWI, combining their superior predictive performance with the principled uncertainty quantification analogous to that of Gaussian processes. The proposed method obtains state-of-the-art performance on several benchmark datasets.