A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methods

Veit David Wild, Sahra Ghalebikesabi, Dino Sejdinovic, Jeremias Knoblauch

Keywords: Wasserstein gradient flow, generalised variational inference, deep ensembles, Bayesian deep learning, variational Bayes

TL;DR: We establish the first unified theory connecting Bayesian, variational Bayesian, and ensemble methods for deep learning by leveraging Wasserstein Gradient Flows.

Abstract: 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.

Submission Number: 3633