A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methods
This provides a foundational understanding for uncertainty quantification in deep learning, connecting disparate approaches and enabling new algorithms with proven convergence.
The paper establishes a rigorous mathematical link between Bayesian, variational Bayesian, and ensemble methods by reformulating deep learning optimization as a convex problem in probability measure space, leading to a unified theory and new ensembling schemes with convergence guarantees.
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.