Variational inference via Wasserstein gradient flows
This work addresses the lack of algorithmic guarantees in VI, a key computational approach for large-scale Bayesian inference, though it is incremental as it builds on existing gradient flow theory.
The authors tackled the problem of providing theoretical guarantees for variational inference (VI) by proposing methods based on gradient flows on the Bures-Wasserstein space, resulting in strong guarantees for log-concave posteriors akin to MCMC.
Along with Markov chain Monte Carlo (MCMC) methods, variational inference (VI) has emerged as a central computational approach to large-scale Bayesian inference. Rather than sampling from the true posterior $π$, VI aims at producing a simple but effective approximation $\hat π$ to $π$ for which summary statistics are easy to compute. However, unlike the well-studied MCMC methodology, algorithmic guarantees for VI are still relatively less well-understood. In this work, we propose principled methods for VI, in which $\hat π$ is taken to be a Gaussian or a mixture of Gaussians, which rest upon the theory of gradient flows on the Bures--Wasserstein space of Gaussian measures. Akin to MCMC, it comes with strong theoretical guarantees when $π$ is log-concave.