Variational inference via radial transport

Luca Ghafourpour, Sinho Chewi, Alessio Figalli, Aram-Alexandre Pooladian

TL;DR: We study variational inference over the space of radial families, providing convergence guarantees for our algorithm, radVI, which can be applied to any off-the-shelf Gaussian VI method to enhance the approximating family beyond the Gaussian family.

Abstract: In variational inference (VI), the practitioner approximates a high-dimensional distribution $\pi$ with a simple surrogate one, often a (product) Gaussian distribution. However, in many cases of practical interest, Gaussian distributions might not capture the correct radial profile of $\pi$, resulting in poor coverage. In this work, we approach the VI problem from the perspective of optimizing over these radial profiles. Our algorithm $\texttt{radVI}$ is a cheap, effective add-on to many existing VI schemes, such as Gaussian (mean-field) VI and Laplace approximation. We provide theoretical convergence guarantees for our algorithm, owing to recent developments in optimization over the Wasserstein space—the space of probability distributions endowed with the Wasserstein distance—and new regularity properties of radial transport maps in the style of Caffarelli (2000).

Submission Number: 840