Wasserstein variational gradient descent: From semi-discrete optimal transport to ensemble variational inference

TL;DR

This paper introduces a novel particle-based variational inference method leveraging semi-discrete optimal transport, enabling more flexible posterior approximations through ensemble variational inference.

Contribution

It proposes a new divergence based on semi-discrete optimal transport, connecting optimal transport theory with particle-based variational inference.

Findings

Provides a new divergence measure for variational inference.

Develops an algorithm that maps particles to posterior segments.

Enables ensemble variational inference with optimal transport.

Abstract

Particle-based variational inference offers a flexible way of approximating complex posterior distributions with a set of particles. In this paper we introduce a new particle-based variational inference method based on the theory of semi-discrete optimal transport. Instead of minimizing the KL divergence between the posterior and the variational approximation, we minimize a semi-discrete optimal transport divergence. The solution of the resulting optimal transport problem provides both a particle approximation and a set of optimal transportation densities that map each particle to a segment of the posterior distribution. We approximate these transportation densities by minimizing the KL divergence between a truncated distribution and the optimal transport solution. The resulting algorithm can be interpreted as a form of ensemble variational inference where each particle is associated with a local variational approximation.