Particle Semi-Implicit Variational Inference

TL;DR

This paper introduces Particle Variational Inference (PVI), a novel semi-implicit variational inference method that directly optimizes the ELBO using particle approximations, improving flexibility and performance over existing approaches.

Contribution

PVI employs empirical measures to approximate optimal mixing distributions, avoiding parametric assumptions and enabling direct ELBO optimization in semi-implicit variational inference.

Findings

PVI outperforms existing SIVI methods on various tasks.

Theoretical analysis confirms existence and uniqueness of solutions.

PVI efficiently approximates complex variational distributions.

Abstract

Semi-implicit variational inference (SIVI) enriches the expressiveness of variational families by utilizing a kernel and a mixing distribution to hierarchically define the variational distribution. Existing SIVI methods parameterize the mixing distribution using implicit distributions, leading to intractable variational densities. As a result, directly maximizing the evidence lower bound (ELBO) is not possible, so they resort to one of the following: optimizing bounds on the ELBO, employing costly inner-loop Markov chain Monte Carlo runs, or solving minimax objectives. In this paper, we propose a novel method for SIVI called Particle Variational Inference (PVI) which employs empirical measures to approximate the optimal mixing distributions characterized as the minimizer of a free energy functional. PVI arises naturally as a particle approximation of a Euclidean--Wasserstein gradient flow and, unlike prior works, it directly optimizes the ELBO whilst making no parametric assumption about the mixing distribution. Our empirical results demonstrate that PVI performs favourably compared to other SIVI methods across various tasks. Moreover, we provide a theoretical analysis of the behaviour of the gradient flow of a related free energy functional: establishing the existence and uniqueness of solutions as well as propagation of chaos results.