Particle Semi-Implicit Variational Inference

Jen Ning Lim, Adam Michael Johansen

Keywords: variational inference, gradient flow

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.

Primary Area: Probabilistic methods (for example: variational inference, Gaussian processes)

Submission Number: 1654