Long-time asymptotics of noisy SVGD outside the population limit

TL;DR

This paper analyzes the long-term behavior of a noisy variant of Stein Variational Gradient Descent (SVGD), showing it converges to the target distribution and avoids variance collapse, with trajectories resembling a McKean-Vlasov process.

Contribution

It provides the first theoretical characterization of the asymptotic behavior of noisy SVGD and demonstrates its advantages over standard SVGD in avoiding variance collapse.

Findings

Noisy SVGD's limit set is well-defined and approaches the target distribution as iterations increase.

Noisy SVGD avoids the variance collapse observed in standard SVGD.

Trajectories of noisy SVGD resemble a McKean-Vlasov process.

Abstract

Stein Variational Gradient Descent (SVGD) is a widely used sampling algorithm that has been successfully applied in several areas of Machine Learning. SVGD operates by iteratively moving a set of interacting particles (which represent the samples) to approximate the target distribution. Despite recent studies on the complexity of SVGD and its variants, their long-time asymptotic behavior (i.e., after numerous iterations ) is still not understood in the finite number of particles regime. We study the long-time asymptotic behavior of a noisy variant of SVGD. First, we establish that the limit set of noisy SVGD for large is well-defined. We then characterize this limit set, showing that it approaches the target distribution as increases. In particular, noisy SVGD provably avoids the variance collapse observed for SVGD. Our approach involves demonstrating that the trajectories of noisy SVGD closely resemble those described by a McKean-Vlasov process.