Stein Variational Gradient Descent as Gradient Flow

TL;DR

This paper provides the first theoretical analysis of Stein Variational Gradient Descent (SVGD), revealing its convergence properties and its interpretation as a gradient flow of the KL divergence under a new metric.

Contribution

It introduces a novel theoretical framework for SVGD, analyzing its weak convergence and asymptotic behavior via a gradient flow perspective.

Findings

SVGD's asymptotic behavior is characterized by a gradient flow of the KL divergence.

The paper establishes weak convergence properties of SVGD.

New results on Stein operator and Stein's identity using weak derivatives.

Abstract

Stein variational gradient descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate given distributions, based on an efficient gradient-based update that guarantees to optimally decrease the KL divergence within a function space. This paper develops the first theoretical analysis on SVGD, discussing its weak convergence properties and showing that its asymptotic behavior is captured by a gradient flow of the KL divergence functional under a new metric structure induced by Stein operator. We also provide a number of results on Stein operator and Stein's identity using the notion of weak derivative, including a new proof of the distinguishability of Stein discrepancy under weak conditions.