Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

TL;DR

This paper provides explicit quantitative local convergence rates in strong norms for Stein Variational Gradient Descent with Riesz kernels, advancing understanding of its efficiency in sampling from target distributions.

Contribution

It establishes the first explicit polynomial convergence rates in L^2-norm for SVGD in the mean-field limit under smoothness and closeness assumptions.

Findings

Convergence rates depend on dimension and regularity parameters.

Rates are shown to be sharp in certain regimes.

Numerical experiments support the theoretical results.

Abstract

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in $L^2$-norm, we obtain explicit polynomial convergence rates in $L^2$-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies.