Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

2026-05-10

AI Summary

This work addresses the absence of quantitative convergence rates for the final iterate of Mean-Field Stein Variational Gradient Descent (SVGD) dynamics in strong norms. Focusing on Riesz-type kernels, the authors establish local strong convergence on the d-dimensional torus under the assumption that the initial density is smooth and close to the target distribution in the L² norm. By integrating tools from Wasserstein gradient flows, kernel mean discrepancy, harmonic analysis, and partial differential equations, they derive the first explicit polynomial L² convergence rate that depends on both dimension and regularity parameters, and demonstrate its sharpness in specific regimes. For the Coulomb singular kernel case, the analysis recovers the known global exponential convergence result. Numerical experiments corroborate the theoretical findings.

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.

Problem

• Stein Variational Gradient Descent
• mean-field limit
• quantitative convergence
• local convergence
• strong norms

Innovation

• Stein Variational Gradient Descent
• mean-field limit
• quantitative convergence rate
• Riesz kernel
• strong norm convergence