Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow
This work provides the first quantitative convergence rates for the last iterate of mean-field SVGD, addressing a key gap for practitioners and theorists in sampling and variational inference.
The paper establishes quantitative local convergence rates for mean-field Stein Variational Gradient Flow, achieving explicit polynomial rates in L^2-norm under smoothness and closeness assumptions, with sharpness in certain regimes and recovery of global exponential convergence for Coulomb singularities.
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.