给出均值场斯坦因变分梯度流的局部收敛速率，揭示其在高维环面上的定量行为。
Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow
- 基于瑞兹型核，在环面上建立强范数下局部收敛分析。
- 在初始密度与目标接近时，获得依赖维数和正则性的多项式收敛率。
- 理论结果被数值实验验证，适用于高维采样与概率推断场景。
斯坦因变分梯度下降（SVGD）是一种通过得分函数从目标概率测度中采样的确定性粒子方法。在均值场和连续时间极限下，已知该流弱收敛于目标分布，但对最后迭代的收敛速率尚无量化结果。本文在 $d$-维环面上，针对瑞兹型相互作用核，建立了该动力学在强范数下的定量局部收敛性。假设初始密度与目标在 $L^2$-范数下光滑且接近，我们获得了依赖于维度及核、初始化与目标正则性参数的显式多项式收敛速率。进一步证明这些速率在某些情形下是紧的，并通过数值实验验证理论。在核具库仑奇异性这一特殊情况下，恢复了先前工作中的全局指数收敛结果。分析受近期关于核均值差异的 Wasserstein 梯度流研究启发。
原文摘要 · Abstract (English)
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.
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