We derive finite-particle rates for the regularized Stein variational gradient descent (R-SVGD) algorithm introduced by He et al. (2024) that corrects the constant-order bias of the SVGD by applying a resolvent-type preconditioner to the kernelized Wasserstein gradient. For the resulting interacting $N$-particle system, we establish explicit non-asymptotic bounds for time-averaged (annealed) empirical measures, illustrating convergence in the \emph{true} (non-kernelized) Fisher information and, under a $\mathrm{W}_1\mathrm{I}$ condition on the target, corresponding $\mathrm{W}_1$ convergence for a large class of smooth kernels. Our analysis covers both continuous- and discrete-time dynamics and yields principled tuning rules for the regularization parameter, step size, and averaging horizon that quantify the trade-off between approximating the Wasserstein gradient flow and controlling finite-particle estimation error.

翻译：本文推导了He等人(2024)提出的正则化Stein变分梯度下降(R-SVGD)算法的有限粒子收敛率。该算法通过对核化Wasserstein梯度应用预解型预条件子，修正了SVGD的常数阶偏差。对于由此产生的$N$粒子相互作用系统，我们建立了时间平均(退火)经验测度的显式非渐近界，证明了其在\emph{真实}(非核化)Fisher信息度量下的收敛性，并在目标分布满足$\mathrm{W}_1\mathrm{I}$条件时，对一大类光滑核证明了相应的$\mathrm{W}_1$收敛性。我们的分析同时涵盖连续时间与离散时间动力学，并为正则化参数、步长和平均时域提供了基于原理的调参规则，量化了逼近Wasserstein梯度流与控制有限粒子估计误差之间的权衡关系。