We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernel Stein Discrepancy ($\mathsf{KSD}$) and Wasserstein-2 metrics. Our key insight is the observation that the time derivative of the relative entropy between the joint density of $N$ particle locations and the $N$-fold product target measure, starting from a regular initial distribution, splits into a dominant `negative part' proportional to $N$ times the expected $\mathsf{KSD}^2$ and a smaller `positive part'. This observation leads to $\mathsf{KSD}$ rates of order $1/\sqrt{N}$, providing a near optimal double exponential improvement over the recent result by~\cite{shi2024finite}. Under mild assumptions on the kernel and potential, these bounds also grow linearly in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence. For the case of `bilinear + Mat\'ern' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.

本文针对斯坦变分梯度下降（SVGD）算法在核斯坦误差（KSD）和Wasserstein-2度量下的有限粒子收敛速率问题，提出了新的见解。研究表明，相对熵的时间导数可以分解为与N成正比的主要负部分及较小的正部分，从而获得收敛速率为$1/\sqrt{N}$，较之前研究显著提高。此外，该方法在维度上也有线性增长，从而进一步推动了Wasserstein-2的收敛性研究。 

斯坦变分梯度下降中改进的有限粒子收敛速率