Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent

TL;DR

This paper establishes improved finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in terms of Kernelized Stein Discrepancy and Wasserstein-2 metrics, demonstrating near optimal rates and polynomial dimension dependence.

Contribution

The paper provides the first finite-particle convergence rates for SVGD in KSD and Wasserstein metrics, with a novel analysis splitting the entropy derivative into dominant and smaller parts, leading to near optimal rates.

Findings

KSD convergence rate of order 1/√N in continuous and discrete time.

Wasserstein-2 convergence under bilinear kernel modifications.

Polynomial growth of bounds in the dimension d.

Abstract

We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy ($\mathsf{KSD}$) and Wasserstein-2 metrics. Our key insight is 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 $1/\sqrt{N}$, in both continuous and discrete time, providing a near optimal (in the sense of matching the corresponding i.i.d. rates) double exponential improvement over the recent result by Shi and Mackey (2024). Under mild assumptions on the kernel and potential, these bounds also grow polynomially in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence in continuous time. 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.