On the Convergence of SVGD in KL Divergence via Approximate Gradient Flow

Abstract

This study investigates the convergence of Stein variational gradient descent (SVGD), which is used to approximate a target distribution based on a gradient flow on the space of probability distributions. The existing studies mainly focus on the convergence in the kernel Stein discrepancy, which doesn't imply weak convergence in many practical settings. To address this issue, we propose to introduce a novel analytical approach called $(\epsilon,\delta)$-approximate gradient flow, extending conventional concepts of approximation error for the Wasserstein gradient. With this approach, we show the sub-linear convergence of SVGD in Kullback--Leibler divergence under the discrete time and infinite particle settings. Finally, we validate our theoretical findings through several numerical experiments.