A Non-Asymptotic Analysis for Stein Variational Gradient Descent
This work addresses theoretical gaps in understanding SVGD's convergence behavior, which is important for researchers in machine learning and statistics, though it is incremental as it builds on existing analysis.
The paper tackles the analysis of the Stein Variational Gradient Descent (SVGD) algorithm for approximating target probability distributions, providing a finite-time analysis with a descent lemma, convergence rates for the Stein Fisher divergence, and results linking finite particle systems to the population limit.
We study the Stein Variational Gradient Descent (SVGD) algorithm, which optimises a set of particles to approximate a target probability distribution $π\propto e^{-V}$ on $\mathbb{R}^d$. In the population limit, SVGD performs gradient descent in the space of probability distributions on the KL divergence with respect to $π$, where the gradient is smoothed through a kernel integral operator. In this paper, we provide a novel finite time analysis for the SVGD algorithm. We provide a descent lemma establishing that the algorithm decreases the objective at each iteration, and rates of convergence for the average Stein Fisher divergence (also referred to as Kernel Stein Discrepancy). We also provide a convergence result of the finite particle system corresponding to the practical implementation of SVGD to its population version.