- Aniket Das
- Dheeraj Nagaraj

## Abstract

Stein Variational Gradient Descent (SVGD) is a popular nonparametric variational inference algorithm which simulates an interacting particle system to approximate a target distribution. While SVGD has demonstrated promising empirical performance across various domains, and its population (i.e, infinite-particle) limit dynamics is well studied, the behavior of SVGD in the finite-particle regime is much less understood. In this work, we design two computationally efficient variants of SVGD, namely VP-SVGD and RB-SVGD, with provably fast finite-particle convergence rates. By introducing the notion of \emph{virtual particles}, we develop novel stochastic approximations of population-limit SVGD dynamics in the space of probability measures, which is exactly implementable using only a finite number of particles. Our algorithms can be viewed as specific random-batch approximations of SVGD, which are computationally more efficient than ordinary SVGD. We establish that the $n$ particles output by VP-SVGD and RB-SVGD, run for $T$ steps, are i.i.d samples from a distribution whose Kernel Stein Discrepancy to the target is at most $O(T^{\nicefrac{-1}{6}})$ under standard assumptions. Our results hold under a mild growth condition on the potential function, which is significantly weaker than the isoperimetric assumptions (e.g. Poincare Inequality) or information-transport conditions (e.g. Talagrand's Inequality $\mathsf{T}_1$) generally considered in prior works. As a corollary, we consider the convergence of the empirical measure (of the particles output by VP-SVGD and RB-SVGD) to the target distribution and demonstrate a \emph{double exponential improvement} over the best known finite-particle analysis of SVGD.

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