Provably Fast Finite Particle Variants of SVGD via Virtual Particle Stochastic Approximation
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Stein Variational Gradient Descent (SVGD) is a popular variational inference algorithm which simulates an interacting particle system to approximately sample from a target distribution, with impressive empirical performance across various domains. Theoretically, its population (i.e, infinite-particle) limit dynamics is well studied but 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 (which is conceptually elegant) and GB-SVGD (which is empirically effective), with provably fast finite-particle convergence rates. We introduce the notion of virtual particles and develop novel stochastic approximations of population-limit SVGD dynamics in the space of probability measures, which are exactly implementable using 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 show that the n particles output by VP-SVGD and GB-SVGD, run for T steps with batch-size K, are at-least as good as i.i.d samples from a distribution whose Kernel Stein Discrepancy to the target is at most O(d^1/3(KT)^1/6) under standard assumptions. Our results also hold under a mild growth condition on the potential function, which is much weaker than the isoperimetric (e.g. Poincare Inequality) or information-transport conditions (e.g. Talagrand's Inequality 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 GB-SVGD) to the target distribution and demonstrate a double exponential improvement over the best known finite-particle analysis of SVGD.
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