Long-time asymptotics of noisy SVGD outside the population limit

Victor Priser, PASCAL BIANCHI, Adil Salim

Keywords: Stochastic approximation, sampling, convergence, interacting particles system, dynamical systems, Stein Variational Gradient Descent, McKean-Vlasov equation

TL;DR: We study the long-time convergence of a noisy Stein Variational Gradient Descent (SVGD) algorithm in the finite particle regime.

Abstract: Stein Variational Gradient Descent (SVGD) is a widely used sampling algorithm that has been successfully applied in several areas of Machine Learning. SVGD operates by iteratively moving a set of $n$ interacting particles (which represent the samples) to approximate the target distribution. Despite recent studies on the complexity of SVGD and its variants, their long-time asymptotic behavior (i.e., after numerous iterations $k$) is still not understood in the finite number of particles regime. We study the long-time asymptotic behavior of a noisy variant of SVGD. First, we establish that the limit set of noisy SVGD for large $k$ is well-defined. We then characterize this limit set, showing that it approaches the target distribution as $n$ increases. In particular, noisy SVGD avoids the variance collapse observed for SVGD. Our approach involves demonstrating that the trajectories of noisy SVGD closely resemble those described by a McKean-Vlasov process.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

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Submission Number: 6643