Long-time asymptotics of noisy SVGD outside the population limit
This addresses a theoretical gap in sampling algorithms for machine learning practitioners, though it appears incremental as it builds on existing SVGD variants.
The paper tackles the problem of understanding the long-time asymptotic behavior of noisy Stein Variational Gradient Descent (SVGD) in the finite-particle regime, showing that it avoids variance collapse and approaches the target distribution as the number of particles increases.
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 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 ) 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 is well-defined. We then characterize this limit set, showing that it approaches the target distribution as increases. In particular, noisy SVGD provably 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.