A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1
This work addresses the limited theoretical understanding of SVGD, providing foundational convergence guarantees for researchers in machine learning and statistics, though it is incremental as it builds on existing SVGD theory.
The paper tackles the theoretical convergence of Stein Variational Gradient Descent (SVGD) for sampling from non-logconcave target distributions under Talagrand's inequality T1, establishing convergence and a dimension-dependent complexity bound in terms of Kernelized Stein Discrepancy without assuming bounded KSD along the trajectory.
Stein Variational Gradient Descent (SVGD) is an algorithm for sampling from a target density which is known up to a multiplicative constant. Although SVGD is a popular algorithm in practice, its theoretical study is limited to a few recent works. We study the convergence of SVGD in the population limit, (i.e., with an infinite number of particles) to sample from a non-logconcave target distribution satisfying Talagrand's inequality T1. We first establish the convergence of the algorithm. Then, we establish a dimension-dependent complexity bound in terms of the Kernelized Stein Discrepancy (KSD). Unlike existing works, we do not assume that the KSD is bounded along the trajectory of the algorithm. Our approach relies on interpreting SVGD as a gradient descent over a space of probability measures.