Improved Analysis of Stein Variational Gradient Descent

Stein Variational Gradient Descent (SVGD) is a particle-based variational inference method that transports a collection of particles toward a target distribution through deterministic interacting updates. Existing convergence analyses of SVGD can broadly be divided into two regimes: Mean-Field SVGD, which studies the population (infinite-particle) limit of the algorithm, and Finite-Particle SVGD, which directly analyzes the practical algorithm with finitely many interacting particles. Despite substantial recent progress, existing theories in both regimes typically rely on restrictive assumptions that significantly limit their applicability. For instance, available convergence guarantees for Mean-Field SVGD may fail to cover even commonly used target distributions such as Gaussian mixtures, while existing finite-particle analyses can impose conditions that exclude even Gaussian target distributions. In this work, we develop an improved convergence analysis of SVGD that addresses these limitations in both the mean-field and finite-particle settings. Our results establish convergence under weaker assumptions than those required in prior work, thereby covering a broader class of practically relevant target distributions. Moreover, our analysis yields improved convergence guarantees over existing results. Together, these results provide a more general and sharper theoretical understanding of SVGD and help narrow the gap between its theoretical guarantees and its practical applicability.