Riesz-Kernel Stein Variational Gradient Descent: Renormalized Entropy and Long-Time Particle Limits
arXiv:2607.14527
The paper studies Stein variational gradient descent with singular periodic Riesz kernels, removing self-interaction, and proves that the particle system’s empirical distribution converges to the target distribution as the number of particles and averaging time grow, providing explicit error bounds under entropy conditions.
Abstract
Stein variational gradient descent (SVGD) transports interacting particles toward a target distribution through deterministic kernelized dynamics. Singular Riesz kernels are attractive because they can provide quantitative population-level convergence, but at the finite-particle level the corresponding Stein energy has infinite self-interaction. We study periodic Riesz SVGD with self-interaction removed and prove a many-particle, long-time sampling theorem. Throughout the range in which the singular Stein energy is locally integrable, under a uniform bound on the initial relative entropy per particle, the time-averaged empirical-measure law converges weakly to the point mass \(δ_Ï\) at the target as the particle number and any diverging averaging horizon tend to infinity. We also show that the empirical-measure laws induced by invariant particle laws of finite relative entropy converge weakly to \(δ_Ï\), without a uniform entropy bound. Below the logarithmic singularity threshold, we obtain an explicit algebraic finite-particle error bound. These results extend the joint-entropy approach for smooth-kernel SVGD to singular interactions.
31 pages. No figures. Many-particle and long-time convergence for Stein variational gradient descent with singular periodic Riesz kernels