Quantitative Target Convergence and Uniform-in-Time Propagation of Chaos for Langevin-Regularized SVGD

• URL: http://arxiv.org/abs/2608.28827v1
• Date: Fri, 28 Aug 2026 19:56:49 GMT
• Title: Quantitative Target Convergence and Uniform-in-Time Propagation of Chaos for Langevin-Regularized SVGD
• Abstract summary: We show convergence to the target and uniform-in-time propagation of chaos for Langevin-regularized Stein descent.<n>For propagation of chaos, we develop two complementary finite-time approaches.
• Score: 2.457924087844968
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We establish quantitative convergence to the target and uniform-in-time propagation of chaos for Langevin-regularized Stein variational gradient descent. The Stein interaction need not be small relative to the confining Langevin drift and does not generally yield a contractive particle coupling. At the mean-field level, the Stein and Langevin components dissipate the same relative entropy in the kernel-induced Stein and $2$-Wasserstein geometries, producing the squared kernel Stein discrepancy and relative Fisher information. Under a log-Sobolev inequality for the target, this yields exponential last-iterate convergence. We also derive a finite-particle entropy identity relative to the product target, giving exponential-in-time convergence of the empirical measure up to polynomial sampling errors. For propagation of chaos, we develop two complementary finite-time approaches. A synchronous coupling, combined with exponential moment estimates for the nonlinear mean-field diffusion, yields explicit single-exponential bounds in Wasserstein distance and kernel Stein discrepancy (KSD). Moving-product entropy gives joint-law relative entropy control relative to the evolving mean-field product law and, through entropy superadditivity and concentration, fixed-marginal relative entropy and total variation bounds and empirical KSD estimates. Under an additional $T_2$ inequality for the initial law, it also yields Wasserstein bounds. Combining these finite-time estimates with target convergence at a logarithmic cutoff time gives polynomial uniform-in-time propagation of chaos rates in expectation for empirical KSD and $W_2^2$, and for fixed-marginal total variation and $W_2^2$. All bounds control the last iterate in physical time. We also compare the two finite-time mechanisms and identify regimes in which each gives the sharper polynomial exponent.

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