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

arXiv: https://arxiv.org/abs/2608.28827 · HTML: https://mlsanity.com/paper/2608.28827/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 44 references that are in the map.
- First posted on arXiv 28 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Annealed Stein Variational Gradient Descent](https://mlsanity.com/paper/2101.09815/) (2021)
- [Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent](https://mlsanity.com/paper/2409.08469/) (2024)
- [Finite-Particle Rates for Regularized Stein Variational Gradient Descent](https://mlsanity.com/paper/2602.05172/) (2026)
- [Long-time asymptotics of noisy SVGD outside the population limit](https://mlsanity.com/paper/2406.11929/) (2024)
- [Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow](https://mlsanity.com/paper/2605.09456/) (2026)
- [Uniform-in-Time Weak Propagation-of-Chaos in Shallow Neural Networks](https://mlsanity.com/paper/2605.22010/) (2026)
- [Uniform-in-time Propagation-of-Chaos for Stein Variational Gradient Descent](https://mlsanity.com/paper/2607.00149/) (2026)
- [Sharp propagation of chaos for mean field Langevin dynamics, control, and games](https://mlsanity.com/paper/2603.10988/) (2026)
- [Riesz-Kernel Stein Variational Gradient Descent: Renormalized Entropy and Long-Time Particle Limits](https://mlsanity.com/paper/2607.14527/) (2026)

## Abstract (arXiv)

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 T2 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 W22, and for fixed-marginal total variation and W22. 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.