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

Sayan Banerjee; Dohyeon Kim

stat.MLAug 28, 2026

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

Authors: Sayan Banerjee, Dohyeon Kim

Organizations: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, 318 Hanes Hall, CB# 3260, Chapel Hill, NC 27599-3260, USA. · Department of Computing and Mathematical Sciences, California Institute of Technology, 1200 East California Boulevard, MC 305-16, Pasadena, CA 91125, USA.

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.

Figures & tables

Work	Dynamics	Principal conditions	Quantitative conclusion
SPOS [ 43 ]	Closely related noisy particle-optimization/SVGD system	Strong monotonicity and a positive contraction margin	Continuous-time one-particle uniform-in-time $W_{1}=O(N^{-1/2})$ , together with discrete-time approximation bounds
Priser–Bianchi–Salim [ 37 ]	Decreasing-step noisy-SVGD recursion associated with the continuous Langevin-regularized dynamics ( 1.2 )	Qualitative well-posedness and long-time assumptions	Characterization of the empirical-measure limit set and its large- $N$ target consistency; no explicit rate in $N$
Deterministic cutoff [ 3 ]	Deterministic SVGD	Target–kernel-specific finite- and long-time estimates	Uniform-in-averaging-time rates in general discrepancies; uniform-in-physical-time parametric rates for special finite-rank target–kernel pairs
Hong–Liu–Yang [ 22 ]	One-dimensional stochastic-SVGD-type dynamics with superlinear polynomial kernels	Pseudo/local monotonicity and sufficiently high polynomial moments	Quantitative strong tagged-particle propagation of chaos on every fixed $[0,T]$ ; no target-convergence or infinite-horizon result
Current paper	Langevin-regularized SVGD ( 1.2 )	Target LSI, smooth bounded kernels, and concentration assumptions for strong finite-time bounds; no small-interaction condition	Polynomial uniform-in-physical-time rates for expected empirical KSD and $W_{2}^{2}$ , and for fixed-marginal TV and $W_{2}^{2}$ ; finite-time fixed-marginal KL control

Table 1: Comparison with closely related work and the main-text results of the present paper.

Discrepancy and kernel condition	Finite-time estimate	Uniform-in-time conclusion
Empirical Langevin KSD KSD kernel condition	$\displaystyle\mathbb{E}\mathrm{KSD}_{\pi}(\mu_{t}^{N},\mu_{t})$ $\displaystyle\lesssim N^{-1/2}e^{\left(\kappa\wedge(L_{\mathrm{cpl}}/2)\right)t}$	$\displaystyle\sup_{t\geq 0}\mathbb{E}\mathrm{KSD}_{\pi}(\mu_{t}^{N},\mu_{t})$ $\displaystyle\lesssim N^{-\varepsilon\alpha/[2(\kappa\wedge(L_{\mathrm{cpl}}/2)+\varepsilon\alpha)]}+N^{-1/2}$
Empirical $W_{2}^{2}$ Wasserstein kernel condition	$\displaystyle\mathbb{E}W_{2}^{2}(\mu_{t}^{N},\mu_{t})$ $\displaystyle\lesssim N^{-1}(e^{L_{\mathrm{cpl}}t}-1)+r_{N,d}$	$\displaystyle\sup_{t\geq 0}\mathbb{E}W_{2}^{2}(\mu_{t}^{N},\mu_{t})$ $\displaystyle\lesssim N^{-2\varepsilon\alpha/(L_{\mathrm{cpl}}+2\varepsilon\alpha)}+r_{N,d}$
Fixed- $k$ $W_{2}^{2}$ Wasserstein kernel condition	$\displaystyle W_{2}^{2}(P_{N,t}^{(k)},\mu_{t}^{\otimes k})$ $\displaystyle\lesssim\frac{k}{N}(e^{L_{\mathrm{cpl}}t}-1)$	$\displaystyle\sup_{t\geq 0}W_{2}^{2}(P_{N,t}^{(k)},\mu_{t}^{\otimes k})$ $\displaystyle\leq C_{k}\left\{N^{-2\varepsilon\alpha/(L_{\mathrm{cpl}}+2\varepsilon\alpha)}+N^{-1}\right\}$
Fixed- $k$ total variation Either kernel condition	$\displaystyle\\|P_{N,t}^{(k)}-\mu_{t}^{\otimes k}\\|_{\mathrm{TV}}$ $\displaystyle\lesssim\sqrt{\frac{k}{N}}\,(e^{\kappa t}-1)^{1/2}$	$\displaystyle\sup_{t\geq 0}\\|P_{N,t}^{(k)}-\mu_{t}^{\otimes k}\\|_{\mathrm{TV}}$ $\displaystyle\leq C_{k}\left\{N^{-\varepsilon\alpha/(\kappa+2\varepsilon\alpha)}+N^{-1/2}\right\}$
Entropy-derived $W_{2}^{2}$ empirical and fixed- $k$ Wasserstein kernel condition and $\mu_{0}\in T_{2}(C_{0})$	$\displaystyle\mathbb{E}W_{2}^{2}(\mu_{t}^{N},\mu_{t})$ $\displaystyle\lesssim N^{-1}e^{L_{\mathrm{ent}}t}+r_{N,d}$ $\displaystyle W_{2}^{2}(P_{N,t}^{(k)},\mu_{t}^{\otimes k})\lesssim\frac{k}{N}e^{L_{\mathrm{ent}}t}$	$\displaystyle\sup_{t\geq 0}\mathbb{E}W_{2}^{2}(\mu_{t}^{N},\mu_{t})$ $\displaystyle\lesssim N^{-2\varepsilon\alpha/(L_{\mathrm{ent}}+2\varepsilon\alpha)}+r_{N,d}$ $\displaystyle\sup_{t\geq 0}W_{2}^{2}(P_{N,t}^{(k)},\mu_{t}^{\otimes k})$ $\displaystyle\leq C_{k}\left\{N^{-2\varepsilon\alpha/(L_{\mathrm{ent}}+2\varepsilon\alpha)}+N^{-1}\right\}$

Table 2: Main metric propagation of chaos bounds.

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Riesz-Kernel Stein Variational Gradient Descent: Renormalized Entropy and Long-Time Particle Limits

Trevor Teolis, Maarten V. de Hoop

Rice University

Kernel Methods Stein Variational Gradient Descent

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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_π

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_π

, 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.

Abstract

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