# Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

arXiv: https://arxiv.org/abs/2605.09456 · HTML: https://mlsanity.com/paper/2605.09456/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 21 references that are in the map.
- First posted on arXiv 10 May 2026.

## Builds on (most cited first, 14 of 14)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [A Non-Asymptotic Analysis for Stein Variational Gradient Descent](https://mlsanity.com/paper/2006.09797/) (2020)
- [Scaling limit of the Stein variational gradient descent: the mean field regime](https://mlsanity.com/paper/1805.04035/) (2018)
- [A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1](https://mlsanity.com/paper/2106.03076/) (2021)
- [Stochastic Gradient MCMC with Repulsive Forces](https://mlsanity.com/paper/1812.00071/) (2018)
- [Stein Variational Gradient Descent: many-particle and long-time asymptotics](https://mlsanity.com/paper/2102.12956/) (2021)
- [Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent](https://mlsanity.com/paper/2409.08469/) (2024)
- [Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies](https://mlsanity.com/paper/2603.01977/) (2026)
- [Finite-Particle Rates for Regularized Stein Variational Gradient Descent](https://mlsanity.com/paper/2602.05172/) (2026)
- [A Note on the Convergence of Mirrored Stein Variational Gradient Descent under (L0,L1)-Smoothness Condition](https://mlsanity.com/paper/2206.09709/) (2022)
- [Long-time asymptotics of noisy SVGD outside the population limit](https://mlsanity.com/paper/2406.11929/) (2024)

## Built on by (most cited first, 3 of 3)

- [Uniform-in-time Propagation-of-Chaos for Stein Variational Gradient Descent](https://mlsanity.com/paper/2607.00149/) (2026)
- [Riesz-Kernel Stein Variational Gradient Descent: Renormalized Entropy and Long-Time Particle Limits](https://mlsanity.com/paper/2607.14527/) (2026)
- [Quantitative Target Convergence and Uniform-in-Time Propagation of Chaos for Langevin-Regularized SVGD](https://mlsanity.com/paper/2608.28827/) (2026)

## Abstract (arXiv)

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the d-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in L2-norm, we obtain explicit polynomial convergence rates in L2-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies.