# Scaling limit of the Stein variational gradient descent: the mean field regime

arXiv: https://arxiv.org/abs/1805.04035 · HTML: https://mlsanity.com/paper/1805.04035/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2018.

## Built on by (most cited first, 30 of 35)

- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
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- [A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1](https://mlsanity.com/paper/2106.03076/) (2021)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Regularized Stein Variational Gradient Flow](https://mlsanity.com/paper/2211.07861/) (2022)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition](https://mlsanity.com/paper/2206.00508/) (2022)
- [Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance](https://mlsanity.com/paper/2302.11024/) (2023)
- [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)
- [Towards Understanding the Dynamics of Gaussian-Stein Variational Gradient Descent](https://mlsanity.com/paper/2305.14076/) (2023)
- [Grassmann Stein Variational Gradient Descent](https://mlsanity.com/paper/2202.03297/) (2022)
- [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024)
- [Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes](https://mlsanity.com/paper/2306.15630/) (2023)
- [Ensemble Inference Methods for Models With Noisy and Expensive Likelihoods](https://mlsanity.com/paper/2104.03384/) (2021)
- [Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce data](https://mlsanity.com/paper/2210.17230/) (2022)
- [A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling](https://mlsanity.com/paper/2202.12927/) (2022)
- [Provably Fast Finite Particle Variants of SVGD via Virtual Particle Stochastic Approximation](https://mlsanity.com/paper/2305.17558/) (2023)
- [Relative Entropy Gradient Sampler for Unnormalized Distributions](https://mlsanity.com/paper/2110.02787/) (2021)
- [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)
- [Homotopy Relaxation Training Algorithms for Infinite-Width Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/2309.15244/) (2023)
- [Ensemble-Based Annealed Importance Sampling](https://mlsanity.com/paper/2401.15645/) (2024)
- [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)
- [Low-rank variational Bayes correction to the Laplace method](https://mlsanity.com/paper/2111.12945/) (2021)
- [Stein Boltzmann Sampling: A Variational Approach for Global Optimization](https://mlsanity.com/paper/2402.04689/) (2024)
- [ELBOing Stein: Variational Bayes with Stein Mixture Inference](https://mlsanity.com/paper/2410.22948/) (2024)

## Abstract (arXiv)

We study an interacting particle system in Rd motivated by Stein variational gradient descent [Q. Liu and D. Wang, NIPS 2016], a deterministic algorithm for sampling from a given probability density with unknown normalization. We prove that in the large particle limit the empirical measure of the particle system converges to a solution of a non-local and nonlinear PDE. We also prove global existence, uniqueness and regularity of the solution to the limiting PDE. Finally, we prove that the solution to the PDE converges to the unique invariant solution in long time limit.