# Sampling from the Mean-Field Stationary Distribution

arXiv: https://arxiv.org/abs/2402.07355 · HTML: https://mlsanity.com/paper/2402.07355/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 83 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 30 of 30)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [On Linear Optimization over Wasserstein Balls](https://mlsanity.com/paper/2004.07162/) (2020)
- [On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method](https://mlsanity.com/paper/2011.03176/) (2020)
- [Uniform-in-time propagation of chaos for mean field Langevin dynamics](https://mlsanity.com/paper/2212.03050/) (2022)
- [Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction](https://mlsanity.com/paper/2306.07221/) (2023)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Mean-field Variational Inference via Wasserstein Gradient Flow](https://mlsanity.com/paper/2207.08074/) (2022)
- [Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets](https://mlsanity.com/paper/1906.08530/) (2019)
- [Independent projections of diffusions: Gradient flows for variational inference and optimal mean field approximations](https://mlsanity.com/paper/2309.13332/) (2023)
- [Mean Field Optimization Problem Regularized by Fisher Information](https://mlsanity.com/paper/2302.05938/) (2023)
- [Mean-field underdamped Langevin dynamics and its spacetime discretization](https://mlsanity.com/paper/2312.16360/) (2023)

## Built on by (most cited first, 7 of 7)

- [Learning Multi-Index Models with Neural Networks via Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2408.07254/) (2024)
- [Improved Particle Approximation Error for Mean Field Neural Networks](https://mlsanity.com/paper/2405.15767/) (2024)
- [Efficient Reinforcement Learning for Global Decision Making in the Presence of Local Agents at Scale](https://mlsanity.com/paper/2403.00222/) (2024)
- [A Particle Algorithm for Mean-Field Variational Inference](https://mlsanity.com/paper/2412.20385/) (2024)
- [Mirror Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2505.02621/) (2025)
- [Uniform-in-Time Weak Propagation-of-Chaos in Shallow Neural Networks](https://mlsanity.com/paper/2605.22010/) (2026)
- [Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity](https://mlsanity.com/paper/2501.14993/) (2025)

## Abstract (arXiv)

We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term. Our main insight is to decouple the two key aspects of this problem: (1) approximation of the mean-field SDE via a finite-particle system, via uniform-in-time propagation of chaos, and (2) sampling from the finite-particle stationary distribution, via standard log-concave samplers. Our approach is conceptually simpler and its flexibility allows for incorporating the state-of-the-art for both algorithms and theory. This leads to improved guarantees in numerous settings, including better guarantees for optimizing certain two-layer neural networks in the mean-field regime. A key technical contribution is to establish a new uniform-in-N log-Sobolev inequality for the stationary distribution of the mean-field Langevin dynamics.