# Mean-field underdamped Langevin dynamics and its spacetime discretization

arXiv: https://arxiv.org/abs/2312.16360 · HTML: https://mlsanity.com/paper/2312.16360/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 59 references that are in the map.
- First posted on arXiv 26 Dec 2023.

## Builds on (most cited first, 27 of 27)

- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (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)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Mean Field Analysis of Neural Networks: A Central Limit Theorem](https://mlsanity.com/paper/1808.09372/) (2018)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Stochastic Particle Gradient Descent for Infinite Ensembles](https://mlsanity.com/paper/1712.05438/) (2017)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond](https://mlsanity.com/paper/1906.11985/) (2019)
- [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019)
- [A Lyapunov Analysis of Momentum Methods in Optimization](https://mlsanity.com/paper/1611.02635/) (2016)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [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)
- [Nesterov's method with decreasing learning rate leads to accelerated stochastic gradient descent](https://mlsanity.com/paper/1908.07861/) (2019)
- [Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case](https://mlsanity.com/paper/2301.08039/) (2023)
- [Ergodicity of the underdamped mean-field Langevin dynamics](https://mlsanity.com/paper/2007.14660/) (2020)
- [Accelerated Stochastic Optimization Methods under Quasar-convexity](https://mlsanity.com/paper/2305.04736/) (2023)
- [Mean Field Optimization Problem Regularized by Fisher Information](https://mlsanity.com/paper/2302.05938/) (2023)

## Built on by (most cited first, 2 of 2)

- [Sampling from the Mean-Field Stationary Distribution](https://mlsanity.com/paper/2402.07355/) (2024)
- [Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity](https://mlsanity.com/paper/2501.14993/) (2025)

## Abstract (arXiv)

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.