# Kernel-based potential mean-field games with unbiased random Fourier U-statistics

arXiv: https://arxiv.org/abs/2605.29371 · HTML: https://mlsanity.com/paper/2605.29371/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 15 of 15)

- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [On the Error of Random Fourier Features](https://mlsanity.com/paper/1506.02785/) (2015)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [Numerical Methods for Mean Field Games and Mean Field Type Control](https://mlsanity.com/paper/2106.06231/) (2021)
- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)
- [FastMMD: Ensemble of Circular Discrepancy for Efficient Two-Sample Test](https://mlsanity.com/paper/1405.2664/) (2014)
- [Computational-Statistical Trade-off in Kernel Two-Sample Testing with Random Fourier Features](https://mlsanity.com/paper/2407.08976/) (2024)
- [A kernel-based method for Schr"odinger bridges](https://mlsanity.com/paper/2310.14522/) (2023)
- [A deep learning approach to multi-marginal optimal transport via Hilbert space embeddings of probability measures](https://mlsanity.com/paper/2507.09206/) (2025)
- [First order Mean Field Games with density constraints: Pressure equals Price](https://mlsanity.com/paper/1507.02019/) (2015)
- [Solving Monge problem by Hilbert space embeddings of probability measures](https://mlsanity.com/paper/2412.03478/) (2024)

## Built on by (most cited first, 1 of 1)

- [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026)

## Abstract (arXiv)

We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For population near-minimizers we prove convergence to the terminal-constrained problem as the penalty diverges, and show that the same limit is recovered almost surely when the learned controls are evaluated on independent finite samples under explicit coupling conditions on the penalty, random-feature count and sample size. The framework includes the kernel-MMD-penalty Schr"odinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schr"odinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.