Kernel-based potential mean-field games with unbiased random Fourier U-statistics
Abstract
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{ö}dinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schr{ö}dinger 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.
Related
- CAFE+FNO: Fourier Kernel Generation via Multiplicative Feature Composition
- Limit theory for Lipschitz-localized statistics in random geometric models
- Mesoscopic eigenvalue statistics for correlated random matrices
- Loop Equations Characterize Random Matrix Statistics
- Heat Kernel Estimates for Random Walks with Killing on Transient Graphs
BibTeX
@misc{nakano2026kernelbased,
title = {{Kernel-based potential mean-field games with unbiased random Fourier $U$-statistics}},
author = {Yumiharu Nakano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.29371},
eprint = {2605.29371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.29371},
}
RIS
TY - UNPB
TI - Kernel-based potential mean-field games with unbiased random Fourier U-statistics
AU - Yumiharu Nakano
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2605.29371
UR - https://arxiv.org/abs/2605.29371
AB - 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{ö}dinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schr{ö}dinger 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.
ER -