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

• URL: http://arxiv.org/abs/2605.29371v1
• Date: Thu, 28 May 2026 05:15:25 GMT
• Title: Kernel-based potential mean-field games with unbiased random Fourier $U$-statistics
• Abstract summary: 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 penalties.<n>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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 this subclass we prove a sample-level almost-sure convergence theorem and an explicit almost-sure rate of convergence, under coupled rate conditions on the penalty parameter, the random-feature count, the sample size, and the optimization tolerance. The framework includes the kernel-MMD-penalty Schrödinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schrödinger 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.

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