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# Mean-field nonparametric estimation of interacting particle systems

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:2242-2275, 2022.

#### Abstract

This paper concerns the nonparametric estimation problem of the distribution-state dependent drift vector field in an interacting $N$-particle system. Observing single-trajectory data for each particle, we derive the mean-field rate of convergence for the maximum likelihood estimator (MLE), which depends on both Gaussian complexity and Rademacher complexity of the function class. In particular, when the function class contains $\alpha$-smooth H{ö}lder functions, our rate of convergence is minimax optimal on the order of $N^{-\frac{\alpha}{d+2\alpha}}$. Combining with a Fourier analytical deconvolution estimator, we derive the consistency of MLE for the external force and interaction kernel in the McKean-Vlasov equation.