Mean-Field Nonparametric Estimation of Interacting Particle Systems

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 d-variate α-Hölder smooth functions, our rate of convergence α is minimax optimal on the order of N^{− d}+2^α . Combining with a Fourier analytical deconvolution argument, we derive the consistency of MLE for the external force and interaction kernel in the McKean-Vlasov equation.