## 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 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.

Original language | English (US) |
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Pages (from-to) | 2242-2275 |

Number of pages | 34 |

Journal | Proceedings of Machine Learning Research |

Volume | 178 |

State | Published - 2022 |

Event | 35th Conference on Learning Theory, COLT 2022 - London, United Kingdom Duration: Jul 2 2022 → Jul 5 2022 |

## Keywords

- Mckean-Vlasov equation
- interacting particle system
- learning interaction kernel
- maximum likelihood estimation
- mean-field regime

## ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability