Mean-field nonparametric estimation of interacting particle systems

Rentian Yao, Xiaohui Chen, Yun Yang
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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v178-yao22a, title = {Mean-field nonparametric estimation of interacting particle systems}, author = {Yao, Rentian and Chen, Xiaohui and Yang, Yun}, booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory}, pages = {2242--2275}, year = {2022}, editor = {Loh, Po-Ling and Raginsky, Maxim}, volume = {178}, series = {Proceedings of Machine Learning Research}, month = {02--05 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v178/yao22a/yao22a.pdf}, url = {https://proceedings.mlr.press/v178/yao22a.html}, 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.} }
Endnote
%0 Conference Paper %T Mean-field nonparametric estimation of interacting particle systems %A Rentian Yao %A Xiaohui Chen %A Yun Yang %B Proceedings of Thirty Fifth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2022 %E Po-Ling Loh %E Maxim Raginsky %F pmlr-v178-yao22a %I PMLR %P 2242--2275 %U https://proceedings.mlr.press/v178/yao22a.html %V 178 %X 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.
APA
Yao, R., Chen, X. & Yang, Y.. (2022). Mean-field nonparametric estimation of interacting particle systems. Proceedings of Thirty Fifth Conference on Learning Theory, in Proceedings of Machine Learning Research 178:2242-2275 Available from https://proceedings.mlr.press/v178/yao22a.html.

Related Material