[Paper Review] Nonparametric estimation for interacting particle systems : McKean-Vlasov models

Laetitia Della Maestra, Marc Hoffmann|arXiv (Cornell University)|Nov 7, 2020

Stochastic processes and financial applicationsEconomics, Econometrics and Finance68 references37 citations

TL;DR

The paper develops nonparametric kernel estimators for the solution of McKean-Vlasov nonlinear Fokker-Planck equations and for the drift, based on observing a trajectory of N interacting particles, with adaptive bandwidth selection and minimax optimality results.

ABSTRACT

We consider a system of $N$ interacting particles, governed by transport and diffusion, that converges in a mean-field limit to the solution of a McKean-Vlasov equation. From the observation of a trajectory of the system over a fixed time horizon, we investigate nonparametric estimation of the solution of the associated nonlinear Fokker-Planck equation, together with the drift term that controls the interactions, in a large population limit $N \ ightarrow \\infty$. We build data-driven kernel estimators and establish oracle inequalities, following Lepski's principle. Our results are based on a new Bernstein concentration inequality in McKean-Vlasov models for the empirical measure around its mean, possibly of independent interest. We obtain adaptive estimators over anisotropic H\\"older smoothness classes built upon the solution map of the Fokker-Planck equation, and prove their optimality in a minimax sense. In the specific case of the Vlasov model, we derive an estimator of the interaction potential and establish its consistency.

Motivation & Objective

• Motivate statistical inference for systems of N interacting particles converging to McKean-Vlasov equations.
• Develop data-driven kernel estimators for the evolving density mu_t(x) and the drift b(t,x,mu_t).
• Establish oracle inequalities and adaptivity results using Lepski’s principle.
• Provide a Bernstein concentration inequality in McKean-Vlasov models to support concentration and estimation results.
• Explore estimation of interaction in the Vlasov model and consistency results.

Proposed method

• Construct kernel estimators for mu_t(x) and for the product b(t,x,mu_t) mu_t(x) via smoothing of the empirical measure pi^N and the density mu_h^N.
• Use a quotient estimator for b(t,x,mu_t) as hat{b} = hat{pi} / (hat{mu} ∨ varpi).
• Apply Goldenshluger-Lepski method to select bandwidths adaptively for mu and b with oracle inequalities (Theorems 7 and 9).
• Develop minimax adaptive estimation results over anisotropic Hölder spaces built from the solution map of the nonlinear parabolic equation.
• Derive a Bernstein concentration inequality for empirical measures in McKean-Vlasov models to control deviations.
• Provide identification and consistent estimation results for the interaction F in the Vlasov case.
• ],
• research_questions':['How can one nonparametrically estimate the evolving density mu_t(x) from trajectory data of an interacting particle system?','How can one nonparametrically estimate the drift b(t,x,mu_t) governing particle interactions from data?','Can adaptive (minimax) estimation be achieved over anisotropic Hölder smoothness classes for mu_t and b in McKean-Vlasov models?','What concentration and deviation inequalities can be established for empirical measures in McKean-Vlasov settings to support estimation?','In the Vlasov case, can one consistently identify and estimate the interaction function F from observed data?'],
• key_findings':['The paper constructs data-driven kernel estimators for mu_t(x) and for b(t,x,mu_t) and proves oracle inequalities that bound estimation error by the best bias-variance tradeoff over bandwidths.','Oracle bounds show the estimators achieve near-optimal bias-variance tradeoffs across anisotropic Hölder classes.','A Bernstein concentration inequality is established for the empirical measure around its mean in McKean-Vlasov models, enabling sharp nonparametric results.','Adaptive minimax results are developed for mu_t and b over anisotropic time-space Hölder spaces, with optimality statements.','In the Vlasov setting, the interaction F can be consistently estimated using a Fourier-type estimator, enabling recovery of the interaction from data.','The analysis relies on a new differentiability framework (k-linear differentiability) of the drift in the measure argument to control empirical process deviations.'],
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Experimental results

Research questions

• RQ1 How can one nonparametrically estimate the evolving density mu_t(x) from trajectory data of an interacting particle system?
• RQ2 How can one nonparametrically estimate the drift b(t,x,mu_t) governing particle interactions from data?
• RQ3 Can adaptive (minimax) estimation be achieved over anisotropic Hölder smoothness classes for mu_t and b in McKean-Vlasov models?
• RQ4 What concentration and deviation inequalities can be established for empirical measures in McKean-Vlasov settings to support estimation?
• RQ5 In the Vlasov case, can one consistently identify and estimate the interaction function F from observed data?

Key findings

• The paper constructs data-driven kernel estimators for mu_t(x) and for b(t,x,mu_t) and proves oracle inequalities that bound estimation error by the best bias-variance tradeoff over bandwidths.
• Oracle bounds show the estimators achieve near-optimal bias-variance tradeoffs across anisotropic Hölder classes.
• A Bernstein concentration inequality is established for the empirical measure around its mean in McKean-Vlasov models, enabling sharp nonparametric results.
• Adaptive minimax results are developed for mu_t and b over anisotropic time-space Hölder spaces, with optimality statements.
• In the Vlasov setting, the interaction F can be consistently estimated using a Fourier-type estimator, enabling recovery of the interaction from data.
• The analysis relies on a new differentiability framework (k-linear differentiability) of the drift in the measure argument to control empirical process deviations.