Optimal minimax rate of learning nonlocal interaction kernels
Xiong Wang, Inbar Seroussi, Fei Lu
TL;DR
The paper establishes that learning the nonlocal, radial interaction kernel φ in a particle system achieves the classical minimax rate $M^{- rac{2β}{2β+1}}$ under a coercivity condition with finite $N$ and for $β≥¼$ (via Sobolev embedding). It introduces a novel tamed least squares estimator (tLSE) that attains this rate across broad exchangeable distributions, handling nonlocal dependencies without relying on covering arguments. The analysis hinges on sharp fourth-moment bounds for the normal vectors and exponential left-tail bounds for the smallest eigenvalue of the normal matrix, ensuring reliable model selection in the tLSE. A matching lower bound via Fano–Tsybakov demonstrates the rate is minimax-optimal. Collectively, the work provides a simple, robust framework to obtain optimal minimax rates for nonparametric regression problems with nonlocal dependencies, and highlights the role of the coercivity condition and Sobolev embeddings in controlling variance and nonlocal bias.
Abstract
Nonparametric estimation of nonlocal interaction kernels is crucial in various applications involving interacting particle systems. The inference challenge, situated at the nexus of statistical learning and inverse problems, arises from the nonlocal dependency. A central question is whether the optimal minimax rate of convergence for this problem aligns with the rate of $M^{-\frac{2β}{2β+1}}$ in classical nonparametric regression, where $M$ is the sample size and $β$ represents the regularity index of the radial kernel. Our study confirms this alignment for systems with a finite number of particles. We introduce a tamed least squares estimator (tLSE) that achieves the optimal convergence rate when $β\geq 1/4$ for a broad class of exchangeable distributions by leveraging random matrix theory and Sobolev embedding. The upper minimax rate relies on fourth-moment bounds for normal vectors and nonasymptotic bounds for the left tail probability of the smallest eigenvalue of the normal matrix. The lower minimax rate is derived using the Fano-Tsybakov hypothesis testing method. Our tLSE method offers a straightforward approach for establishing the optimal minimax rate for models with either local or nonlocal dependency.