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Nonparametric estimation of nonlocal interaction kernels is crucial invarious applications involving interacting particle systems. The inferencechallenge, situated at the nexus of statistical learning and inverse problems,comes from the nonlocal dependency. A central question is whether the optimalminimax rate of convergence for this problem aligns with the rate of$M^{-\frac{2\beta}{2\beta+1}}$ in classical nonparametric regression, where $M$is the sample size and $\beta$ represents the smoothness exponent of the radialkernel. Our study confirms this alignment for systems with a finite number ofparticles.  We introduce a tamed least squares estimator (tLSE) that attains the optimalconvergence rate for a broad class of exchangeable distributions. The tLSEbridges the smallest eigenvalue of random matrices and Sobolev embedding. Thisestimator relies on nonasymptotic estimates for the left tail probability ofthe smallest eigenvalue of the normal matrix. The lower minimax rate is derivedusing the Fano-Tsybakov hypothesis testing method. Our findings reveal thatprovided the inverse problem in the large sample limit satisfies a coercivitycondition, the left tail probability does not alter the bias-variance tradeoff,and the optimal minimax rate remains intact. Our tLSE method offers astraightforward approach for establishing the optimal minimax rate for modelswith either local or nonlocal dependency.

Optimal minimax rate of learning interaction kernels