  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,
comes from the nonlocal dependency. A central question is whether the optimal
minimax rate of convergence for this problem aligns with the rate of
$M^{-\frac{2\beta}{2\beta+1}}$ in classical nonparametric regression, where $M$
is the sample size and $\beta$ represents the smoothness exponent 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 attains the optimal
convergence rate for a broad class of exchangeable distributions. The tLSE
bridges the smallest eigenvalue of random matrices and Sobolev embedding. This
estimator relies on nonasymptotic estimates 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 findings reveal that
provided the inverse problem in the large sample limit satisfies a coercivity
condition, the left tail probability does not alter the bias-variance tradeoff,
and the optimal minimax rate remains intact. Our tLSE method offers a
straightforward approach for establishing the optimal minimax rate for models
with either local or nonlocal dependency.
