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.

翻译：非局部交互核的非参数估计在涉及交互粒子系统的各类应用中至关重要。这一推理性挑战处于统计学习与反问题的交叉点，其复杂性源于非局部依赖性。一个核心问题是：该问题的最优极小化极大收敛速率是否与经典非参数回归中$M^{-\frac{2\beta}{2\beta+1}}$的速率一致，其中$M$为样本量，$\beta$表示径向核的光滑性指数。我们的研究证实，对于有限粒子数系统，该一致性成立。我们引入了一种驯化最小二乘估计量（tLSE），它能够在一类广泛的可交换分布中达到最优收敛速率。tLSE方法桥接了随机矩阵的最小特征值与Sobolev嵌入。该估计量依赖于法线矩阵最小特征值左尾概率的非渐近估计。下界极小化极大速率通过Fano-Tsybakov假设检验方法推导得出。研究结果表明，只要大样本极限下的反问题满足强制条件，左尾概率不会改变偏差-方差权衡，且最优极小化极大速率保持不变。我们的tLSE方法为建立具有局部或非局部依赖性的模型的最优极小化极大速率提供了一种简洁途径。