A novel linear integration rule called $\textit{control neighbors}$ is proposed in which nearest neighbor estimates act as control variates to speed up the convergence rate of the Monte Carlo procedure. The main result is the $\mathcal{O}(n^{-1/2} n^{-1/d})$ convergence rate -- where $n$ stands for the number of evaluations of the integrand and $d$ for the dimension of the domain -- of this estimate for Lipschitz functions, a rate which, in some sense, is optimal. Several numerical experiments validate the complexity bound and highlight the good performance of the proposed estimator.

翻译：提出了一种新型线性积分规则——$\textit{控制近邻法}$，该方法利用最近邻估计作为控制变量，以加速蒙特卡洛过程的收敛速度。主要成果是对于Lipschitz函数，该估计量达到$\mathcal{O}(n^{-1/2} n^{-1/d})$收敛速率——其中$n$表示被积函数评估次数，$d$表示域维度——该速率在某种意义上具有最优性。多项数值实验验证了这一复杂度界，并凸显了所提估计量的优异性能。