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：提出一种新型线性积分规则，称为“控制近邻法”。该方法利用最近邻估计量作为控制变量，以加速蒙特卡洛过程的收敛速度。核心成果是证明了该估计量对于Lipschitz函数的收敛速率为$\mathcal{O}(n^{-1/2} n^{-1/d})$——其中$n$表示被积函数的评估次数，$d$表示定义域的维度——这一速率在某种意义上已达最优。多项数值实验验证了该复杂度边界，并凸显了所提估计量的优异性能。