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 on metric spaces. The main result is the $\mathcal{O}(n^{-1/2} n^{-s/d})$ 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 H\"older functions with regularity $s \in (0,1]$, 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$\textit{控制近邻}$（control neighbors），该方法利用最近邻估计量作为控制变量，在度量空间上加速蒙特卡洛过程的收敛速率。主要结果是对正则性$s \in (0,1]$的Hölder函数，该估计量达到$\mathcal{O}(n^{-1/2} n^{-s/d})$的收敛速率——其中$n$表示被积函数评估次数，$d$表示定义域维度。该速率在某种意义上是最优的。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良性能。