Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence

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.