Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence

• DOI: 10.48550/arxiv.2305.06151
• Publication Date: 2023-01-01
• Authors: Rémi Leluc; François Portier; Johan Segers; Aigerim Zhuman
• Keywords: Control variates; Quasi-Monte Carlo method

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 Monte Carlo procedure on metric spaces. The main result $\mathcal{O}(n^{-1/2} n^{-s/d})$ -- where $n$ stands for number evaluations integrand and $d$ dimension domain this estimate H\"older functions with regularity $s \in (0,1]$, a which, some sense, optimal. Several numerical experiments validate complexity bound highlight good performance estimator.