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Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence

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arxiv 2305.06151 v3 pith:7OCKGPWJ submitted 2023-05-10 math.NA cs.NAmath.STstat.COstat.TH

Rémi Leluc , François Portier , Johan Segers , Aigerim Zhuman This is my paper · ORCID
classification math.NAcs.NAmath.STstat.COstat.TH
keywords controlconvergenceratecarlointegrationmonteneighborsoptimal
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abstract

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.

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Cited by 4 Pith papers

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