Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence
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A novel linear integration rule called 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 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.
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