Local antithetic sampling with scrambled nets

TL;DR

This paper introduces a novel antithetic sampling method combined with randomized quasi-Monte Carlo to improve the convergence rate of integral approximation, achieving better error bounds for smooth functions.

Contribution

It extends classical antithetic sampling to RQMC, resulting in a modest but notable improvement in the RMSE convergence rate for smooth functions.

Findings

Achieves RMSE rate of O(n^{-3/2-1/d+ε}) for smooth functions.

Combines antithetic sampling with RQMC for variance reduction.

Provides theoretical analysis of improved convergence rates.

Abstract

We consider the problem of computing an approximation to the integral $I=\int_{[0,1]^d}f(x) dx$. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of $O(n^{-1/2})$ from $n$ independent random function evaluations. By contrast, quasi-Monte Carlo (QMC) sampling using carefully equispaced evaluation points can attain the rate $O(n^{-1+\varepsilon})$ for any $\varepsilon>0$ and randomized QMC (RQMC) can attain the RMSE $O(n^{-3/2+\varepsilon})$, both under mild conditions on $f$. Classical variance reduction methods for MC can be adapted to QMC. Published results combining QMC with importance sampling and with control variates have found worthwhile improvements, but no change in the error rate. This paper extends the classical variance reduction method of antithetic sampling and combines it with RQMC. One such method is shown to bring a modest improvement in the RMSE rate, attaining $O(n^{-3/2-1/d+\varepsilon})$ for any $\varepsilon>0$, for smooth enough $f$.