Accelerated convergence of error quantiles using robust randomized quasi Monte Carlo methods

We aim to calculate an expectation μ(F)=E(F(U)) for functions F:[0,1]^d↦R using a family of estimators μˆ_B with a budget of B evaluation points. The standard Monte Carlo method achieves a root mean squared risk of order 1/B, both for a fixed square integrable function F and for the worst-case risk over the class F of functions with ‖F‖_L2≤1. Using a sequence of Randomized Quasi Monte Carlo (RQMC) methods, in contrast, we achieve faster convergence σ_B≪1/B for the risk σ_B=σ_B(F) when fixing a function F, compared to the worst-case risk which is still of order 1/B. We address the convergence of quantiles of the absolute error, namely, for a given confidence level 1−δ this is the minimal ε such that P(|μˆ_B(F)−μ(F)|>ε)≤δ holds. We show that a judicious choice of a robust aggregation method coupled with RQMC methods allows reaching improved convergence rates for ε depending on δ and B when fixing a function F. This study includes a review on concentration bounds for the empirical mean as well as sub-Gaussian mean estimates and is supported by numerical experiments, ranging from bounded F to heavy-tailed F(U), the latter being well suited to functions F with a singularity. The different methods we have tested are available in a Julia package.