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

Abstract

We aim to calculate an expectation $\mu \left(F\right)=E\left(F\right(U\left)\right)$ for functions $F:{[0,1]}^d\mapsto R$ using a family of estimators ${\hat{\mu }}_B$ with a budget of B evaluation points. The standard Monte Carlo method achieves a root mean squared risk of order $1/\sqrt{B}$, both for a fixed square integrable function F and for the worst-case risk over the class $F$ of functions with ${\Vert F\Vert }_{L_2}\le 1$. Using a sequence of Randomized Quasi Monte Carlo (RQMC) methods, in contrast, we achieve faster convergence ${\sigma }_B\ll 1/\sqrt{B}$ for the risk ${\sigma }_B={\sigma }_B\left(F\right)$ when fixing a function F, compared to the worst-case risk which is still of order $1/\sqrt{B}$. We address the convergence of quantiles of the absolute error, namely, for a given confidence level $1-\delta$ this is the minimal ε such that $P\left(\right|{\hat{\mu }}_B\left(F\right)-\mu \left(F\right)|>\epsilon )\le \delta$ 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 $F\left(U\right)$, the latter being well suited to functions F with a singularity. The different methods we have tested are available in a Julia package.