Optimal confidence for Monte Carlo integration of smooth functions

Abstract

We study the information-based complexity of approximating integrals of smooth functions at absolute precision e > 0 with confidence level 1 − δ ∈ (0, 1) using function evaluations within randomized algorithms. The probabilistic error criterion is new in the context of integrating smooth functions. In previous research, Monte Carlo integration was studied in terms of the expected error (or the root mean squared error), for which linear methods achieve optimal rates of the error e(n) in terms of the number n of function evaluations. In our context, usually methods that provide optimal confidence properties exhibit non-linear features. The optimal probabilistic error rate e(n,δ) for multivariate functions from classical isotropic Sobolev spaces ${W_{p}^{r}}(G)$ with sufficient smoothness on bounded Lipschitz domains $G \subset {\mathbb R}^{d}$ is determined. It turns out that the integrability index p has an effect on the influence of the uncertainty δ in the complexity. In the limiting case p = 1, we see that deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the additional effort for diminishing the uncertainty. Finally, we add a discussion about this problem for function spaces with mixed smoothness.