Optimal confidence for Monte Carlo integration of smooth functions

TL;DR

This paper analyzes the complexity of Monte Carlo integration for smooth functions, determining optimal error rates and how integrability and smoothness affect the effort needed to achieve confidence levels.

Contribution

It establishes the optimal error rates for multivariate smooth functions and explores how integrability and smoothness influence the complexity of achieving confidence.

Findings

Higher smoothness reduces effort for lower uncertainty.

Deterministic methods are optimal for p=1, no improvement via randomization.

Integrability index p affects the influence of confidence level on complexity.

Abstract

We study the complexity of approximating integrals of smooth functions at absolute precision $\varepsilon > 0$ with confidence level $1 - \delta \in (0,1)$. The optimal error rate 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 $\delta$ 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.