Linear Monte Carlo quadrature with optimal confidence intervals

TL;DR

This paper introduces a new linear Monte Carlo method called stratified control variates (SCV) for numerical integration in Sobolev spaces, achieving optimal probabilistic error rates in high smoothness regimes without parameter tuning.

Contribution

The paper proposes the SCV method that attains optimal error bounds for linear algorithms in high smoothness settings and analyzes its limitations in low smoothness regimes.

Findings

Linear methods with SCV achieve optimal error rates in high smoothness regimes.

In low smoothness regimes, linear methods exhibit polynomial error dependence on $\delta^{-1}$.

Non-linear algorithms outperform linear ones in low smoothness regimes by achieving logarithmic dependence on $\delta^{-1}$.

Abstract

We study the numerical integration of functions from isotropic Sobolev spaces $W_p^s([0,1]^d)$ using finitely many function evaluations within randomized algorithms, aiming for the smallest possible probabilistic error guarantee $\varepsilon > 0$ at confidence level $1-\delta \in (0,1)$. For spaces consisting of continuous functions, non-linear Monte Carlo methods with optimal confidence properties have already been known, in few cases even linear methods that succeed in that respect. In this paper we promote a new method called stratified control variates (SCV) and by it show that already linear methods achieve optimal probabilistic error rates in the high smoothness regime without the need to adjust algorithmic parameters to the uncertainty $\delta$. We also analyse a version of SCV in the low smoothness regime where $W_p^s([0,1]^d)$ may contain functions with singularities. Here, we observe a polynomial dependence of the error on $\delta^{-1}$ which cannot be avoided for linear methods. This is worse than what is known to be possible using non-linear algorithms where only a logarithmic dependence on $\delta^{-1}$ occurs if we tune in for a specific value of $\delta$.