Higher-order stochastic integration through cubic stratification

TL;DR

This paper introduces two new unbiased estimators for high-dimensional integrals that leverage cubic stratification and numerical derivatives, achieving optimal error rates for smooth functions and demonstrating good practical performance.

Contribution

The paper presents two novel unbiased estimators for multivariate integrals using cubic stratification, one optimal for smooth functions and another computationally cheaper for boundary-vanishing functions.

Findings

Achieves optimal error rate of n^{-1/2 - r/s} for r-times differentiable functions.

Demonstrates good numerical performance even with moderate sample sizes.

Provides estimators that are unbiased and rely on cubic stratification and numerical derivatives.

Abstract

We propose two novel unbiased estimators of the integral $\int_{[0,1]^{s}}f(u) du$ for a function $f$, which depend on a smoothness parameter $r\in\mathbb{N}$. The first estimator integrates exactly the polynomials of degrees $p<r$ and achieves the optimal error $n^{-1/2-r/s}$ (where $n$ is the number of evaluations of $f$) when $f$ is $r$ times continuously differentiable. The second estimator is computationally cheaper but it is restricted to functions that vanish on the boundary of $[0,1]^s$. The construction of the two estimators relies on a combination of cubic stratification and control ariates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of $n$.