It has been known for a long time that stratification is one possible strategy to obtain higher convergence rates for the Monte Carlo estimation of integrals over the hyper-cube $[0, 1]^s$ of dimension $s$. However, stratified estimators such as Haber's are not practical as $s$ grows, as they require $\mathcal{O}(k^s)$ evaluations for some $k\geq 2$. We propose an adaptive stratification strategy, where the strata are derived from a a decision tree applied to a preliminary sample. We show that this strategy leads to higher convergence rates, that is, the corresponding estimators converge at rate $\mathcal{O}(N^{-1/2-r})$ for some $r>0$ for certain classes of functions. Empirically, we show through numerical experiments that the method may improve on standard Monte Carlo even when $s$ is large.

翻译：长期以来，分层策略被认为是提高超立方体 $[0, 1]^s$（维度为 $s$）上积分蒙特卡洛估计收敛速度的可能途径之一。然而，随着 $s$ 的增大，诸如哈伯（Haber）估计器之类的分层估计器因需要 $\mathcal{O}(k^s)$ 次评估（其中 $k\geq 2$）而变得不实用。本文提出了一种自适应分层策略，其分层结构源自对初步样本应用决策树。我们证明，该策略能够实现更高的收敛速度，即对于特定函数类，相应的估计器以 $\mathcal{O}(N^{-1/2-r})$ 的速率收敛（其中 $r>0$）。通过数值实验，我们经验性地表明，即使 $s$ 较大，该方法仍可能优于标准蒙特卡洛方法。