Adaptive stratified Monte Carlo using decision trees

Nicolas Chopin; Hejin Wang; Mathieu Gerber

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arXiv · 2501.04842

Adaptive stratified Monte Carlo using decision trees

Nicolas Chopin·Hejin Wang·Mathieu Gerber

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Abstract

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.

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Nicolas Chopin, Hejin Wang, Mathieu Gerber. 2025-01-08. Adaptive stratified Monte Carlo using decision trees. https://arxiv.org/abs/2501.04842

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