Adaptive stratified Monte Carlo using decision trees

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.