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On the asymptotics of random forests. (English) Zbl 1337.62063
Summary: The last decade has witnessed a growing interest in random forest models which are recognized to exhibit good practical performance, especially in high-dimensional settings. On the theoretical side, however, their predictive power remains largely unexplained, thereby creating a gap between theory and practice. In this paper, we present some asymptotic results on random forests in a regression framework. Firstly, we provide theoretical guarantees to link finite forests used in practice (with a finite number \(M\) of trees) to their asymptotic counterparts (with \(M = \infty\)). Using empirical process theory, we prove a uniform central limit theorem for a large class of random forest estimates, which holds in particular for L. Breiman’s original forests [Mach. Learn. 45, No. 1, 5–32 (2001; Zbl 1007.68152)]. Secondly, we show that infinite forest consistency implies finite forest consistency and thus, we state the consistency of several infinite forests. In particular, we prove that \(q\) quantile forests – close in spirit to Breiman’s [loc. cit.] forests but easier to study – are able to combine inconsistent trees to obtain a final consistent prediction, thus highlighting the benefits of random forests compared to single trees.

62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI
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