On the asymptotics of random forests

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. The aim of this paper is twofold. 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 = ∞). Using empirical process theory, we prove a uniform central limit theorem for a large class of random forest estimates, which holds in particular for Breiman’s original forests. 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 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. Index Terms — Random forests, randomization, consistency, central limit theorem, empirical process, number of trees, q-quantile. 2010 Mathematics Subject Classification: 62G05, 62G20.