Rates of convergence for random forests via generalized U-statistics∗

Abstract

Random forests are among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical suc-cess, however, until recently, few theoretical results were available to de-scribe their performance and behavior. In this work we push beyond recent work on consistency and asymptotic normality by establishing rates of convergence for random forests and other supervised learning ensembles. We develop the notion of generalized U-statistics and show that within this framework, random forest predictions can remain asymptotically normal for larger subsample sizes and under weaker conditions than previously es-tablished. Moreover, we provide Berry-Esseen bounds in order to quantify the rate at which this convergence occurs, making explicit the roles of the subsample size and the number of trees in determining the distribution of random forest predictions. When these generalized estimators are reduced to their classical U-statistic form, the rates we establish are faster than any available in the existing literature.