Open Access
2022 Rates of convergence for random forests via generalized U-statistics
Wei Peng, Tim Coleman, Lucas Mentch
Author Affiliations +
Electron. J. Statist. 16(1): 232-292 (2022). DOI: 10.1214/21-EJS1958


Random forests are among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical success, however, until recently, few theoretical results were available to describe 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 established. 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.

Funding Statement

This work was partially supported by NSF DMS-1712041.


We would like to thank Larry Wasserman for helpful conversations and feedback, and anonymous reviewers for helpful suggestions.


Download Citation

Wei Peng. Tim Coleman. Lucas Mentch. "Rates of convergence for random forests via generalized U-statistics." Electron. J. Statist. 16 (1) 232 - 292, 2022.


Received: 1 May 2021; Published: 2022
First available in Project Euclid: 6 January 2022

MathSciNet: MR4359361
zbMATH: 1493.62071
Digital Object Identifier: 10.1214/21-EJS1958

Primary: 62E17 , 62E20
Secondary: 60F05

Keywords: bagging , Berry-Esseen , CART , random forests , U-statistics

Vol.16 • No. 1 • 2022
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