The Annals of Statistics

Consistency of random forests

Erwan Scornet, Gérard Biau, and Jean-Philippe Vert

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Random forests are a learning algorithm proposed by Breiman [ Mach. Learn. 45 (2001) 5–32] that combines several randomized decision trees and aggregates their predictions by averaging. Despite its wide usage and outstanding practical performance, little is known about the mathematical properties of the procedure. This disparity between theory and practice originates in the difficulty to simultaneously analyze both the randomization process and the highly data-dependent tree structure. In the present paper, we take a step forward in forest exploration by proving a consistency result for Breiman’s [ Mach. Learn. 45 (2001) 5–32] original algorithm in the context of additive regression models. Our analysis also sheds an interesting light on how random forests can nicely adapt to sparsity.

Article information

Ann. Statist. Volume 43, Number 4 (2015), 1716-1741.

Received: May 2014
Revised: February 2015
First available in Project Euclid: 17 June 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Random forests randomization consistency additive model sparsity dimension reduction


Scornet, Erwan; Biau, Gérard; Vert, Jean-Philippe. Consistency of random forests. Ann. Statist. 43 (2015), no. 4, 1716--1741. doi:10.1214/15-AOS1321.

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Supplemental materials

  • Supplement to “Consistency of random forests”. Proofs of technical results.