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Variance reduction in purely random forests

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  • Robin Genuer

Abstract

Random forests (RFs), introduced by Leo Breiman in 2001, are a very effective statistical method. The complex mechanism of the method makes theoretical analysis difficult. Therefore, simplified versions of RF, called purely RFs (PRF), which can be theoretically handled more easily, have been considered. In this paper, we study the variance of such forests. First, we show a general upper bound which emphasises the fact that a forest reduces the variance. We then introduce a simple variant of PRFs, that we call purely uniformly RFs. For this variant and in the context of regression problems with a one-dimensional predictor space, we show that both random trees and RFs reach minimax rate of convergence. In addition, we prove that compared with random trees, RFs improve accuracy by reducing the estimator variance by a factor of three-fourths.

Suggested Citation

  • Robin Genuer, 2012. "Variance reduction in purely random forests," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(3), pages 543-562.
    • Handle: RePEc:taf:gnstxx:v:24:y:2012:i:3:p:543-562
    DOI: 10.1080/10485252.2012.677843
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    Cited by:

    1. Sylvain Arlot & Robin Genuer, 2016. "Comments on: A random forest guided tour," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 228-238, June.
    2. Jaouad Mourtada & Stéphane Gaïffas & Erwan Scornet, 2021. "AMF: Aggregated Mondrian forests for online learning," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(3), pages 505-533, July.
    3. Zhexiao Lin & Fang Han, 2022. "On regression-adjusted imputation estimators of the average treatment effect," Papers 2212.05424, arXiv.org, revised Jan 2023.
    4. Gérard Biau & Erwan Scornet, 2016. "A random forest guided tour," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 197-227, June.
    5. Rogelio Ochoa-Barragán & Aurora del Carmen Munguía-López & José María Ponce-Ortega, 2024. "A hybrid machine learning-mathematical programming optimization approach for municipal solid waste management during the pandemic," Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, Springer, vol. 26(7), pages 17653-17672, July.

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