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April 2012 Characterizing $L_{2}$Boosting
John Ehrlinger, Hemant Ishwaran
Ann. Statist. 40(2): 1074-1101 (April 2012). DOI: 10.1214/12-AOS997

Abstract

We consider $L_{2}$Boosting, a special case of Friedman’s generic boosting algorithm applied to linear regression under $L_{2}$-loss. We study $L_{2}$Boosting for an arbitrary regularization parameter and derive an exact closed form expression for the number of steps taken along a fixed coordinate direction. This relationship is used to describe $L_{2}$Boosting’s solution path, to describe new tools for studying its path, and to characterize some of the algorithm’s unique properties, including active set cycling, a property where the algorithm spends lengthy periods of time cycling between the same coordinates when the regularization parameter is arbitrarily small. Our fixed descent analysis also reveals a repressible condition that limits the effectiveness of $L_{2}$Boosting in correlated problems by preventing desirable variables from entering the solution path. As a simple remedy, a data augmentation method similar to that used for the elastic net is used to introduce $L_{2}$-penalization and is shown, in combination with decorrelation, to reverse the repressible condition and circumvents $L_{2}$Boosting’s deficiencies in correlated problems. In itself, this presents a new explanation for why the elastic net is successful in correlated problems and why methods like LAR and lasso can perform poorly in such settings.

Citation

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John Ehrlinger. Hemant Ishwaran. "Characterizing $L_{2}$Boosting." Ann. Statist. 40 (2) 1074 - 1101, April 2012. https://doi.org/10.1214/12-AOS997

Information

Published: April 2012
First available in Project Euclid: 18 July 2012

zbMATH: 1274.62420
MathSciNet: MR2985944
Digital Object Identifier: 10.1214/12-AOS997

Subjects:
Primary: 62J05
Secondary: 62J99

Keywords: Critical direction , gradient-correlation , regularization , repressibility , solution path

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 2 • April 2012
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