The Annals of Statistics

Some sharp performance bounds for least squares regression with L1 regularization

Tong Zhang

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We derive sharp performance bounds for least squares regression with L1 regularization from parameter estimation accuracy and feature selection quality perspectives. The main result proved for L1 regularization extends a similar result in [Ann. Statist. 35 (2007) 2313–2351] for the Dantzig selector. It gives an affirmative answer to an open question in [Ann. Statist. 35 (2007) 2358–2364]. Moreover, the result leads to an extended view of feature selection that allows less restrictive conditions than some recent work. Based on the theoretical insights, a novel two-stage L1-regularization procedure with selective penalization is analyzed. It is shown that if the target parameter vector can be decomposed as the sum of a sparse parameter vector with large coefficients and another less sparse vector with relatively small coefficients, then the two-stage procedure can lead to improved performance.

Article information

Ann. Statist. Volume 37, Number 5A (2009), 2109-2144.

First available in Project Euclid: 15 July 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 62G05: Estimation
Secondary: 62J05: Linear regression

L_1 regularization Lasso regression sparsity variable selection parameter estimation


Zhang, Tong. Some sharp performance bounds for least squares regression with L 1 regularization. Ann. Statist. 37 (2009), no. 5A, 2109--2144. doi:10.1214/08-AOS659.

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