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August 2011 Oracle inequalities and optimal inference under group sparsity
Karim Lounici, Massimiliano Pontil, Sara van de Geer, Alexandre B. Tsybakov
Ann. Statist. 39(4): 2164-2204 (August 2011). DOI: 10.1214/11-AOS896

Abstract

We consider the problem of estimating a sparse linear regression vector β* under a Gaussian noise model, for the purpose of both prediction and model selection. We assume that prior knowledge is available on the sparsity pattern, namely the set of variables is partitioned into prescribed groups, only few of which are relevant in the estimation process. This group sparsity assumption suggests us to consider the Group Lasso method as a means to estimate β*. We establish oracle inequalities for the prediction and 2 estimation errors of this estimator. These bounds hold under a restricted eigenvalue condition on the design matrix. Under a stronger condition, we derive bounds for the estimation error for mixed (2, p)-norms with 1 ≤ p ≤ ∞. When p=∞, this result implies that a thresholded version of the Group Lasso estimator selects the sparsity pattern of β* with high probability. Next, we prove that the rate of convergence of our upper bounds is optimal in a minimax sense, up to a logarithmic factor, for all estimators over a class of group sparse vectors. Furthermore, we establish lower bounds for the prediction and 2 estimation errors of the usual Lasso estimator. Using this result, we demonstrate that the Group Lasso can achieve an improvement in the prediction and estimation errors as compared to the Lasso.

An important application of our results is provided by the problem of estimating multiple regression equations simultaneously or multi-task learning. In this case, we obtain refinements of the results in [In Proc. of the 22nd Annual Conference on Learning Theory (COLT) (2009)], which allow us to establish a quantitative advantage of the Group Lasso over the usual Lasso in the multi-task setting. Finally, within the same setting, we show how our results can be extended to more general noise distributions, of which we only require the fourth moment to be finite. To obtain this extension, we establish a new maximal moment inequality, which may be of independent interest.

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Karim Lounici. Massimiliano Pontil. Sara van de Geer. Alexandre B. Tsybakov. "Oracle inequalities and optimal inference under group sparsity." Ann. Statist. 39 (4) 2164 - 2204, August 2011. https://doi.org/10.1214/11-AOS896

Information

Published: August 2011
First available in Project Euclid: 26 October 2011

zbMATH: 1306.62156
MathSciNet: MR2893865
Digital Object Identifier: 10.1214/11-AOS896

Subjects:
Primary: 62J05
Secondary: 62C20 , 62F07

Keywords: group lasso , group sparsity , minimax risk , moment inequality , Oracle inequalities , penalized least squares , Statistical learning

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 4 • August 2011
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