The Annals of Statistics

Oracle inequalities and optimal inference under group sparsity

Karim Lounici, Massimiliano Pontil, Sara van de Geer, and Alexandre B. Tsybakov

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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.

Article information

Ann. Statist. Volume 39, Number 4 (2011), 2164-2204.

First available: 26 October 2011

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Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 62J05: Linear regression
Secondary: 62C20: Minimax procedures 62F07: Ranking and selection

Oracle inequalities group Lasso minimax risk penalized least squares moment inequality group sparsity statistical learning


Lounici, Karim; Pontil, Massimiliano; van de Geer, Sara; Tsybakov, Alexandre B. Oracle inequalities and optimal inference under group sparsity. The Annals of Statistics 39 (2011), no. 4, 2164--2204. doi:10.1214/11-AOS896.

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