Oracle Inequalities and Optimal Inference under Group Sparsity

Karim Lounici (1) , Alexandre Tsybakov (1, 2) , Massimiliano Pontil (3) , Sara A. van de Geer (4)

1 CREST - Centre de Recherche en Économie et Statistique
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 Department of Computer Science
4 Seminar für Statistik

We consider the problem of estimating a sparse linear regression vector $\beta^*$ 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 $\beta^*$. We establish oracle inequalities for the prediction and $\ell_2$ estimation errors of this estimator. These bounds hold under a restricted eigenvalue condition on the design matrix. Under a stronger coherence condition, we derive bounds for the estimation error for mixed $(2,p)$-norms with $1\le p\leq \infty$. When $p=\infty$, this result implies that a threshold version of the Group Lasso estimator selects the sparsity pattern of $\beta^*$ 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 $\ell_2$ 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 properties as compared to the Lasso. An important application of our results is provided by the problem of estimating multiple regression equation simultaneously or multi-task learning. In this case, our result lead to refinements of the results in \cite{colt2009} and allow one to establish the 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.

Mots clés: Minimax risk; Penalized least squares; Moment inequality; Group sparsity; Statistical learning; Oracle inequalities; Group Lasso

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Karim Lounici, Alexandre Tsybakov, Massimiliano Pontil, Sara A. van de Geer. Oracle Inequalities and Optimal Inference under Group Sparsity. 2010. ⟨hal-00501509⟩