Oracle Inequalities and Optimal Inference under Group Sparsity

We consider the problem of estimating a sparse linear regression vector under a gaussiannoise model, for the purpose of both prediction and model selection. We assume that priorknowledge is available on the sparsity pattern, namely the set of variables is partitioned intoprescribed groups, only few of which are relevant in the estimation process. This group sparsityassumption suggests us to consider the Group Lasso method as a means to estimate . Weestablish oracle inequalities for the prediction and `2 estimation errors of this estimator. Thesebounds hold under a restricted eigenvalue condition on the design matrix. Under a strongercoherence condition, we derive bounds for the estimation error for mixed (2, p)-norms with1 p 1. When p = 1, this result implies that a threshold version of the Group Lassoestimator selects the sparsity pattern of with high probability. Next, we prove that the rateof 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 boundsfor the prediction and `2 estimation errors of the usual Lasso estimator. Using this result, wedemonstrate that the Group Lasso can achieve an improvement in the prediction and estimationproperties as compared to the Lasso.An important application of our results is provided by the problem of estimating multipleregression equation simultaneously or multi-task learning. In this case, our results lead toref nements of the results in [22] and allow one to establish the quantitative advantage of theGroup Lasso over the usual Lasso in the multi-task setting. Finally, within the same setting, weshow how our results can be extended to more general noise distributions, of which we onlyrequire the fourth moment to be f nite. To obtain this extension, we establish a new maximalmoment inequality, which may be of independent interest.