Oracle Inequalities for a Group Lasso Procedure Applied to Generalized Linear Models in High Dimension

We present a group lasso procedure for generalized linear models (GLMs) and we study the properties of this estimator applied to sparse high-dimensional GLMs. Under general conditions on the covariates and on the joint distribution of the pair covariates, we provide oracle inequalities promoting group sparsity of the covariables. We get convergence rates for the prediction and estimation error and we show the ability of this estimator to recover good sparse approximation of the true model. Then, we extend this procedure to the case of an elastic net penalty. At last, we apply these results to the so-called Poisson regression model (the output is modeled as a Poisson process whose intensity relies on a linear combination of the covariables). The group lasso method enables to select few groups of meaningful variables among the set of inputs.