Non-asymptotic Oracle Inequalities for the Lasso and Group Lasso in high dimensional logistic model

We consider the problem of estimating a function $f_{0}$ in logistic regression model. We propose to estimate this function $f_{0}$ by a sparse approximation build as a linear combinaison of elements of a given dictionary of $p$ functions. This sparse approximation is selected by the Lasso or Group Lasso procedure. In this context, we state non asymptotique oracle inequalities for Lasso and Group Lasso under restricted eigenvalues assumption as introduced in \cite{BRT}. Those theoretical results are illustrated through a simulation study.