Penalized Least Square in Sparse Setting with Convex Penalty and Non Gaussian Errors

This paper consider the penalized least squares estimators with convex penalties or regularization norms. We provide sparsity oracle inequalities for the prediction error for a general convex penalty and for the particular cases of Lasso and Group Lasso estimators in a regression setting. The main contribution is that our oracle inequalities are established for the more general case where the observations noise is issued from probability measures that satisfy a weak spectral gap (or Poincare) inequality instead of Gaussian distributions. We illustrate our results on a heavy tailed example and a sub Gaussian one; we especially give the explicit bounds of the oracle inequalities for these two special examples.