Abstract: 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 Poincaré) 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.

Key words: penalized least squares, Gaussian errors, convex penalty