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The Lasso has attracted the attention of many authors these last years. Whilemany efforts have been made to prove that the Lasso behaves like a variableselection procedure at the price of strong (though unavoidable) assumptions onthe geometric structure of these variables, much less attention has been paidto the analysis of the performance of the Lasso as a regularization algorithm.Our first purpose here is to provide a conceptually very simple result in thisdirection. We shall prove that, provided that the regularization parameter isproperly chosen, the Lasso works almost as well as the deterministic Lasso.This result does not require any assumption at all, neither on the structure ofthe variables nor on the regression function. Our second purpose is tointroduce a new estimator particularly adapted to deal with infinite countabledictionaries. This estimator is constructed as an l0-penalized estimator amonga sequence of Lasso estimators associated to a dyadic sequence of growingtruncated dictionaries. The selection procedure automatically chooses the bestlevel of truncation of the dictionary so as to make the best tradeoff betweenapproximation, l1-regularization and sparsity. From a theoretical point ofview, we shall provide an oracle inequality satisfied by this selected Lassoestimator. The oracle inequalities established for the Lasso and the selectedLasso estimators shall enable us to derive rates of convergence on a wide classof functions, showing that these estimators perform at least as well as greedyalgorithms. Besides, we shall prove that the rates of convergence achieved bythe selected Lasso estimator are optimal in the orthonormal case by boundingfrom below the minimax risk on some Besov bodies. Finally, some theoreticalresults about the performance of the Lasso for infinite uncountabledictionaries will be studied in the specific framework of neural networks. Allthe oracle inequalities presented in this paper are obtained via theapplication of a single general theorem of model selection among a collectionof nonlinear models which is a direct consequence of the Gaussian concentrationinequality. The key idea that enables us to apply this general theorem is tosee l1-regularization as a model selection procedure among l1-balls.

An l1-Oracle Inequality for the Lasso