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An l1-Oracle Inequality for the Lasso

The Lasso has attracted the attention of many authors these last years. While many efforts have been made to prove that the Lasso behaves like a variable selection procedure at the price of strong (though unavoidable) assumptions on the geometric structure of these variables, much less attention has been paid to 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 this direction. We shall prove that, provided that the regularization parameter is properly chosen, the Lasso works almost as well as the deterministic Lasso. This result does not require any assumption at all, neither on the structure of the variables nor on the regression function. Our second purpose is to introduce a new estimator particularly adapted to deal with infinite countable dictionaries. This estimator is constructed as an l0-penalized estimator among a sequence of Lasso estimators associated to a dyadic sequence of growing truncated dictionaries. The selection procedure automatically chooses the best level of truncation of the dictionary so as to make the best tradeoff between approximation, l1-regularization and sparsity. From a theoretical point of view, we shall provide an oracle inequality satisfied by this selected Lasso estimator. The oracle inequalities established for the Lasso and the selected Lasso estimators shall enable us to derive rates of convergence on a wide class of functions, showing that these estimators perform at least as well as greedy algorithms. Besides, we shall prove that the rates of convergence achieved by the selected Lasso estimator are optimal in the orthonormal case by bounding from below the minimax risk on some Besov bodies. Finally, some theoretical results about the performance of the Lasso for infinite uncountable dictionaries will be studied in the specific framework of neural networks. All the oracle inequalities presented in this paper are obtained via the application of a single general theorem of model selection among a collection of nonlinear models which is a direct consequence of the Gaussian concentration inequality. The key idea that enables us to apply this general theorem is to see l1-regularization as a model selection procedure among l1-balls.