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We introduce a computationally effective algorithm for a linear modelselection consisting of three steps: screening--ordering--selection (SOS).Screening of predictors is based on the thresholded Lasso that is l_1 penalizedleast squares. The screened predictors are then fitted using least squares (LS)and ordered with respect to their t statistics. Finally, a model is selectedusing greedy generalized information criterion (GIC) that is l_0 penalized LSin a nested family induced by the ordering. We give non-asymptotic upper boundson error probability of each step of the SOS algorithm in terms of bothpenalties. Then we obtain selection consistency for different (n, p) scenariosunder conditions which are needed for screening consistency of the Lasso. Forthe traditional setting (n >p) we give Sanov-type bounds on the errorprobabilities of the ordering--selection algorithm. Its surprising consequenceis that the selection error of greedy GIC is asymptotically not larger than ofexhaustive GIC. We also obtain new bounds on prediction and estimation errorsfor the Lasso which are proved in parallel for the algorithm used in practiceand its formal version.

Combined l_1 and greedy l_0 penalized least squares for linear model selection