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Model selection is difficult to analyse yet theoretically and empiricallyimportant, especially for high-dimensional data analysis. Recently the leastabsolute shrinkage and selection operator (Lasso) has been applied in thestatistical and econometric literature. Consis- tency of Lasso has beenestablished under various conditions, some of which are difficult to verify inpractice. In this paper, we study model selection from the perspective ofgeneralization ability, under the framework of structural risk minimization(SRM) and Vapnik-Chervonenkis (VC) theory. The approach emphasizes the balancebetween the in-sample and out-of-sample fit, which can be achieved by usingcross-validation to select a penalty on model complexity. We show that an exactrelationship exists between the generalization ability of a model and modelselection consistency. By implementing SRM and the VC inequality, we show thatLasso is L2-consistent for model selection under assumptions similar to thoseimposed on OLS. Furthermore, we derive a probabilistic bound for the distancebetween the penalized extremum estimator and the extremum estimator withoutpenalty, which is dominated by overfitting. We also propose a new measurementof overfitting, GR2, based on generalization ability, that converges to zero ifmodel selection is consistent. Using simulations, we demonstrate that theproposed CV-Lasso algorithm performs well in terms of model selection andoverfitting control.

Model selection consistency from the perspective of generalization ability and VC theory with an application to Lasso