AIC for the Lasso in generalized linear models

Abstract

The Lasso is a popular regularization method that can simultaneously do estimation and model selection. It contains a regularization parameter, and several information criteria have been proposed for selecting its proper value. While any of them would assure consistency in model selection, we have no appropriate rule to choose between the criteria. Meanwhile, a finite correction to the AIC has been provided in a Gaussian regression setting. The finite correction is theoretically assured from the viewpoint not of the consistency but of minimizing the prediction error and does not have the above-mentioned difficulty. Our aim is to derive such a criterion for the Lasso in generalized linear models. Towards this aim, we derive a criterion from the original definition of the AIC, that is, an asymptotically unbiased estimator of the Kullback-Leibler divergence. This becomes the finite correction in the Gaussian regression setting, and so our criterion can be regarded as its generalization. Our criterion can be easily obtained and requires fewer computational tasks than does cross-validation, but simulation studies and real data analyses indicate that its performance is almost the same as or superior to that of cross-validation. Moreover, our criterion is extended for a class of other regularization methods.