AIC for the group Lasso in generalized linear models

When covariates are assumed to be clustered in groups in regression problems, the group Lasso is useful, because it tends to drive all the weights in one group to zero together. The group Lasso includes a tuning parameter which controls a penalty level, and an unbiased estimator of the true prediction error is derived as a Cp-type criterion to select an appropriate value of the tuning parameter. However, in general, the Cp-type criterion cannot be derived in generalized linear regressions such as a logistic regression. Hence, this paper obtains the AIC for the group Lasso based on its original definition under the framework of generalized linear models. In terms of computational cost, our criterion is clearly better than the cross validation, but it is shown through simulation studies that the performance of the our criterion is almost the same as or better than that of the cross validation.