Generalized degrees of freedom measure the complexity of a modeling procedure; a modeling procedure is a combination of model selection and model fitting. In this manuscript, we consider two definitions of generalized degrees of freedom for a modeling procedure under the L1 loss function, and investigate the connections between those two definitions. We also propose the extended Akaike information criterion, the adaptive model selection, and the extended generalized cross-validation under the L1 loss function. Finally, we extend the results to M-estimation.
