On the Degrees of Freedom of some Lasso procedures

The effective degrees of freedom of penalized regression models quantify the actual amount of information used to generate predictions, playing a pivotal role in model evaluation and selection. Although a closed-form estimator is available for the Lasso penalty, adaptive extensions of widely used penalized approaches, including the Adaptive Lasso and Adaptive Group Lasso, have remained without analogous theoretical characterization. This paper presents the first unbiased estimator of the effective degrees of freedom for these methods, along with their main theoretical properties, for both orthogonal and non-orthogonal designs, derived within Stein's unbiased risk estimation framework. The resulting expressions feature inflation terms influenced by the regularization parameter, coefficient signs, and least-squares estimates. These advances enable more accurate model selection criteria and unbiased prediction error estimates, illustrated through synthetic and real data. These contributions offer a rigorous theoretical foundation for understanding model complexity in adaptive regression, bridging a critical gap between theory and practice.