A computationally fast alternative to cross-validation in penalized Gaussian graphical models

We study the problem of selecting a regularization parameter in penalized Gaussian graphical models. When the goal is to obtain a model with good predictive power, cross-validation is the gold standard. We present a new estimator of Kullback–Leibler loss in Gaussian Graphical models which provides a computationally fast alternative to cross-validation. The estimator is obtained by approximating leave-one-out-cross-validation. Our approach is demonstrated on simulated data sets for various types of graphs. The proposed formula exhibits superior performance, especially in the typical small sample size scenario, compared to other available alternatives to cross-validation, such as Akaike's information criterion and Generalized approximate cross-validation. We also show that the estimator can be used to improve the performance of the Bayesian information criterion when the sample size is small.