  Regularized estimators in the context of group variables have been applied
successfully in model and feature selection in order to preserve
interpretability. We formulate a Distributionally Robust Optimization (DRO)
problem which recovers popular estimators, such as Group Square Root Lasso
(GSRL). Our DRO formulation allows us to interpret GSRL as a game, in which we
learn a regression parameter while an adversary chooses a perturbation of the
data. We wish to pick the parameter to minimize the expected loss under any
plausible model chosen by the adversary - who, on the other hand, wishes to
increase the expected loss. The regularization parameter turns out to be
precisely determined by the amount of perturbation on the training data allowed
by the adversary. In this paper, we introduce a data-driven (statistical)
criterion for the optimal choice of regularization, which we evaluate
asymptotically, in closed form, as the size of the training set increases. Our
easy-to-evaluate regularization formula is compared against cross-validation,
showing good (sometimes superior) performance.
