Model selection consistency of generalized information criteria for m-estimators in high dimensions: a unified framework
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Regularized M-estimators are widely used in science, due to their ability to fit a simpler, low- dimensional model in high-dimensional scenarios. Some of the recent efforts on the subject have focused on the creation of a unified framework, and the establishment of sufficient conditions for consistency and model selection consistency. We use that same general setting to derive sufficient conditions for Model Selection consistency of the GIC and the Pathconsistency of regularized M-estimators. Here, Pathconsistency means that a sequence of submodels contains the true model with probability converging to one. This allows the practical use of the GIC for Model Selection in high-dimensional scenarios. We illustrate those conditions on some examples, including the estimation of the support of sparse and group sparse vectors, with various loss functions.