Regularized m-estimators are widely used due to their ability of recovering a low-dimensional model in high-dimensional scenarios. Some recent efforts on this subject focused on creating a unified framework for establishing oracle bounds, and deriving conditions for support recovery. Under this same framework, we propose a new Generalized Information Criteria (GIC) that takes into consideration the sparsity pattern one wishes to recover. We obtain non-asymptotic model selection bounds and sufficient conditions for model selection consistency of the GIC. Furthermore, we show that the GIC can also be used for selecting the regularization parameter within a regularized $m$-estimation framework, which allows practical use of the GIC for model selection in high-dimensional scenarios. We provide examples of group LASSO in the context of generalized linear regression and low rank matrix regression.

翻译：正则化M估计因能够在高维场景中恢复低维模型而被广泛使用。近期相关研究致力于构建统一框架以建立预言界并推导支撑恢复条件。在此框架下，我们提出了一种新的广义信息准则，该准则考虑了待恢复的稀疏模式。我们得到了该准则的非渐近模型选择界以及模型选择一致性的充分条件。进一步证明，该准则还可用于正则化M估计框架内的正则化参数选择，从而支持其在高维模型选择中的实际应用。我们以广义线性回归与低秩矩阵回归中的群组LASSO为例进行说明。