On model selection consistency of regularized M-estimators
This work addresses the challenge of reliable model selection in high-dimensional statistics, which is crucial for applications in science and engineering, but it is incremental as it builds on existing regularization methods.
The authors tackled the problem of ensuring model selection consistency for regularized M-estimators in high-dimensional settings, and they developed a general framework that identifies geometric decomposability and irrepresentability as key properties for achieving this consistency.
Regularized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Usually the low-dimensional structure is encoded by the presence of the (unknown) parameters in some low-dimensional model subspace. In such settings, it is desirable for estimates of the model parameters to be \emph{model selection consistent}: the estimates also fall in the model subspace. We develop a general framework for establishing consistency and model selection consistency of regularized M-estimators and show how it applies to some special cases of interest in statistical learning. Our analysis identifies two key properties of regularized M-estimators, referred to as geometric decomposability and irrepresentability, that ensure the estimators are consistent and model selection consistent.