Sparsistency of l1-Regularized M-Estimators

Yen-Huan Li, Jonathan Scarlett, Pradeep Ravikumar, Volkan Cevher

Abstract:   We consider the model selection consistency or sparsistency of a broad set of ℓ1-regularized M-estimators for linear and non-linear statistical models in a unified fashion. For this purpose, we propose the local structured smoothness condition (LSSC) on the loss function. We provide a general result giving deterministic sufficient conditions for sparsistency in terms of the regularization parameter, ambient dimension, sparsity level, and number of measurements. We show that several important statistical models have M-estimators that indeed satisfy the LSSC, and as a result, the sparsistency guarantees for the corresponding ℓ1-regularized M-estimators can be derived as simple applications of our main theorem.

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  • Sparsistency of l1-Regularized M-Estimators (pdf, arXiv, software)
    Y. Li, J. Scarlett, P. Ravikumar, V. Cevher.
    In International Conference on Artificial Intelligence and Statistics (AISTATS), 2015. (Oral)