Sparsistency of \ell_1-Regularized M-Estimators

Yen-Huan Li, Jonathan Scarlett, Pradeep Ravikumar, Volkan Cevher
Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, PMLR 38:644-652, 2015.

Abstract

We consider the model selection consistency or sparsistency of a broad set of \ell_1-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 \ell_1-regularized M-estimators can be derived as simple applications of our main theorem.

Cite this Paper

BibTeX
@InProceedings{pmlr-v38-li15f, title = {{Sparsistency of \ell_1-Regularized M-Estimators}}, author = {Li, Yen-Huan and Scarlett, Jonathan and Ravikumar, Pradeep and Cevher, Volkan}, booktitle = {Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics}, pages = {644--652}, year = {2015}, editor = {Lebanon, Guy and Vishwanathan, S. V. N.}, volume = {38}, series = {Proceedings of Machine Learning Research}, address = {San Diego, California, USA}, month = {09--12 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v38/li15f.pdf}, url = {https://proceedings.mlr.press/v38/li15f.html}, abstract = {We consider the model selection consistency or sparsistency of a broad set of \ell_1-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 \ell_1-regularized M-estimators can be derived as simple applications of our main theorem.} }
Endnote
%0 Conference Paper %T Sparsistency of \ell_1-Regularized M-Estimators %A Yen-Huan Li %A Jonathan Scarlett %A Pradeep Ravikumar %A Volkan Cevher %B Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2015 %E Guy Lebanon %E S. V. N. Vishwanathan %F pmlr-v38-li15f %I PMLR %P 644--652 %U https://proceedings.mlr.press/v38/li15f.html %V 38 %X We consider the model selection consistency or sparsistency of a broad set of \ell_1-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 \ell_1-regularized M-estimators can be derived as simple applications of our main theorem.
RIS
TY - CPAPER TI - Sparsistency of \ell_1-Regularized M-Estimators AU - Yen-Huan Li AU - Jonathan Scarlett AU - Pradeep Ravikumar AU - Volkan Cevher BT - Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics DA - 2015/02/21 ED - Guy Lebanon ED - S. V. N. Vishwanathan ID - pmlr-v38-li15f PB - PMLR DP - Proceedings of Machine Learning Research VL - 38 SP - 644 EP - 652 L1 - http://proceedings.mlr.press/v38/li15f.pdf UR - https://proceedings.mlr.press/v38/li15f.html AB - We consider the model selection consistency or sparsistency of a broad set of \ell_1-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 \ell_1-regularized M-estimators can be derived as simple applications of our main theorem. ER -
APA
Li, Y., Scarlett, J., Ravikumar, P. & Cevher, V.. (2015). Sparsistency of \ell_1-Regularized M-Estimators. Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 38:644-652 Available from https://proceedings.mlr.press/v38/li15f.html.

Related Material