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Square-root lasso: pivotal recovery of sparse signals via conic programming

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  • A. Belloni
  • V. Chernozhukov
  • L. Wang

Abstract

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation σ nor does it need to pre-estimate σ. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate σ{(s/n) log p}-super-1/2 in the prediction norm, and thus matching the performance of the lasso with known σ. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods. Copyright 2011, Oxford University Press.

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Bibliographic Info

Article provided by Biometrika Trust in its journal Biometrika.

Volume (Year): 98 (2011)
Issue (Month): 4 ()
Pages: 791-806

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Handle: RePEc:oup:biomet:v:98:y:2011:i:4:p:791-806

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Cited by:
  1. Alexandre Belloni & Victor Chernozhukov & Iván Fernández-Val & Christian Hansen, 2013. "Program evaluation with high-dimensional data," CeMMAP working papers CWP57/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  2. Wang, Lie, 2013. "The L1 penalized LAD estimator for high dimensional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 135-151.
  3. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2011. "Inference for high-dimensional sparse econometric models," CeMMAP working papers CWP41/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  4. Mehmet Caner & Anders Bredahl Kock, 2013. "Oracle Inequalities for Convex Loss Functions with Non-Linear Targets," CREATES Research Papers 2013-51, School of Economics and Management, University of Aarhus.
  5. Eric Gautier & Alexandre Tsybakov, 2013. "Pivotal estimation in high-dimensional regression via linear programming," Papers 1303.7092, arXiv.org, revised Apr 2013.
  6. Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2013. "Robust inference in high-dimensional approximately sparse quantile regression models," CeMMAP working papers CWP70/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  7. Zhu, Ying, 2013. "Sparse Linear Models and Two-Stage Estimation in High-Dimensional Settings with Possibly Many Endogenous Regressors," MPRA Paper 49846, University Library of Munich, Germany.
  8. Alexandre Belloni & Victor Chernozhukov & Lie Wang, 2013. "Pivotal estimation via square-root lasso in nonparametric regression," CeMMAP working papers CWP62/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  9. Alexandre Belloni & Victor Chernozhukov & Iván Fernández-Val & Christian Hansen, 2013. "Program evaluation with high-dimensional data," CeMMAP working papers CWP77/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  10. Anders Bredahl Kock, 2013. "Oracle inequalities for high-dimensional panel data models," CREATES Research Papers 2013-20, School of Economics and Management, University of Aarhus.
  11. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2011. "Estimation of treatment effects with high-dimensional controls," CeMMAP working papers CWP42/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  12. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Central limit theorems and multiplier bootstrap when p is much larger than," CeMMAP working papers CWP45/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  13. Eric Gautier & Alexandre B, Tsybakov, 2013. "Pivotal Estimation in High-Dimensional Regression via Linear Programming," Working Papers 2013-40, Centre de Recherche en Economie et Statistique.
  14. Olga Klopp, 2012. "Noisy Low-rank Matrix Completion with General Sampling Distribution," Working Papers 2012-06, Centre de Recherche en Economie et Statistique.

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