Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming

36 Pages Posted: 17 Aug 2011 Last revised: 9 Oct 2011

See all articles by Alexandre Belloni

Alexandre Belloni

Massachusetts Institute of Technology (MIT) - Operations Research Center

Victor Chernozhukov

Massachusetts Institute of Technology (MIT) - Department of Economics; New Economic School

Lie Wang

Massachusetts Institute of Technology (MIT), Department of Mathematics

Date Written: June 13, 2011

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 or 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}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.

Suggested Citation

Belloni, Alexandre and Chernozhukov, Victor and Wang, Lie, Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming (June 13, 2011). MIT Department of Economics Working Paper No. 11-16, Available at SSRN: https://ssrn.com/abstract=1910753 or http://dx.doi.org/10.2139/ssrn.1910753

Alexandre Belloni

Massachusetts Institute of Technology (MIT) - Operations Research Center ( email )

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Victor Chernozhukov (Contact Author)

Massachusetts Institute of Technology (MIT) - Department of Economics ( email )

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Lie Wang

Massachusetts Institute of Technology (MIT), Department of Mathematics ( email )

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