Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming
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Victor11-16.pdf
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10bd5b2a8c6adc1efbc97925c726a2d4
Author(s)
Belloni, Alexandre
Chernozhukov, Victor
Wang, Lie
Date Issued
June 13, 2011
Publisher
Cambridge, MA: Department of Economics; Massachusetts Institute of Technology
Series/Report no.
Working paper (Massachusetts Institute of Technology, Department of Economics);11-16
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.
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