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

Author(s)
Bellini, A.; Chernozhukov, Victor V.; Wang, Lie
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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}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.
Date issued
2011-12
URI
http://hdl.handle.net/1721.1/71663
Journal
Biometrika
Publisher
Oxford University Press
Citation
Belloni, A., V. Chernozhukov, and L. Wang. "Square-root lasso: pivotal recovery of sparse signals via conic programming." Biometrika (2011) 98 (4): 791-806.
Version: Author's final manuscript
ISSN
1464-3510
0006-3444

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