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


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


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.

Suggested Citation

  • A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
  • Handle: RePEc:oup:biomet:v:98:y:2011:i:4:p:791-806

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    References listed on IDEAS

    1. Gareth O. Roberts & Jeffrey S. Rosenthal, 1999. "Convergence of Slice Sampler Markov Chains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 643-660.
    2. Francesco Bartolucci & Luisa Scaccia & Antonietta Mira, 2006. "Efficient Bayes factor estimation from the reversible jump output," Biometrika, Biometrika Trust, vol. 93(1), pages 41-52, March.
    3. Sylvia Fruhwirth-Schnatter, 2004. "Estimating marginal likelihoods for mixture and Markov switching models using bridge sampling techniques," Econometrics Journal, Royal Economic Society, vol. 7(1), pages 143-167, June.
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