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

• 成果类型: Article
• 署名作者: Belloni, A.; Chernozhukov, V.; Wang, L.
• 署名单位: Duke University; Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT)
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asr043
• 发表日期: 2011
• 页码: 791806
• 关键词: selection; regression; inequalities; Consistency

摘要

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 Sigma nor does it need to pre-estimate Sigma. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate Sigma{(s/n) log p}(1/2) in the prediction norm, and thus matching the performance of the lasso with known Sigma. 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.