Improved bounds for square-root lasso and square-root slope

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8 Citations (Scopus)

Abstract

Extending the results of Bellec, Lecué and Tsybakov [1] to the setting of sparse high-dimensional linear regression with unknown variance, we show that two estimators, the Square-Root Lasso and the Square-Root Slope can achieve the optimal minimax prediction rate, which is (s/n) log (p/s), up to some constant, under some mild conditions on the design matrix. Here, n is the sample size, p is the dimension and s is the sparsity parameter. We also prove optimality for the estimation error in the lq-norm, with q ∈ [1, 2] for the Square-Root Lasso, and in the l2 and sorted l1 norms for the Square-Root Slope. Both estimators are adaptive to the unknown variance of the noise. The Square-Root Slope is also adaptive to the sparsity s of the true parameter. Next, we prove that any estimator depending on s which attains the minimax rate admits an adaptive to s version still attaining the same rate. We apply this result to the Square-root Lasso. Moreover, for both estimators, we obtain valid rates for a wide range of confidence levels, and improved concentration properties as in [1] where the case of known variance is treated. Our results are non-asymptotic.

Original languageEnglish
Pages (from-to)741-766
Number of pages26
JournalElectronic Journal of Statistics
Volume12
Issue number1
DOIs
Publication statusPublished - 2018
Externally publishedYes

Keywords

  • Adaptivity
  • High-dimensional statistics
  • Minimax rates
  • Sparse linear regression
  • Square-root estimators

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