Slope meets lasso: Improved oracle bounds and optimality

Pierre C. Bellec, Guillaume Lecué, Alexandre B. Tsybakov

Research output: Contribution to journalArticlepeer-review

30 Scopus citations


We show that two polynomial time methods, a Lasso estimator with adaptively chosen tuning parameter and a Slope estimator, adaptively achieve the minimax prediction and2 estimation rate (s/n)log(p/s) in high-dimensional linear regression on the class of s-sparse vectors in Rp. This is done under the Restricted Eigenvalue (RE) condition for the Lasso and under a slightly more constraining assumption on the design for the Slope. The main results have the form of sharp oracle inequalities accounting for the model misspecification error. The minimax optimal bounds are also obtained for theq estimation errors with 1 ≤ q ≤ 2 when the model is well specified. The results are nonasymptotic, and hold both in probability and in expectation. The assumptions that we impose on the design are satisfied with high probability for a large class of random matrices with independent and possibly anisotropically distributed rows. We give a comparative analysis of conditions, under which oracle bounds for the Lasso and Slope estimators can be obtained. In particular, we show that several known conditions, such as the RE condition and the sparse eigenvalue condition are equivalent if the 2-norms of regressors are uniformly bounded.

Original languageEnglish (US)
Pages (from-to)3603-3642
Number of pages40
JournalAnnals of Statistics
Issue number6B
StatePublished - 2018

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • High-dimensional statistics
  • Lasso
  • Minimax rates
  • Slope
  • Sparse linear regression


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