ℓ1-regularized linear regression: persistence and oracle inequalities

By Peter L. Bartlett, Shahar Mendelson, Joe Neeman. Probability theory and related fields, 2011

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Non-negative least squares for high-dimensional linear models: consistency and sparse recovery without regularization (2012, background): “The main result of the following section is a bound on the excess error of NNLS that resembles the so-called slow rate bound of the lasso [3, 8, 24].”
Assumptionless consistency of the Lasso (2013, background): “Incidentally, the above theorem is related to the notion of persistence defined in [17] and thoroughly investigated in [1].”
Prediction error of cross-validated Lasso (2015, background): “ive, especially since it has been observed in several recent papers that risk consistency in Lasso (with deterministic value of the tuning parameter) holds without any conditions on the design matrix [1, 3, 6, 27]. The goal of this paper is to address these issues and prove a new and better upper bo”
Learning without Concentration (2014, background): “To illustrate the clear advantages Theorem 3.1 has over Theorem 1.1, we will present one example in which α∗N and β ∗ N may be computed in a relatively straightforward way: the persistence problem in n1 [Greenshtein and Ritov 2004; Greenshtein 2006; Bartlett et al. 2012].”