Rate Minimaxity of the Lasso and Dantzig Selector for the lq Loss in lr Balls

By Fei Ye, Cun-Hui Zhang. 2010

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Double/De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers (2018, background): “The cone invertibility factor is a generalization of the restricted eigenvalue condition of Bickel et al. (2009), proposed by Ye and Zhang (2010).”
Minimax risks for sparse regressions: Ultra-high-dimensional phenomenons (2010, unknown): “Φ2k,+( √ Σ) is close to zero. 5.2. Fixed design 5.2.1. Known variance The minimax prediction risk with known variance has been studied in Raskutti et al. [39] and Rigollet and Tsybakov [40] (see also [1, 47]). For any design X and any 1 ≤ k ≤ n, these authors have proved that the minimax risk RF [k,”
SLOPE is Adaptive to Unknown Sparsity and Asymptotically Minimax (2015, methodology): “This is already apparent when scanning the literature on penalized sparse estimation procedures such as the Lasso, SCAD [32], the Dantzig selector [21] and MC+ [58]; see, for example, [7, 11, 22, 23, 27, 37, 45, 53, 55, 57, 59] for a highly incomplete list of references.”
Scaled sparse linear regression (2011, methodology): “In addition, (21) yields sharper oracle inequalities (Ye & Zhang, 2010).”
Slope meets Lasso: Improved oracle bounds and optimality (2016, background): “We are interested here in non-asymptotic bounds and therefore the results in [23, 29] obtained in some asymptotics do not fit in our context.”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, background): “…condition (van de Geer and Bühlmann (2009)), the restricted eigenvalue condition (Bickel et al. (2009)) or the cone invertibility factors condition (Ye and Zhang (2010)) are automatically satisfied with probability 1 − c1 exp(−c2n), where constants c1 and c2 depend on the spectra of Cov(X) and σ.”
Robust regression via mutivariate regression depth (2017, background): “he order of κ. In contrast, Theorem 3.2 removes these two assumptions and the convergence rates hold uniformly for all β ∈ Θs. When ǫ = 0, the rates obtained in Theorem 3.2 are all minimax optimal by [37, 26, 34]. Though most lower bound results in the literature for sparse linear regression are for”
High-dimensional generalizations of asymmetric least squares regression and their applications (2016, methodology): “The estimation accuracy of the Lasso has been extensively studied in the literature; see, for example, Negahban et al. (2012) and Ye and Zhang (2010).”
Confidence intervals for low dimensional parameters in high dimensional linear models (2011, methodology): “It is worthwhile to note that F1.ξ, S/ κ2.ξ, S/ RE22.ξ, S/ .38/ always holds and lower bounds of these quantities can be expressed in terms of sparse eigenvalues (Ye and Zhang, 2010).”
High-dimensional simultaneous inference with the bootstrap (2016, background): “|)) is a consequence of (A1) and the 1 minimax rate of the Lasso (Ye and Zhang 2010), and upper bounds of the form max j ‖Z j‖2/|ZTj X j | = OP (n−1/2), implying the requirement in Theorem 3 and the uniform n−1/2 rate for the standard error of b̂ j , can be found in Zhang and Zhang (2014) and van…”
High-dimensional instrumental variables regression and confidence sets (2011, background): “r own instruments and not necessarily the actual set of exogenous potential regressors) but for brevity we do not make the dependence explicit. Similar but diﬀerent quantities have been introduced in Ye and Zhang (2010) under the name of cone invertibility factors. They diﬀer from the sensitivities ”
Causal Dantzig: Fast inference in linear structural equation models with hidden variables under additive interventions (2017, methodology): “Similarly, as in Ye and Zhang (2010), define z∗ := ‖Ẑ − Ĝβ0‖∞ and let S denote the active set of β0.”
High-dimensional regression with unknown variance (2011, background): “The minimax lower bound (6) has been first proved in [61, 62, 79] while (7) is stated in [77].”