Sparsity in penalized empirical risk minimization

By Vladimir Koltchinskii. 2009

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Citations in the corpus

Smoothness, Low Noise and Fast Rates (2010, background): “Bounds similar to (7) have been derived using specialized arguments [12, 30, 5]—here we demonstrate that bounds of these forms can be obtained under simple conditions, using the generic framework we suggest.”
SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR (2008, methodology): “Modified versions of Lasso estimators (nonquadratic terms and/or penalties slightly different from 1) for nonparametric regression with random design are suggested and studied under prediction loss in [14] and [25].”
Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming (2010, background): “odels. Ann. Statist., 36(2):587613, 2008. [9] B. Y. Jing, Q. M. Shao, and Q. Y. Wang. Self-normalized Cram´er type large deviations for independent random variables. Ann. Probab., 31:2167–2215, 2003. [10] V. Koltchinskii. Sparsity in penalized empirical risk minimization. Ann. Inst. H. Poincar Proba”
On the conditions used to prove oracle results for the Lasso (2009, background): “[12] and [2] show that φ(L, S, 2s) ≥ (1 − Lθweak−RIP(S, 2s))Λ(S, 2s).”
PERFORMANCE GUARANTEES FOR INDIVIDUALIZED TREATMENT RULES (2011, methodology): “With regards quadratic loss, Koltchinskii [15] requires the response Y is bounded, while both Bunea, Tsybakov and Wegkamp [4] and van de Geer [33] assumed the existence of a sparse θ ∈ R such that E( θ − Q0)(2) is upper bounded by a quantity that decreases to 0 at a certain rate as n → ∞ (by permitt”
ℓ[subscript 1]-penalized quantile regression in high-dimensional sparse models (2011, background): “These results therefore contribute to the rate results obtained for l1-penalized mean regression and related estimators in the fundamental papers of [7, 12, 23, 29, 36, 38].”
Sparsity oracle inequalities for the Lasso (2007, background): “…to logarithmic factor) rates of convergence of f on the Besov classes without establishing oracle inequalities. b Finally we mention the papers [17, 18, 27] that analyze in the same spirit as we do below the sparsity issue for estimators that diﬀer from λ pen in that the goodness-of-ﬁt term in the…”
On the asymptotic properties of the group lasso estimator for linear models (2008, background): “…Bühlmann, 2006; Wainwright, 2006, 2007; Zhao and Yu, 2006), consistency and oracle properties (see, e.g. Meinshausen and Yu, 2006; Bickel et al., 2007; Bunea et al., 2007a,b; Zhang, 2007; Koltchinskii, 2005; Zhang and Huang, 2007), and persistence (Greenshtein and Ritov, 2006; Greenshtein, 2006).”
The Dantzig selector and sparsity oracle inequalities (2009, methodology): “We will establish several “sparsity oracle inequalities” for the Dantzig selector that are akin to recent inequalities proved in Bunea, Tsybakov and Wegkamp (2007), van de Geer (2008) and Koltchinskii (2009) in the case of ℓ1- or ℓp-penalized empirical risk minimization.”
Oracle inequalities in empirical risk minimization and sparse recovery problems (2011, background): “We will prove several ”sparsity oracle inequalities” for the Dantzig selector in spirit of recent results of Bunea, Tsybakov and Wegkamp [25], van de Geer [46], Koltchinskii [65] in the case of l1- or lp-penalized empirical risk minimization.”
Exponential Screening and optimal rates of sparse estimation (2010, methodology): “…procedures in terms of the ℓ 0 norm (number of non-zero components) of the estimated vectors or their approximations (see Bickel et al., 2009; Bunea et al., 2007a,b; Candes and Tao, 2007; Koltchinskii, 2008, 2009a,b; van de Geer, 2008; Zhang and Huang, 2008; Zhang, 2009, and references therein).”
The Lasso, correlated design, and improved oracle inequalities (2011, background): “See also [8], and see [13] for a discussion of the relation between restricted eigenvalues and compatibility.”
Some sharp performance bounds for least squares regression with L1 regularization (2009, methodology): “In addition to the above work, prediction error bounds were also obtained in [13, 14] and [11] for general loss functions and random design.”
Aggregation by exponential weighting, sharp PAC-Bayesian bounds and sparsity (2008, methodology): “The oracle inequality of Theorem 6 can be compared with the analogous SOI obtained for the Lasso and Dantzig selector under deterministic design [6, 3]. Similar oracle inequalities for the case of random design x1,...,xn can be found in [7, 34, 23 ].”
Adaptive Forward-Backward Greedy Algorithm for Sparse Learning with Linear Models (2008, methodology): “The prediction performance of this method has been considered in [6, 2, 1, 9].”
Statistical significance in high-dimensional linear models (2012, background): “ taking the minimum on the right-handside over all p 1 vectors , we replace it by a constrained minimum, typically over a cone. This leads to the concept of restricted eigenvalues (Bickel et al.,2009;Koltchinskii, 2009a,b;Raskutti et al.,2010) or weaker forms such as the compatibility constants (van”
High-dimensional additive hazards models and the Lasso (2011, background): “For the proof of oracle inequalities with a fast logM/n rate, the restricted eigenvalue (RE) condition introduced in [5] and [15, 16] is of importance.”
SPARSITY IN MULTIPLE KERNEL LEARNING (2010, methodology): “The following simple proposition easily follows from some statements in Koltchinskii (2009a, 2009b), (2008) (where the case of simple dictionaries consisting of one-dimensional spaces H j was considered).”
Some theoretical results on the Grouped Variables Lasso (2008, unknown): “AMS 2000 Subject Classifications: Primary: 62J05, 62J07, Secondary: 62H20, 62F30.”