A Bennett concentration inequality and its application to suprema of empirical processes

By Olivier Bousquet. 2002

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Subfields: Applied Mathematics, Statistics and Probability

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Citations in the corpus

On the optimal estimation of probability measures in weak and strong topologies (2013, neutral): “√ 2n2α+ 6 √ 2en θ Zsup g ,g2∈G ρ(g 1,g 2) α log N(G,ρ,ǫ) √ 1−θ dǫ. Since αis arbitrary, taking inﬁmum over α>0 yields the result. The following version of Talagrand’s inequality is due to Bousquet [7], which we quote from [37, Theorem 7.5]. Theorem A.3 (Talagrand’s Inequality). Let (X,P) be a pro”
Variance-based Regularization with Convex Objectives (2016, methodology): “ Rsup n (F) 2 log3 nM+ M2 n (t+loglogn) with probability at least 1−e−t. Next, we give an upper bound for (EPb n (f−E[f]))2. We use the following version of Talagrand’s inequality due to Bousquet [12, 14]. (See also Bartlett et al. [3, Thm 2.1].) Lemma B.2. Let r>0 and F be a class of functions”
Local Rademacher complexities (2005, methodology): “ for the sake of completeness. 34 P. L. BARTLETT, O. BOUSQUET AND S. MENDELSON Recall the following improvement of Rio’s [29] version of Talagrand’s concentration inequality, which is due to Bousquet [7, 8]. Theorem A.1. Let c>0, let Xi be independent random variables distributed according to Pan”
High-dimensional additive modeling (2008, background): “Theorem 5. (Bousquet’s Concentration Theorem [ 4 ]) Suppose”
Efficient Policy Learning (2017, methodology): “, Bartlett, Bousquet, and Mendelson (2005) and Giné and Koltchinskii (2006).”
Asymptotic analysis of high-dimensional LAD regression with Lasso (2016, methodology): “A popular approach is the Lasso (Tibshirani (1996)) that uses the `1 penalty (Chen and Donoho (1995)).”
Concentration Inequalities (2008, methodology): “In this section we focus on so-called “entropy method”, based on logarithmic Sobolev inequalities developed by Ledoux [20, 21], see also Bobkov and Ledoux [22], Massart [23], Rio [10], Boucheron, Lugosi, and Massart [25], [26], and Bousquet [27, 28].”
Estimation of Censored Quantile Regression for Panel Data With Fixed Effects (2013, methodology): “To this end, we make use of Bousquet’s version of Talagrand’s inequality (see Bousquet 2002).”
Risk bounds for statistical learning (2007, methodology): “More precisely, we shall use the following version of it due to Bousquet [4] which has the advantage of providing explicit constants and of dealing with one-sided suprema.”
Convexity, Classification, and Risk Bounds (2006, background): “The best estimates on the constants are due to Bousquet (2002).”
Direct importance estimation for covariate shift adaptation (2008, background): “For a class F of measurable functions such that ∀f ∈ F , ∥f∥∞ ≤ 1, the following bound holds, which we refer to as the Bousquet bound (Bousquet, 2002):”
HIGH-DIMENSIONAL GENERALIZED LINEAR MODELS AND THE LASSO (2008, methodology): “The proof of the main theorems is based on the concentration inequality of Bousquet (2002).”
Theory of classification : a survey of some recent advances (2005, methodology): “A general result, applicable under general noise conditions, was proven by Bartlett, Bousquet and Mendelson [25].”
Policy Learning With Observational Data (2017, methodology): “, Bartlett, Bousquet, and Mendelson (2005) and Giné and Koltchinskii (2006).”