Concentration around the mean for maxima of empirical processes

By Thierry Klein, E. Rio. 2005

256 citations recorded by the source corpus.

In this paper we give optimal constants in Talagrand's concentration inequalities for maxima of empirical processes associated to independent and eventually nonidentically distributed random variables. Our approach is based on the entropy method introduced by Ledoux.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/bdacefd229c98c1246072b81810291d34e9ca79d

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Citations in the corpus

Oracle inequalities in empirical risk minimization and sparse recovery problems (2011, methodology): “In particular, we will frequently use the bounds proved by Bousquet [22] and Klein [54] (in fact, Klein and Rio [55] provide an improved version of this inequality).”
A No-Free-Lunch Theorem for MultiTask Learning (2020, methodology): “tically distributed data in Appendix A. The main technical modiﬁcation of the proof, compared to the i.i.d. case, is employing a generalization of Bousquet’s inequality for non-identical data, due to [KR05]. Lemma 1. (Uniform Bernstein inequality for non-identical distributions) For any m2N and 2(0;”
Nonparametric estimation for pure jump Lévy processes based on high frequency data. (2009, background): “Note that for p = 1, (12) is a simple variance inequality: (13) ∆E(|θ̂∆(u) − θ∆(u)|(2)) ≤ 1 n∆ (m2 + ∆m 2 1) = 1 n∆2 E(Z(2) 1 ).”
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression (2018, methodology): “Under the condition Ξ (M) n,|S| = o(1), for large enough n, Γn(S) ≥ λmin(Σn)/2. The constants could possibly be improved here. This is an oracle inequality because there is no assumption on β0 and the bound adapts to the true sparsity of β0. The proof can be found in Section S.4.4 (Proposition S.4.1”
On the Bennett–Hoeffding inequality (2009, unknown): “Contents Contents 1”
Adaptive functional linear regression (2011, methodology): “Moreover, the key argument used in the proof of Lemma D.3 is the following inequality due to Talagrand [1996] (see e.g. Klein and Rio [2005]).”
Adaptive Laguerre density estimation for mixed Poisson models (2015, unknown): “Consider the estimator f̂l̂ defined by (10) and (13).”
On concentration of self-bounding functions (2009, methodology): “The proofs, just like for the abovementioned inequalities, is based on the entropy method pioneered by Ledoux[8] and further developed, among others, by Boucheron, Lugosi, Massart[3], Boucheron, Bousquet, Lugosi, Massart[1], Bousquet[4], Klein[6], Massart[10], Rio[16], Klein and Rio[ 7 ].”
Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications (2016, methodology): “An alternative simple data splitting approach by reducing the U-statistics to sums of iid random vectors can give the exact rate for bounding the moments in the non-degenerate case [69, 53, 43, 29].”
Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections (2008, methodology): “We need to use Talagrand’s exponential inequality with sharp constants [3,21] in the proofs, but to do this, we have to estimate the expectation of suprema of certain empirical processes that appear in the centering of Talagrand’s inequality.”
Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles (2009, background): “The version from [13], presented below, provides the best-known constants in this inequality (we will however not take advantage of explicit constants).”
Exact Matrix Completion via Convex Optimization (2008, background): “We note that very precise values of the numerical constant K are known and are small; see [21].”
Adaptive density estimation: a curse of support? ✩ (2009, methodology): “1 of Houdré and Reynaud-Bouret (2003) combined with the appropriate choice of constants derived by Klein and Rio (2005): for any u > 0 and any τ > 0,”
A tail inequality for suprema of unbounded empirical processes with applications to Markov chains (2007, result): “Theorem 3 (Klein, Rio, [7], Theorems 1.”
Compressive Sensing with structured random matrices (2012, background): “The constants in the deviation and concentration inequalities were successfully improved in [85, 107, 108, 14, 77].”