Open Access
July 2003 Concentration inequalities using the entropy method
Stéphane Boucheron, Gábor Lugosi, Pascal Massart
Ann. Probab. 31(3): 1583-1614 (July 2003). DOI: 10.1214/aop/1055425791


We investigate a new methodology, worked out by Ledoux and Massart, to prove concentration-of-measure inequalities. The method is based on certain modified logarithmic Sobolev inequalities. We provide some very simple and general ready-to-use inequalities. One of these inequalities may be considered as an exponential version of the Efron--Stein inequality. The main purpose of this paper is to point out the simplicity and the generality of the approach. We show how the new method can recover many of Talagrand's revolutionary inequalities and provide new applications in a variety of problems including Rademacher averages, Rademacher chaos, the number of certain small subgraphs in a random graph, and the minimum of the empirical risk in some statistical estimation problems.


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Stéphane Boucheron. Gábor Lugosi. Pascal Massart. "Concentration inequalities using the entropy method." Ann. Probab. 31 (3) 1583 - 1614, July 2003.


Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1051.60020
MathSciNet: MR1989444
Digital Object Identifier: 10.1214/aop/1055425791

Primary: 28A35 , 60C05 , 60E15
Secondary: 05C80

Keywords: Concentration inequalities , Empirical processes , random graphs.

Rights: Copyright © 2003 Institute of Mathematical Statistics

Vol.31 • No. 3 • July 2003
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