Concentration inequalities using the entropy method

• Stéphane Boucheron ^[3] ; Gábor Lugosi ^[1] ; Pascal Massart ^[2]
  1. [1] Universitat Pompeu Fabra

     Universitat Pompeu Fabra

     Barcelona, España

     
  2. [2] University of Paris-Sud

     University of Paris-Sud

     Arrondissement de Palaiseau, Francia

     
  3. [3] CNRS-Université Paris-Sud

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 3, 2003, págs. 1583-1614
• Idioma: inglés
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• Resumen

  • 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.