Concentration Inequalities St´ephane Boucheron1, G´abor Lugosi2, and Olivier Bousquet3 1 Universit´e de Paris-Sud, Laboratoire d'Informatique B^atiment 490, F-91405 Orsay Cedex, France
[email protected] WWW home page: http://www.lri.fr/~bouchero 2 Department of Economics, Pompeu Fabra University Ramon Trias Fargas 25-27, 08005 Barcelona, Spain
[email protected] WWW home page: http://www.econ.upf.es/~lugosi 3 Max-Planck Institute for Biological Cybernetics Spemannstr. 38, D-72076 Tubingen,¨ Germany
[email protected] WWW home page: http://www.kyb.mpg.de/~bousquet Abstract. Concentration inequalities deal with deviations of functions of independent random variables from their expectation. In the last decade new tools have been introduced making it possible to establish simple and powerful inequalities. These inequalities are at the heart of the mathematical analysis of various problems in machine learning and made it possible to derive new efficient algorithms. This text attempts to summarize some of the basic tools. 1 Introduction The laws of large numbers of classical probability theory state that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. More recent results reveal that such a behavior is shared by a large class of general functions of independent random variables. The purpose of these notes is to give an introduction to some of these general concentration inequalities. The inequalities discussed in these notes bound tail probabilities of general functions of independent random variables.