Concentration inequalities for functions of independent variables

Following the entropy method this paper presents general concentration inequalities, which can be applied to combinatorial optimization and empirical processes. The inequalities give improved concentration results for optimal travelling salesmen tours, Steiner trees and the eigenvalues of random symmetric matrices. 1 Introduction Since its appearance in 1995 Talagrand’s convex distance inequality [17] has been very successful as a tool to prove concentration results for functions of independent variables in cases which were previously inaccessible, or could be handled only with great di¢ culties. The now classical applications (see McDiarmid [12] and Steele [15]) include concentration inequalities for con…guration functions, such as the length of the longest increasing subsequence in a sample, or for geometrical constructions, such as the length of an optimal travelling salesman tour or an optimal Steiner tree. Another recently emerged technique to prove concentration results is the entropy method. Originating in the work of Leonard Gross on logarithmic Sobolev inequalities for Gaussian measures [6], the method has been developed and re…ned by Ledoux, Bobkov, Massart, Boucheron, Lugosi, Rio, Bousquet and others ( see [8], [10], [11], [2], [3], etc) to become an important tool in the study of empirical processes and learning theory. In [2, Boucheron at al] a general theorem on con…guration functions is presented, which improves on the results obtained from the convex distance inequality. In [3, Boucheron et al] more results of this type are given and a weak version of the convex distance inequality itself is derived. Technically the core of the entropy method is a tensorisation inequality bounding the entropy of an n variable function Z in terms of a sum of entropies