Symmetrization approach to concentration inequalities for empirical processes

Abstract

We introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional. As an application we give a new easy proof of Talagrand's concentration inequality for empirical processes, where besides symmetrization we use only Talagrand's concentration inequality on the discrete cube $\{0,1\}^n.$ As another application of this technique we prove new Vapnik--Chervonenkis type inequalities. For example, for VC-classes of functions we prove a classical inequality of Vapnik and Chervonenkis only with normalization by the sum of variance and sample variance.