  We propose a general theorem providing upper bounds for the risk of an
empirical risk minimizer (ERM).We essentially focus on the binary
classification framework. We extend Tsybakov's analysis of the risk of an ERM
under margin type conditions by using concentration inequalities for
conveniently weighted empirical processes. This allows us to deal with ways of
measuring the ``size'' of a class of classifiers other than entropy with
bracketing as in Tsybakov's work. In particular, we derive new risk bounds for
the ERM when the classification rules belong to some VC-class under margin
conditions and discuss the optimality of these bounds in a minimax sense.
