Risk bounds for statistical learning
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 other ways of measuring the size of a class of classifiers 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 those bounds in a minimax sense.