Risk bounds for Statistical Learning ( Running Title : 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 classication framework. We extend Tsybakovs 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 sizeof a class of classiers than entropy with bracketing as in Tsybakovs work. In particular we derive new risk bounds for the ERM when the classication rules belong to some VC-class under margin conditions and discuss the optimality of those bounds in a minimax sense.