## Risk bounds for statistical learning.(English)Zbl 1108.62007

Summary: 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 A. B. Tsybakov’s [ibid. 32, No. 1, 135–166 (2004; Zbl 1105.62353)] 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 Vapuis-Chervonenkis class under margin conditions and discuss the optimality of these bounds in a minimax sense.

### MSC:

 62B10 Statistical aspects of information-theoretic topics 62H30 Classification and discrimination; cluster analysis (statistical aspects) 68T05 Learning and adaptive systems in artificial intelligence 60E15 Inequalities; stochastic orderings 94A17 Measures of information, entropy 62F15 Bayesian inference

Zbl 1105.62353
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