## Local Rademacher complexities and oracle inequalities in risk minimization. (2004 IMS Medallion Lecture). (With discussions and rejoinder).(English)Zbl 1118.62065

Summary: Let $${\mathcal F}$$ be a class of measurable functions $$f:S\mapsto[0,1]$$ defined on a probability space $$(S,{\mathcal A},P)$$. Given a sample $$(X_1, \dots,X_n)$$ of i.i.d. random variables taking values in $$S$$ with common distribution $$P$$, let $$P_n$$ denote the empirical measure based on $$(X_1, \dots,X_n)$$. We study an empirical risk minimization problem $$P_nf\to \min$$, $$f\in{\mathcal F}$$. Given a solution $$\widehat f_n$$ of this problem, the goal is to obtain very general upper bounds on its excess risk ${\mathcal E}_P(\widehat f_n):=P\widehat f_n-\inf_{f\in{\mathcal F}}Pf,$ expressed in terms of relevant geometric parameters of the class $${\mathcal F}$$. Using concentration inequalities and other empirical process tools, we obtain both distribution-dependent and data-dependent upper bounds on the excess risk that are of asymptotically correct order in many examples. The bounds involve localized sup-norms of empirical and Rademacher processes indexed by functions from the class. We use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification.

### MSC:

 62H30 Classification and discrimination; cluster analysis (statistical aspects) 60B99 Probability theory on algebraic and topological structures 68Q32 Computational learning theory 62G08 Nonparametric regression and quantile regression 68T05 Learning and adaptive systems in artificial intelligence 68T10 Pattern recognition, speech recognition
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