  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,...,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,...,X_n). We
study an empirical risk minimization problem $P_nf\to \min$, $f\in
\mathcal{F}$. Given a solution $\hat{f}_n$ of this problem, the goal is to
obtain very general upper bounds on its excess risk
\[\mathcal{E}_P(\hat{f}_n):=P\hat{f}_n-\inf_{f\in \mathcal{F}}Pf,\] expressed
in terms of relevant geometric parameters of the class $\mathcal{F}$. Using
concentration inequalities and other empirical processes 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.
