  The statistical learning problem consists in building a predictive function
$\hat{f}$ based on independent copies of $(X,Y)$ so that $Y$ is approximated by
$\hat{f}(X)$ with minimum (squared) error. Motivated by various applications,
special attention is paid here to the case of extreme (i.e. very large)
observations $X$. Because of their rarity, the contributions of such
observations to the (empirical) error is negligible, and the predictive
performance of empirical risk minimizers can be consequently very poor in
extreme regions. In this paper, we develop a general framework for regression
on extremes. Under appropriate regular variation assumptions regarding the pair
$(X,Y)$, we show that an asymptotic notion of risk can be tailored to summarize
appropriately predictive performance in extreme regions. It is also proved that
minimization of an empirical and nonasymptotic version of this 'extreme risk',
based on a fraction of the largest observations solely, yields good
generalization capacity. In addition, numerical results providing strong
empirical evidence of the relevance of the approach proposed are displayed.
