  We conduct a non asymptotic study of the Cross Validation (CV) estimate of
the generalization risk for learning algorithms dedicated to extreme regions of
the covariates space. In this Extreme Value Analysis context, the risk function
measures the algorithm's error given that the norm of the input exceeds a high
quantile. The main challenge within this framework is the negligible size of
the extreme training sample with respect to the full sample size and the
necessity to re-scale the risk function by a probability tending to zero. We
open the road to a finite sample understanding of CV for extreme values by
establishing two new results: an exponential probability bound on the \Kfold CV
error and a polynomial probability bound on the leave-\textrm{p}-out CV. Our
bounds are sharp in the sense that they match state-of-the-art guarantees for
standard CV estimates while extending them to encompass a conditioning event of
small probability. We illustrate the significance of our results regarding high
dimensional classification in extreme regions via a Lasso-type logistic
regression algorithm. The tightness of our bounds is investigated in numerical
experiments.
