In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for stable predictors in the context of risk assessment. The notion of stability has been first introduced by \cite{DEWA79} and extended by \cite{KEA95}, \cite{BE01} and \cite{KUNIY02} to characterize class of predictors with infinite VC dimension. In particular, this covers $k$-nearest neighbors rules, bayesian algorithm (\cite{KEA95}), boosting,... General loss functions and class of predictors are considered. We use the formalism introduced by \cite{DUD03} to cover a large variety of cross-validation procedures including leave-one-out cross-validation, $k$-fold cross-validation, hold-out cross-validation (or split sample), and the leave-$\upsilon$-out cross-validation. In particular, we give a simple rule on how to choose the cross-validation, depending on the stability of the class of predictors. In the special case of uniform stability, an interesting consequence is that the number of elements in the test set is not required to grow to infinity for the consistency of the cross-validation procedure. In this special case, the particular interest of leave-one-out cross-validation is emphasized.

In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for empirical risk minimizers. In the general setting, we prove sanity-check bounds in the spirit of \cite{KR99} \textquotedblleft\textit{bounds showing that the worst-case error of this estimate is not much worse that of training error estimate} \textquotedblright . General loss functions and class of predictors with finite VC-dimension are considered. We closely follow the formalism introduced by \cite{DUD03} to cover a large variety of cross-validation procedures including leave-one-out cross-validation, $k$% -fold cross-validation, hold-out cross-validation (or split sample), and the leave-$\upsilon$-out cross-validation. In particular, we focus on proving the consistency of the various cross-validation procedures. We point out the interest of each cross-validation procedure in terms of rate of convergence. An estimation curve with transition phases depending on the cross-validation procedure and not only on the percentage of observations in the test sample gives a simple rule on how to choose the cross-validation. An interesting consequence is that the size of the test sample is not required to grow to infinity for the consistency of the cross-validation procedure.