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In this paper, we introduce a new concept of stability for cross-validation,called the $\left( \beta, \varpi \right)$-stability, and use it as a newperspective to build the general theory for cross-validation. The $\left(\beta, \varpi \right)$-stability mathematically connects the generalizationability and the stability of the cross-validated model via the Rademachercomplexity. Our result reveals mathematically the effect of cross-validationfrom two sides: on one hand, cross-validation picks the model with the bestempirical generalization ability by validating all the alternatives on testsets; on the other hand, cross-validation may compromise the stability of themodel selection by causing subsampling error. Moreover, the difference betweentraining and test errors in q\textsuperscript{th} round, sometimes referred toas the generalization error, might be autocorrelated on q. Guided by the ideasabove, the $\left( \beta, \varpi \right)$-stability help us derivd a new classof Rademacher bounds, referred to as the one-round/convoluted Rademacherbounds, for the stability of cross-validation in both the i.i.d.\ andnon-i.i.d.\ cases. For both light-tail and heavy-tail losses, the new boundsquantify the stability of the one-round/average test error of thecross-validated model in terms of its one-round/average training error, thesample sizes $n$, number of folds $K$, the tail property of the loss (encodedas Orlicz-$\Psi_\nu$ norms) and the Rademacher complexity of the model class$\Lambda$. The new class of bounds not only quantitatively reveals thestability of the generalization ability of the cross-validated model, it alsoshows empirically the optimal choice for number of folds $K$, at which theupper bound of the one-round/average test error is lowest, or, to put it inanother way, where the test error is most stable.

$\left( \beta, \varpi \right)$-stability for cross-validation and the choice of the number of folds