We derive an extension of McDiarmid’s inequality for functions f with bounded differences on a high probability set Y (instead of almost surely). The behavior of f outside Y may be arbitrary. The proof is short and elementary, and relies on an extension argument similar to Kirszbraun’s theorem [4].
We present a novel way of obtaining PAC-style bounds on the generalization error of learning algorithms, explicitly using their stability properties. A stable learner is one for which the learned solution does not change much with small changes in the training set. The bounds we obtain do not depend on any measure of the complexity of the hypothesis space (e.g. VC dimension) but rather depend on how the learning algorithm searches this space, and can thus be applied even when the VC dimension is infinite. We demonstrate that regularization networks possess the required stability property and apply our method to obtain new bounds on their generalization performance.  For a fixed probabilityp, 0<p<1, almost every random graphGn,p has chromatic number @math ,