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There is accumulating evidence in the literature that stability of learningalgorithms is a key characteristic that permits a learning algorithm togeneralize. Despite various insightful results in this direction, there seemsto be an overlooked dichotomy in the type of stability-based generalizationbounds we have in the literature. On one hand, the literature seems to suggestthat exponential generalization bounds for the estimated risk, which areoptimal, can be only obtained through stringent, distribution independent andcomputationally intractable notions of stability such as uniform stability. Onthe other hand, it seems that weaker notions of stability such as hypothesisstability, although it is distribution dependent and more amenable tocomputation, can only yield polynomial generalization bounds for the estimatedrisk, which are suboptimal.  In this paper, we address the gap between these two regimes of results. Inparticular, the main question we address here is \emph{whether it is possibleto derive exponential generalization bounds for the estimated risk using anotion of stability that is computationally tractable and distributiondependent, but weaker than uniform stability. Using recent advances inconcentration inequalities, and using a notion of stability that is weaker thanuniform stability but distribution dependent and amenable to computation, wederive an exponential tail bound for the concentration of the estimated risk ofa hypothesis returned by a general learning rule, where the estimated risk isexpressed in terms of either the resubstitution estimate (empirical error), orthe deleted (or, leave-one-out) estimate. As an illustration, we deriveexponential tail bounds for ridge regression with unbounded responses, where weshow how stability changes with the tail behavior of the response variables.

An Exponential Efron-Stein Inequality for Lq Stable Learning Rules