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A recent line of works, initiated by Russo and Xu, has shown that thegeneralization error of a learning algorithm can be upper bounded byinformation measures. In most of the relevant works, the convergence rate ofthe expected generalization error is in the form of O(sqrt{lambda/n}) wherelambda is some information-theoretic quantities such as the mutual informationbetween the data sample and the learned hypothesis. However, such a learningrate is typically considered to be "slow", compared to a "fast rate" of O(1/n)in many learning scenarios. In this work, we first show that the square rootdoes not necessarily imply a slow rate, and a fast rate (O(1/n)) result canstill be obtained using this bound under appropriate assumptions. Furthermore,we identify the key conditions needed for the fast rate generalization error,which we call the (eta,c)-central condition. Under this condition, we giveinformation-theoretic bounds on the generalization error and excess risk, witha convergence rate of O(\lambda/{n}) for specific learning algorithms such asempirical risk minimization. Finally, analytical examples are given to show theeffectiveness of the bounds.

Fast Rate Generalization Error Bounds: Variations on a Theme