Fast Rate Generalization Error Bounds: Variations on a Theme

A recent line of works, initiated by \cite{russo2016controlling} and \cite{xu2017information}, has shown that the generalization error of a learning algorithm can be upper bounded by information measures. In most of the relevant works, the convergence rate of the expected generalization error is in the form of O(\sqrt{\lambda/{n}}) where \lambda is some information-theoretic quantities such as the mutual information between the data sample and the learned hypothesis. However, such a learning rate 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 root does not necessarily imply a slow rate, and a fast rate (O(1/n)) result can still 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 give information-theoretic bounds on the generalization error and excess risk, with a convergence rate of O(\lambda/{n}) for specific learning algorithms such as empirical risk minimization. Finally, analytical examples are given to show the effectiveness of the bounds.