  A recent line of works, initiated by Russo and Xu, 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 or conditional mutual information between the data and the learned
hypothesis. However, such a learning rate is typically considered to be
``slow", compared to a ``fast rate" of $O(\lambda/n)$ 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 result can still be obtained
using this bound under appropriate assumptions. Furthermore, we identify the
critical 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 fast convergence rate for specific learning algorithms such as empirical risk
minimization and its regularized version. Finally, several analytical examples
are given to show the effectiveness of the bounds.
