  We present a general approach, based on exponential inequalities, to derive
bounds on the generalization error of randomized learning algorithms. Using
this approach, we provide bounds on the average generalization error as well as
bounds on its tail probability, for both the PAC-Bayesian and single-draw
scenarios. Specifically, for the case of sub-Gaussian loss functions, we obtain
novel bounds that depend on the information density between the training data
and the output hypothesis. When suitably weakened, these bounds recover many of
the information-theoretic bounds available in the literature. We also extend
the proposed exponential-inequality approach to the setting recently introduced
by Steinke and Zakynthinou (2020), where the learning algorithm depends on a
randomly selected subset of the available training data. For this setup, we
present bounds for bounded loss functions in terms of the conditional
information density between the output hypothesis and the random variable
determining the subset choice, given all training data. Through our approach,
we recover the average generalization bound presented by Steinke and
Zakynthinou (2020) and extend it to the PAC-Bayesian and single-draw scenarios.
For the single-draw scenario, we also obtain novel bounds in terms of the
conditional $\alpha$-mutual information and the conditional maximal leakage.
