  We give a novel, unified derivation of conditional PAC-Bayesian and mutual
information (MI) generalization bounds. We derive conditional MI bounds as an
instance, with special choice of prior, of conditional MAC-Bayesian (Mean
Approximately Correct) bounds, itself derived from conditional PAC-Bayesian
bounds, where `conditional' means that one can use priors conditioned on a
joint training and ghost sample. This allows us to get nontrivial PAC-Bayes and
MI-style bounds for general VC classes, something recently shown to be
impossible with standard PAC-Bayesian/MI bounds. Second, it allows us to get
faster rates of order $O \left(({\text{KL}}/n)^{\gamma}\right)$ for $\gamma >
1/2$ if a Bernstein condition holds and for exp-concave losses (with
$\gamma=1$), which is impossible with both standard PAC-Bayes generalization
and MI bounds. Our work extends the recent work by Steinke and Zakynthinou
[2020] who handle MI with VC but neither PAC-Bayes nor fast rates, the recent
work of Hellstr\"om and Durisi [2020] who extend the latter to the PAC-Bayes
setting via a unifying exponential inequality, and Mhammedi et al. [2019] who
initiated fast rate PAC-Bayes generalization error bounds but handle neither MI
nor general VC classes.
