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We give a novel, unified derivation of conditional PAC-Bayesian and mutualinformation (MI) generalization bounds. We derive conditional MI bounds as aninstance, with special choice of prior, of conditional MAC-Bayesian (MeanApproximately Correct) bounds, itself derived from conditional PAC-Bayesianbounds, where `conditional' means that one can use priors conditioned on ajoint training and ghost sample. This allows us to get nontrivial PAC-Bayes andMI-style bounds for general VC classes, something recently shown to beimpossible with standard PAC-Bayesian/MI bounds. Second, it allows us to getfaster 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 generalizationand 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 recentwork of Hellstr\"om and Durisi [2020] who extend the latter to the PAC-Bayessetting via a unifying exponential inequality, and Mhammedi et al. [2019] whoinitiated fast rate PAC-Bayes generalization error bounds but handle neither MInor general VC classes.

PAC-Bayes, MAC-Bayes and Conditional Mutual Information: Fast rate bounds that handle general VC classes