  The developments of Rademacher complexity and PAC-Bayesian theory have been
largely independent. One exception is the PAC-Bayes theorem of Kakade,
Sridharan, and Tewari (2008), which is established via Rademacher complexity
theory by viewing Gibbs classifiers as linear operators. The goal of this paper
is to extend this bridge between Rademacher complexity and state-of-the-art
PAC-Bayesian theory. We first demonstrate that one can match the fast rate of
Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes
(Wegkamp, 2003; Lecu\'{e} and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018).
We then derive a new fast-rate PAC-Bayes bound in terms of the "flatness" of
the empirical risk surface on which the posterior concentrates. Our analysis
establishes a new framework for deriving fast-rate PAC-Bayes bounds and yields
new insights on PAC-Bayesian theory.
