  We present a unified framework for deriving PAC-Bayesian generalization
bounds. Unlike most previous literature on this topic, our bounds are
anytime-valid (i.e., time-uniform), meaning that they hold at all stopping
times, not only for a fixed sample size. Our approach combines four tools in
the following order: (a) nonnegative supermartingales or reverse
submartingales, (b) the method of mixtures, (c) the Donsker-Varadhan formula
(or other convex duality principles), and (d) Ville's inequality. Our main
result is a PAC-Bayes theorem which holds for a wide class of discrete
stochastic processes. We show how this result implies time-uniform versions of
well-known classical PAC-Bayes bounds, such as those of Seeger, McAllester,
Maurer, and Catoni, in addition to many recent bounds. We also present several
novel bounds. Our framework also enables us to relax traditional assumptions;
in particular, we consider nonstationary loss functions and non-i.i.d. data. In
sum, we unify the derivation of past bounds and ease the search for future
bounds: one may simply check if our supermartingale or submartingale conditions
are met and, if so, be guaranteed a (time-uniform) PAC-Bayes bound.
