  We present a unifying picture of PAC-Bayesian and mutual information-based
upper bounds on the generalization error of randomized learning algorithms. As
we show, Tong Zhang's information exponential inequality (IEI) gives a general
recipe for constructing bounds of both flavors. We show that several important
results in the literature can be obtained as simple corollaries of the IEI
under different assumptions on the loss function. Moreover, we obtain new
bounds for data-dependent priors and unbounded loss functions. Optimizing the
bounds gives rise to variants of the Gibbs algorithm, for which we discuss two
practical examples for learning with neural networks, namely, Entropy- and
PAC-Bayes- SGD. Further, we use an Occam's factor argument to show a
PAC-Bayesian bound that incorporates second-order curvature information of the
training loss.
