  PAC-Bayes is a useful framework for deriving generalization bounds which was
introduced by McAllester ('98). This framework has the flexibility of deriving
distribution- and algorithm-dependent bounds, which are often tighter than
VC-related uniform convergence bounds. In this manuscript we present a
limitation for the PAC-Bayes framework. We demonstrate an easy learning task
that is not amenable to a PAC-Bayes analysis.
  Specifically, we consider the task of linear classification in 1D; it is
well-known that this task is learnable using just $O(\log(1/\delta)/\epsilon)$
examples. On the other hand, we show that this fact can not be proved using a
PAC-Bayes analysis: for any algorithm that learns 1-dimensional linear
classifiers there exists a (realizable) distribution for which the PAC-Bayes
bound is arbitrarily large.

Machine Learning;Machine Learning