  PAC-Bayes has recently re-emerged as an effective theory with which one can
derive principled learning algorithms with tight performance guarantees.
However, applications of PAC-Bayes to bandit problems are relatively rare,
which is a great misfortune. Many decision-making problems in healthcare,
finance and natural sciences can be modelled as bandit problems. In many of
these applications, principled algorithms with strong performance guarantees
would be very much appreciated. This survey provides an overview of PAC-Bayes
bounds for bandit problems and an experimental comparison of these bounds. On
the one hand, we found that PAC-Bayes bounds are a useful tool for designing
offline bandit algorithms with performance guarantees. In our experiments, a
PAC-Bayesian offline contextual bandit algorithm was able to learn randomised
neural network polices with competitive expected reward and non-vacuous
performance guarantees. On the other hand, the PAC-Bayesian online bandit
algorithms that we tested had loose cumulative regret bounds. We conclude by
discussing some topics for future work on PAC-Bayesian bandit algorithms.
