  We consider a policy gradient algorithm applied to a finite-arm bandit
problem with Bernoulli rewards. We allow learning rates to depend on the
current state of the algorithm, rather than use a deterministic time-decreasing
learning rate. The state of the algorithm forms a Markov chain on the
probability simplex. We apply Foster-Lyapunov techniques to analyse the
stability of this Markov chain. We prove that if learning rates are well chosen
then the policy gradient algorithm is a transient Markov chain and the state of
the chain converges on the optimal arm with logarithmic or poly-logarithmic
regret.
