Regret Analysis of a Markov Policy Gradient Algorithm for Multi-arm Bandits

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 using 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 analyze 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 polylogarithmic regret.