A Diffusion Analysis of Policy Gradient for Stochastic Bandits

We study a continuous-time diffusion approximation of policy gradient for $k$-armed stochastic bandits. We prove that with a learning rate $\eta = O(\Delta^2/\log(n))$ the regret is $O(k \log(k) \log(n) / \eta)$ where $n$ is the horizon and $\Delta$ the minimum gap. Moreover, we construct an instance with only logarithmically many arms for which the regret is linear unless $\eta = O(\Delta^2)$.