  Actor-critic style two-time-scale algorithms are one of the most popular
methods in reinforcement learning, and have seen great empirical success.
However, their performance is not completely understood theoretically. In this
paper, we characterize the \emph{global} convergence of an online natural
actor-critic algorithm in the tabular setting using a single trajectory of
samples. Our analysis applies to very general settings, as we only assume
ergodicity of the underlying Markov decision process. In order to ensure enough
exploration, we employ an $\epsilon$-greedy sampling of the trajectory.
  For a fixed and small enough exploration parameter $\epsilon$, we show that
the two-time-scale natural actor-critic algorithm has a rate of convergence of
$\tilde{\mathcal{O}}(1/T^{1/4})$, where $T$ is the number of samples, and this
leads to a sample complexity of $\Tilde{\mathcal{O}}(1/\delta^{8})$ samples to
find a policy that is within an error of $\delta$ from the \emph{global
optimum}. Moreover, by carefully decreasing the exploration parameter
$\epsilon$ as the iterations proceed, we present an improved sample complexity
of $\Tilde{\mathcal{O}}(1/\delta^{6})$ for convergence to the global optimum.
