On the Convergence Rates of Policy Gradient Methods

We consider infinite-horizon discounted Markov decision problems with finite state and action spaces. We show that with direct parametrization in the policy space, the weighted value function, although non-convex in general, is both quasi-convex and quasi-concave. While quasi-convexity helps explain the convergence of policy gradient methods to global optima, quasi-concavity hints at their convergence guarantees using arbitrarily large step sizes that are not dictated by the Lipschitz constant charactering smoothness of the value function. In particular, we show that when using geometrically increasing step sizes, a general class of policy mirror descent methods, including the natural policy gradient method and a projected Q-descent method, all enjoy a linear rate of convergence without relying on entropy or other strongly convex regularization. In addition, we develop a theory of weak gradient-mapping dominance and use it to prove sharper sublinear convergence rate of the projected policy gradient method. Finally, we also analyze the convergence rate of an inexact policy mirror descent method and estimate its sample complexity under a simple generative model.