  We consider infinite-horizon discounted Markov decision problems with finite
state and action spaces and study the convergence rates of the projected policy
gradient method and a general class of policy mirror descent methods, all with
direct parametrization in the policy space. First, we develop a theory of weak
gradient-mapping dominance and use it to prove sharper sublinear convergence
rate of the projected policy gradient method. Then we show that with
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. Finally, we also analyze the convergence
rate of an inexact policy mirror descent method and estimate its sample
complexity under a simple generative model.
