Direct policy gradient methods for reinforcement learning and continuous control problems are a popular approach for a variety of reasons: 1) they are easy to implement without explicit knowledge of the underlying model 2) they are an "end-to-end" approach, directly optimizing the performance metric of interest 3) they inherently allow for richly parameterized policies. A notable drawback is that even in the most basic continuous control problem (that of linear quadratic regulators), these methods must solve a non-convex optimization problem, where little is understood about their efficiency from both computational and statistical perspectives. In contrast, system identification and model based planning in optimal control theory have a much more solid theoretical footing, where much is known with regards to their computational and statistical properties. This work bridges this gap showing that (model free) policy gradient methods globally converge to the optimal solution and are efficient (polynomially so in relevant problem dependent quantities) with regards to their sample and computational complexities.
The paper studies the global convergence for policy gradient methods for linear control problems. 
(1) The topic of this paper seems to have minimal connection with ICRL. It might be more appropriate for this paper to be reviewed at a control/optimization conference, so that all the technical analysis can be evaluated carefully. 
(2) I am not convinced if the main results are novel. The convergence of policy gradient does not rely on the convexity of the loss function, which is known in the community of control and dynamic programming. The convergence of policy gradient is related to the convergence of actor-critic, which is essentially a form of policy iteration. I am not sure if it is a good idea to examine the convergence purely from an optimization perspective.
(3) The main results of this paper seem technical sound. However, the results seem a bit limited because it does not apply to neural-network function approximator. It does not apply to the more general control problem rather than quadratic cost function, which is quite restricted. I might have missed something here. I strongly suggest that these results be submitted to a more suitable venue.