Paper:
Policy gradient methods with actor-critic schemes demonstrate tremendous empirical successes, especially when the actors and critics are parameterized by neural networks. However, it remains less clear whether such "neural" policy gradient methods converge to globally optimal policies and whether they even converge at all. We answer both the questions affirmatively under the overparameterized twolayer neural-network parameterization. In detail, assuming independent sampling, we prove that neural natural policy gradient converges to a globally optimal policy at a sublinear rate. Also, we show that neural vanilla policy gradient converges sublinearly to a stationary point. Meanwhile, by relating the suboptimality of the stationary points to the representation power of neural actor and critic classes, we prove the global optimality of all stationary points under mild regularity conditions. Particularly, we show that a key to the global optimality and convergence is the "compatibility" between the actor and critic, which is ensured by sharing neural architectures and random initializations across the actor and critic. To the best of our knowledge, our analysis establishes the first global optimality and convergence guarantees for neural policy gradient methods.

Review:
This paper studies policy gradient where the policy is parameterized by an extremely wide neural network. The authors assume that the number of nodes ($m$) in the network is extremely large ($T^{8}R^{18}$, $T$ is the total runtime and $R$ is the radius of the function class that the network falls into), and they restrict their convergence analysis of the policy gradient algorithm in a particular function class and claims that the approximation error between the true network and the function class goes to zero when $m$ is large. The paper is written in a rigorous way and the presentation is mostly clear. I have some concerns about many of the assumptions made across the paper that are not explained or verified. This may potentially further decrease the usefulness of the analysis in this paper though the theoretical result with $m=T^8$ is already very impractical. Another major issue with this paper is that the theoretical analysis is not novel in terms of bringing new insights and results to the field given many other papers including global convergence of policy gradient (Agarwal et al., 2019;), convergence of neural TD learning (Cai et el., 2019), theory for overparameterized neural networks (Jacot et al., 2018; Allen-Zhu et al., 2018a;b; Du et al., 2018a;b; Zou et al., 2018; Chizat and Bach, 2018; Jacot et al., 2018, etc.) and other very similar papers. 
There is a prior work by Agarwal et al. (2019) that proves the global convergence of both vanilla policy gradient and natural policy gradient methods. In the related work, the authors distinguish their paper from that of Agarwal et al. (2019) by claiming that they are studying the non-tabular setting and they use the actor-critic scheme. However, the first claim is incorrect because Agarwal et al. (2019) also studied the non-tabular setting (see Section 6 in Agarwal et al. (2019)) and proved an O(1/\sqrt{T}) convergence rate. Moreover, the actor-critic scheme in this paper is just a trivial modification of the nonlinear policy gradient method by calling existing result for TD learning in Cai et al. (2019). Therefore, the contribution of this paper is not so clear given existing papers.
In Algorithm 1, the policy gradient estimator $\widehat{\nabla} J(\pi_{\theta_i})$ also depends on the critic parameter $\omega_i$. It is better to show this dependency in the notation as well.
In Algorithm 1, the temperature parameter $\tau_i$ is updated in natural policy gradient but not in vanilla policy gradient. It seems that $\tau_i$ increases linearly with $i$, which makes the policy defined in eq (3.1) close to a uniform distribution when the time horizon goes to infinity. This seems to offset the update of parameter $\theta_i$.
In the update of natural policy gradient, solving eq (3.8) is really expensive in computation, especially in the setting of this paper where $m$ is chosen as $T^{8}$. It seems impossible to obtain a reasonable solution within the claimed $O(1/T^{1/4})$ runtime.
What is the function $\iota(w)$ in Assumption 4.1?
It would be better for the authors to discuss more about Assumption 4.1. It is unknown why the action-value function $Q^{\pi}$ for all policy can fall into this class. 
The equation in Assumption 4.2 is exceeding the paper margin. Please make sure the paper follows the format guidelines.
Assumption 4.2 seems to be very strong. The remark after the assumption says that this condition is made on the Markov transition kernel. However, this may not be true since the assumption needs to hold for any two arbitrary policies. It is not known what kind of transition kernel $\mathcal{P}$ will satisfy this.
In each step of the neural policy gradient (Algorithm 1), the authors need to call a TD learning (Algorithm 2) to approximate the unknown action-value function $Q_{\omega_i}$ associated with the policy $\pi_{\theta_i}$ at the $i$-th step. It seems that in the learning process of ALgorithm 2, at each iteration, it samples independent data from the stationary state-action distribution which is unknown.  
In the proof of Theorem 4.8, it seems that eq (D.14) and (D.15) are the same. Why it needs to be proved twice? In addition, why the equation after (D.14) holds?
The authors should provide more details about the function $u_{\hat \theta}$ defined in eq (4.4), which seems to approximate the critic function. Specifically, why are there the derivative terms instead of just the inner product term in eq (4.4).
Other comments:
In the last sentence of Section 3.1, “... approximate aompatible function approximation ...”