We study continuous action reinforcement learning problems in which it is crucial that the agent interacts with the environment only through safe policies, i.e., policies that keep the agent in desirable situations, both during training and at convergence. We formulate these problems as constrained Markov decision processes (CMDPs) and present safe policy optimization algorithms that are based on a Lyapunov approach to solve them. Our algorithms can use any standard policy gradient (PG) method, such as deep deterministic policy gradient (DDPG) or proximal policy optimization (PPO), to train a neural network policy, while guaranteeing near-constraint satisfaction for every policy update by projecting either the policy parameter or the selected action onto the set of feasible solutions induced by the state-dependent linearized Lyapunov constraints. Compared to the existing constrained PG algorithms, ours are more data efficient as they are able to utilize both on-policy and off-policy data. Moreover, our action-projection algorithm often leads to less conservative policy updates and allows for natural integration into an end-to-end PG training pipeline. We evaluate our algorithms and compare them with the state-of-the-art baselines on several simulated (MuJoCo) tasks, as well as a real-world robot obstacle-avoidance problem, demonstrating their effectiveness in terms of balancing performance and constraint satisfaction.
Summary:
Authors propose ideas to perform safe RL in continuous actions domain with modifications to Policy Gradient (PG)  algorithms via either constraining the policy parameters or constraining the actions selected by PG with a surrogate/augmented state dependent objective. The paper is well motivated and the experiments (although I have some reservations about the setup) demonstrate efficacy of the proposed method.  
Review:
--> Introduction
I do not agree with the statement that value function based algorithms are restricted to discrete action domains, especially when you rely on  “ignoring function approximation errors” for some of your claims. 
Again in switching from value function to PG is true for traditional RL/Control theory but this is not valid here. ( your methods rely on Q(s, a) which is action-value function, or the constraint in equation 3 is integral of Q over all actions which would be value-function in traditional definition )
Note: this is explained very well towards the end in the Appendix B, but this is a review of the paper and not Appendix B or C. 
--> Section 2
section 2.3 I would strongly advise the authors to rewrite this, this section reads like it was copied as is from the reference [Chow et al 2018].  especially the way Lyapunov function is defined. And the language and arguments are almost same. Some sentences cite the reference but conclusions drawn on these are not cited, are you claiming that these conclusions are original from this paper ?
It is not clear to me how the feasibility of initial pi_0 is ensured ? Did I miss this somewhere ?
→ Section 3
Section 3 is pleasant to read and very easy to understand, however, same cannot be said of the
section 3.1. I had to spend significant time reading 3.1 and I am still not sure I have understood it very well. 
Experiments:
I don’t think halfCheetah-Safe is actually actually an useful experiment, Limiting the joint torques is perfectly understandable, just limiting speed and getting smooth motion could just be an artifact of the simulation environment. Are both constraints applied simultaneously (torque and speed) ? It is unclear from the text. 
I am not sure CPO without linesearch is actually a fair comparison. Line search may actually deem of the actions unsafe and therefore I would presume original CPO do be less prone to constraint violation than the proposed modification in your experiments. Again PPO is more heuristic than TRPO which makes it hard to compare like for like. PPO might give higher rewards but constraint violations may increase as well. An important point for Safe-RL I feel.
Figure 6 
Can you be more specific as what the figure 6 is showing ? Constraint ? is this constraint violation count? or cumulative sum of constraint slack over the whole trajectory ?
 
Not part of assessment :
Unclear Statements:
Page 7, DDPG Vs PPO: explain clearly what you mean by  “covariate shift” or remove the statement altogether. 
Page 7, section 5 second paragraph, “The actions are the linear …. center of mass” I couldn’t understand this ? What do you mean by actions are velocity ?
Minor points (Language, Typos):
page 3, last paragraph,  Chow et al. is repeated, I can see why this happens there but suggest editing to avoid this. [This is also in intro paragraph, there it is just a typo and should be rectified]
Figure 6: Captions labels are incorrect.