This paper shows how policy gradient and Q-Learning may be combined together, improving learning as demonstrated in particular in the Atari Learning Environment. The core idea is to note that entropy-regularized policy gradient leads to a Boltzmann policy based on Q values, thus linking pi & Q together and allowing both policy gradient and Q-Learning updates to be applied.  I think this is a very interesting paper, not just for its results and the proposed algorithm (dubbed PGQ), but mostly because of the links it draws between several techniques, which I found quite insightful.  That being said, I also believe it could have done a better job at clearly exposing these links: I found it somewhat difficult to follow, and it took me a while to wrap my head around it, even though the underlying concepts are not that complex. In particular: - The notation \tilde{Q}^pi is introduced in a way that is not very clear, as "an estimate of the Q-values" while eq. 5 is an exact equality (no estimation) - It is not clear to me what section 3.2 is bringing exactly, I wonder if it could just be removed to expand some other sections with more explanations. - The links to dueling networks (Wang et al, 2016) are in my opinion not explicit and detailed enough (in 3.3 & 4.1): as far as I can tell the proposed architecture ends up being very similar to such networks and thus it would be worth telling more about it (also in experiments my understanding is that the "variant of asynchronous deep Q-learning" being used is essentially such a dueling network, but it is not clearly stated). I also believe it should be mentioned that PGQ can also be seen as combining Q-Learning with n-step expected Sarsa using a dueling network: this kind of example helps better understand the links between methods - Overall I wish section 3.3 was clearer, as it draws some very interesting links, but it is hard to see where this is all going when reading the paper for the first time. One confusing point is w.r.t. to the relationship with section 3.2, that assumes a critic outputting Q values while in 3.3 the critic outputs V. The "mu" distribution also comes somewhat out of nowhere.  I hope the authors can try and improve the readability of the paper in a final version, as well as clarify the points raised in pre-review questions (in particular related to experimental details, the derivation of eq. 4, and the issue of the discounted distribution of states).  Minor remarks: - The function r(s, a) used in the Bellman equation in section 2 is not formally defined. It looks a bit weird because the expectation is on s' and b' but r(s, a) does not depend on them (so either it should be moved out of the expectation, or the expectation should also be over the reward, depending on how r is defined) - The definition of the Boltzmann policy at end of 2.1 is a bit confusing since there is a sum over "a" of a quantity that does not depend (clearly) on "a" - I believe 4.3 is for the tabular case but this is not clearly stated - Any idea why in Fig. 1 the 3 algorithms do not all converge to the same policy? In such a simple toy setting I would expect it to be the case.  Typos: - "we refer to the classic text Sutton & Barto (1998)" => missing "by"? - "Online policy gradient typically require an estimate of the action-values function" => requires & value - "the agent generates experience from interacting the environment" => with the environment - in eq. 12 (first line) there is a comma to remove near the end, just before the dlog pi - "allowing us the spread the influence of a reward" => to spread - "in the off-policy case tabular case" => remove the first case - "The green line is Q-learning where at the step an update is performed" => at each step - In Fig. 2 it says A2C instead of A3C  NB: I did not have time to carefully read Appendix A
Output: en