__ Summary and Contributions__: This paper proposes to use the softmax operator, instead of max, in DDPG/TD3 to correct the overestimation/underestimation bias. Practical algorithms SD2/SD3 are proposed with good empirical studies. In particular, it is empirically shown that the softmax operator helps smooth the loss landscape, and the proposed methods achieve state-of-the-art results on MuJoCo and Box2d.
===========After rebuttal=========
I have read the other reviews and the rebuttal. I don't think the rebuttal address my questions in the original review and I still have some concern in the rigorous analysis and the mathematical writing of the paper. On the other hand, the idea of the paper and its empirical analysis are interesting. I will keep my original score.

__ Strengths__: (1) It is shown on a simple example that the softmax operator helps smooth the loss landscape, thus help in the optimization. It is also empirically shown that the softmax operator reduces the overestimation/underestimation problem compared to DDPG/TD3. An ablation study is also included in the paper. Overall, the empirical investigation of the proposed algorithms in this paper is careful and detailed.
(2) This paper proposes a practical approximation method for estimating the softmax operator, as it includes integration in the definition of the softmax operator.
(3) The proposed algorithms achieve state-of-the-art results.

__ Weaknesses__: 1) There is no analysis of how the softmax can help smooth the loss landscape.
(2) Theorem 3 and Theorem 4 seem to be crude. In Theorem 3, while SD2 helps to reduce the overestimation bias, the bias of T_{SD2} could still be larger as there is no guarantee that SD2 would not underestimate the value. Theorem 4 seems to be flawed. In particular, in the proof of theorem 4, why would Equation (19) hold? Note that softmax_0 is generally taking the average Q value.

__ Correctness__: (1) In example 1, the rate is correct only if \beta is large. If \beta is small, the rate is constant.
(2) Right after theorem 2, what if lnF is negative?
(2) The proofs of theorem 3 and theorem 4 may need some corrections.

__ Clarity__: Overall the paper is easy to follow. The structure is clear and the experiments are clearly described. Some notations are used in the paper without definitions.
(1) Any condition on the action set A in the paper?
(2) In theorem 3/4, what is the definition of bias?
(3) Line 231, "y_i = r + \gamma min( ****)" seems to have a typo
(4) What is the definition of \Tau_{SD3}?

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: In section 4.2, a pratical approximation method is proposed in estimating the softmax operator. How is the performance of this method?

__ Summary and Contributions__: The work provide strong theoretical, analytical, and empirical argument for an improved way to bootstrap values from multiple critics using soft-max as opposed to a cold max operator. A key technique is the use of multiple actors.

__ Strengths__: This paper is simply a pleasure to read. The motivate is quite clear and well-founded, supported by cited references. The method section involve a simple toy domain that illustrates the benefit quite well. The experimental result section shows clear results and thorough discussion over a complete set of standard test domains, providing much confidence that this method would be impactful and should be adopted as a replacement of TD3 for continuous control.
The change is simple, well-motivated, and clearly supported by the evidence.

__ Weaknesses__: I wonder if there is a more fundamental view why soft-max alieviates both the over estimation problem *and* the under estimation problem. If the logits are participating in this weighted averaging in a subtle, but important fashion, such as being an energy model. It would also be good to relate this more broadly in the related works session to ensemble methods. For me personally that connection would be interesting.

__ Correctness__: Yes.

__ Clarity__: It is a pleasure to read.

__ Relation to Prior Work__: Yes, very well motivated.

__ Reproducibility__: Yes

__ Additional Feedback__: I came out quite inspired by this work. Thank you for writing this paper!
- It would help a bit if the text features that SD3 uses double actors a bit more prominently. During my reading, it was note really clear that SD3 uses two actors up untill the experiment section, when the comparison took place.
- Although it is clear to practitioners, sometimes a good highlevel figure for the algorithm would help with illustrating the high-level structure, and differences against related methods. I understand that this is might be difficulty given the limited space for submission.
- Some of the fonts in the figures are hard to read.

__ Summary and Contributions__: This paper analyses the use of a softmax operator applied to the Q-target of a continuous control RL algorithms such as DPG. It provides some theory (bounding the approximation error), insight (smoothing of optimisation landscape and reduction of overoptimism), and performance (outperforming state-of-the-art TD3 on standard control suite tasks).

__ Strengths__: This is a nice paper. The paper is packed with small insightful experiments. It's great to see a careful study that unpicks the contributions of a simple idea to understand its benefits, and then puts it back together in a logical way to arrive at improved overall performance in standard benchmark tasks.

__ Weaknesses__: The paper is what it is - a clear but narrow contribution based upon the introduction of softmax operators into existing algorithms.

__ Correctness__: There were a couple of points I was unclear on:
-The bounds include an integral over the action space. Presumably this then requires that the action space is bounded? I didn't see this stipulated. Nor is it clear to me that for large action spaces that the approximation bounds are necessarily very meaningful. What am I missing?
-I was unclear about some terminology. Does the c > 0 in Theorem 3 refer to the noise clipping in the previous paragraph? Is it then specific to this particular form?
-What is the (1-d) term in the main algorithm box?
-Section 4.2 mentions importance sampling. But the algorithm appears to simply apply the softmax to sampled actions, without any importance sampling that I can see. I think I'm missing something here - what is actually going on?

__ Clarity__: The paper is generally well written.
Section 4.2 is too dense to fully understand, and this led to most of my questions and possible misunderstandings.
It's a shame not to have the main algorithm box in main text.

__ Relation to Prior Work__: Yes this is clear

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This draft extended the idea of using the softmax Bellman operator in the discrete action space to the continuous case. The authors showed both the performance bound as well as the overestimation bias reduction. Empirical results were presented to demonstrate the superior performance of the proposed algorithms.

__ Strengths__: 1. Extending the idea of averaging the Q-functions to the policy gradient methods is novel.
2. The author provided enough empirical evidence to support the claim, including the plots of reward and bias, which is indeed helpful to understand the paper.
3. The performance on the MuJoCo environment is encouraging as it consistently outperforms the previous method.

__ Weaknesses__: I am a bit concerned about the significance of some theoretical results: In Theorem 1, the bound depends on \epsilon, even though \beta goes to infinity. Similarly in Theorem 2, the error converges to a non-zero value. These results are a bit surprising, given that the theorem in Song et al. showed it converges to zero. What would be the factors to lead to this gap?

__ Correctness__: Yes

__ Clarity__: This paper is well written.

__ Relation to Prior Work__: This paper discussed clearly how it is different from previous work.

__ Reproducibility__: Yes

__ Additional Feedback__: I have the following questions for the authors to clarify and respond.
1. For the bias definition in Theorems 3 and 4, is E [T (s')] also dependent on \theta^{true}? If yes, would this be a reasonable assumption?
2. The authors showed that the proposed estimator can simultaneously reduce over- and under-estimation bias. Such results, however, definitely depend on the choice of \beta. Could you elaborate more on how to choose this parameter?
===After Rebuttal ===
Thanks for the clarification. I am a bit concerned that the bounds in Theorems 1 and 2 may not be tight enough, which has also been shared by other reviewers. For instance, if Q(s, a) is set to be a constant w.r.t. a in Theorem 1, max == softmax and C(Q, s, \epsilon) = A. However, the upper bound there goes to a non-trivial positive value.
That being said, I still see the practical benefits of this submission, and hope the authors can address the issue on theoretical rigor.