__ Summary and Contributions__: The paper discusses a decentralized stochastic optimization scheme for generic smooth min-max optimization problems with application to training GANs over large datasets, e.g. Image-net.

__ Strengths__: Decentralized methods are becoming popular to tackle large-scale problems, and this paper studies a fairly general setup with a relatively simple algorithm, and with provable convergence guarantees. The authors tried their method on the state-of-the- art GAN approaches to well-known datasets and even compared different computational environments with high and low latency.

__ Weaknesses__: I do not find any major issue with the paper.

__ Correctness__: I did not closely check the proofs, but the arguments and the sketch looks convincing to me.

__ Clarity__: It is well written.

__ Relation to Prior Work__: Excellent citation.

__ Reproducibility__: Yes

__ Additional Feedback__: As I said, I do not have a major comment. The only part which confused me was the repeated local averaging for t times. Please clarify the text and also state that W^t means W to the power of t (if I am correct!)
Post rebuttal comment:
My opinion is unchanged about this paper.

__ Summary and Contributions__: ------ edit ---------
1) It is expected that in the case where M=1 exactly then the single-machine results can be recovered exactly since the analysis is a generalization of [15]. However, this seems to break down as soon as M=2, so the complexity substantially increases when going from 1 to 2 machines. This may be unavoidable but it means that even with free unlimited communications, it would be slower to use 2 machines rather than 1 with this algorithm. This seems worth commenting on, or knowing whether it's an actual feature of the algorithm or an artifact of the proof.
2) logarithmic complexity in epsilon
From what I understand from the rebuttal, Corollary 1 and Remark 1 indeed only hold for communication graphs with small rho (such as the complete graph). In general the communication complexity could be much worse (\epsilon^{-8} in the case of the ring graph for example). Similarly, Corollary 1 considers multiplication by the "max degree of the network". The rebuttal seems to indicate that this is is the maximum degree of a node in case the communication matrices are stochastic but this not completely clear either and should be made more specific. If this is the case then Remark 1 would almost only hold for the random mixing strategy on the complete graph, which is far from being explicit. In particular, I think multi-consensus would actually largely degrade the result in this case since it would make the actual "max degree" equal to O(M), thus losing all hopes of logarithmic communication complexity. In any case, multi-consensus is tailored for cases when rho is close to 1.
Thus, I still believe that the analysis may not be completely tight, and that Corollary 1 and Remark 1 should be substantially modified since they actually hold in a very restricted communication setting which is not specified in the paper (but which is acknowledged in the rebuttal). It would be interesting to know whether logarithmic communication is achievable in the general setting anyway if M = poly(1/epsilon).
Therefore, I keep my original score. Should the paper be accepted, I believe the authors should rewrite the interpretations of Theorem 1 and be as straightforward as in the rebuttal by clearly emphasizing when the remark holds and when it does not.
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This paper presents a decentralized version of the Parallel Optimistic Stochastic Gradient algorithm in which parameters are approximately averaged (in a decentralized way) before applying each gradient update. A non-asymptotic convergence theorem is given.
The paper is generally well-written and includes (too?) extensive state-of-the art review. Experimental results on ImageNet and CIFAR-10 datasets are given, and seem to indicate the benefits of using a decentralized algorithm.

__ Strengths__: Extension of the optimistic stochastic gradient algorithm to the decentralized setting, with convergence guarantees.
Extensive state of the art
Both low-latency and high-latency settings are investigated, for small and big models. The experiments seem rather convincing to me (though I am not an expert in GANs training).

__ Weaknesses__: Although a convergence theorem is given, the results do not seem tight since they do not seem to recover the single-machine guarantees when M=1 (only one node).
In particular, the results from Corollary 1 and Remark 1 should be compared with the results from [Theorem 1, 15]. In particular, it should be emphasized that the results from [15] are *not* recovered when M=1 (single machine setting), although the algorithm is the same. This suggests that the analysis is not tight, or otherwise should be explained more in details. The two things that prevent this recovery are that: (i) the variance term decreases as $1 / \sqrt{mM}$ instead of $1/(mM)$, and (ii) the step-size has to be chosen smaller in general even though $M=1$ (because of the $\sqrt{m}$ term). Please tell me if I have missed something and this is not the case, but this should me made clearer anyway.

__ Correctness__: - I don't quite get the claimed $O(\log(1/\varepsilon))$ communication complexity at the busiest node. If I am correct, this is the per-iteration complexity, so the overall communication complexity has to be multiplied by N, which is of order $\varepsilon^{-8}$ in Corollary 1.
- Parameter $\rho$ depends on $M$, and in general tends to $1$ as $M$ grows (except from few topologies such as the complete graph). In Remark 2 for example, it is noted that $\rho = 1 - O(1/M^2)$ for the ring graph. In this case, we roughly have that $\log(1/\rho) = O(1/M^2)$ and so it cannot be said that the convergence rate is logarithmic in $\varepsilon$ since $\log_{1/\rho}(x) = \log(x) / \log(1/\rho) = O(M \log(x))$. Have I missed something here? I believe that the derivations leading to the logarithmic communication rate from Corollary 1 and Remark 1 should be made more explicit, as I am not convinced that they hold in the current form.

__ Clarity__: Yes it is, apart from some typos (given at the end of the additional feedback).

__ Relation to Prior Work__: Maybe give more details about the links with extra gradient? It basically looks like extra gradient on u and v at the same time for the min max problem. This is probably worth mentioning.
On a side note, I know that reference pages are not limited (which is a good thing) but more than 5 pages of references for an 8 pages paper (which is not a review paper) seems like a lot to me.

__ Reproducibility__: Yes

__ Additional Feedback__: - I could not find which value is used for parameter $t$ in the experiments. It seems that $t=1$ is used (which is not what theory suggests) but this is not clear. It's somewhat important because the theoretical results seem to be based on approximate but precise averaging at each step, which is far from being the case with $t=1$.
- Random mixing strategies are discussed in Remark 2, but Theorem 1 is presented for fixed matrices $W$. This should be clarified.
- Simply communicating with $W^t$ is generally inefficient. It is much faster to use Chebyshev acceleration (also called multi consensus) as in Scaman et al. [A], and leads to a dependence in $\sqrt{\rho}$ rather than $\rho$ for fixed topologies. It is more complicated for random topologies.
- It seems that the second term in the min defining $\eta$ is always smaller than the first one, which can therefore be dropped. Similarly, since $m$ and $M$ are greater than 1, $\sqrt{mM} < 1$ so the second term in the equation is actually always smaller than the third one (and so it could be removed to improve readability, and $\sigma^2$ be replaced by $2 \sigma^2$).
Overall, I think that the idea of using OSG in a decentralized fashion is great, but the results from Theorem 1 seem to be improvable in many ways. In particular, they do not seem tight since they do not recover the single-machine results when using a single machine, and the logarithmic communication complexity aspect is unclear to me. I would gladly revise my opinion if the authors clarify these 2 points in the rebuttal.
Minor things, typos:
Section 4: "Denote by ..." rather than "Denote ... by"?
Typos: L250: an upper bound.
References:
[A] Scaman, K., Bach, F., Bubeck, S., Lee, Y. T., & Massoulié, L. (2017, July). Optimal Algorithms for Smooth and Strongly Convex Distributed Optimization in Networks. In International Conference on Machine Learning (pp. 3027-3036).

__ Summary and Contributions__: In this paper, the authors propose a decentralized parallel optimistic stochastic gradient method to train generative adversarial networks (GANs). The authors analyze its convergence, and demonstrate its applications in decentralized systems.

__ Strengths__: To the best of my knowledge, this is the first paper that considers decentralized training of GANs. Overall, this paper is well written.

__ Weaknesses__: 1. The convergence is in terms of \bar{z}, the average of local iterates. However, the local iterates can be quite different (namely, lack of consensus). Is there any difficulty in proving consensus of local iterates?
2. In page 7, the authors mention that “An implicit barrier is enforced at the end of each iteration so that every learner advances in a lock-step”. Please explain the meanings of barrier and lock-step.

__ Correctness__: The proposed method and the analysis seem reasonable. The numerical experiments are convincing.

__ Clarity__: Yes.

__ Relation to Prior Work__: Clearly discussed.

__ Reproducibility__: Yes

__ Additional Feedback__: I have read the author response. My concerns have been addressed.

__ Summary and Contributions__: The paper proposed a gradient-based decentralized parallel algorithm for GAN traning. The algorithm seems to be the first method for distributed GAN training procedure.

__ Strengths__: The algorithm is designed for general min-max nonconvex nonconcave problems, but is suitable for GAN.

__ Weaknesses__: *update your review*
Although the proposed algorithm seems to be the first decentralized parallel algorithm for min-max nonconvex nonconcave problems (including GAN problem), the novelty of the proposed algorithm is still not enough. The proposed algorithm and theoretical analysis are proposed, but they both are direct extensions of existing results, which significantly weaken the contribution.

__ Correctness__: Yes

__ Clarity__: Yes

__ Relation to Prior Work__: In some sense.

__ Reproducibility__: Yes

__ Additional Feedback__: