Paper:
Published as a conference paper at ICLR 2018 DEEP GRADIENT COMPRESSION: REDUCING THE COMMUNICATION BANDWIDTH FOR DISTRIBUTED TRAINING
Large-scale distributed training requires significant communication bandwidth for gradient exchange that limits the scalability of multi-node training, and requires expensive high-bandwidth network infrastructure. The situation gets even worse with distributed training on mobile devices (federated learning), which suffers from higher latency, lower throughput, and intermittent poor connections. In this paper, we find 99.9% of the gradient exchange in distributed SGD are redundant, and propose Deep Gradient Compression (DGC) to greatly reduce the communication bandwidth. To preserve accuracy during this compression, DGC employs four methods: momentum correction, local gradient clipping, momentum factor masking, and warm-up training. We have applied Deep Gradient Compression to image classification, speech recognition, and language modeling with multiple datasets including Cifar10, ImageNet, Penn Treebank, and Librispeech Corpus. On these scenarios, Deep Gradient Compression achieves a gradient compression ratio from 270× to 600× without losing accuracy, cutting the gradient size of ResNet-50 from 97MB to 0.35MB, and for DeepSpeech from 488MB to 0.74MB. Deep gradient compression enables large-scale distributed training on inexpensive commodity 1Gbps Ethernet and facilitates distributed training on mobile.

Review:
I think this is a good work that I am sure will have some influence in the near future. I think it should be accepted and my comments are mostly suggestions for improvement or requests for additional information that would be interesting to have.
Generally, my feeling is that this work is a little bit too dense, and would like to encourage the authors in this case to make use of the non-strict ICLR page limit, or move some details to appendix and focus more on more thorough explanations. With increased clarity, I think my rating (7) would be higher.
Several Figures and Tables are never referenced in the text, making it a little harder to properly follow text. Pointing to them from appropriate places would improve clarity I think.
Algorithm 1 line 14: You never seem to explain what is sparse(G). Sec 3.1: What is it exactly that gets communicated? How do you later calculate the Compression Ratio? This should surely be explained somewhere.
Sec 3.2 you mention 1% loss of accuracy. A pointer here would be good, at that point it is not clear if it is in your work later, or in another paper. The efficient momentum correction is great!
As I was reading the paper, I got to the experiments and realized I still don't understand what is it that you refer to as "deep gradient compression". Pointer to Table 1 at the end of Sec 3 would probably be ideal along with some summary comments.
I feel the presentation of experimental results is somewhat disorganized. It is not clear what is immediately clear what is the baseline, that should be somewhere stressed. I find it really confusing why you sometimes compare against Gradient Dropping, sometimes against TernGrad, sometimes against neither, sometimes include Gradient Sparsification with momentum correction (not clear again what is the difference from DGC). I recommend reorganizing this and make it more consistent for sake of clarity. Perhaps show here only some highlights, and point to more in the Appendix.
Sec 5: Here I feel would be good to comment on several other things not mentioned earlier. 
Why do you only work with 99.9% sparsity? Does 99% with 64 training nodes lead to almost dense total updates, making it inefficient in your communication model? If yes, does that suggest a scaling limit in terms of number of training nodes? If not, how important is the 99.9% sparsity if you care about communication cost dominating the total runtime? I would really like to better understand how does this change and what is the point beyond which more sparsity is not practically useful. Put differently, is DGC with 600x size reduction in total runtime any better than DGC with 60x reduction?
Finally, a side remark:
Under eq. (2) you point to something that I think could be more discussed. When you say what you do has the effect of increasing stepsize, why don't you just increase the stepsize? 
There has recently been this works on training ImageNet in 1 hour, then in 24 minutes, latest in 15 minutes... You cite the former, but highlight different part of their work. Broader idea is that this is trend that potentially makes this kind of work less relevant. While I don't think that makes your work bad or misplaced, I think mentioning this would be useful as an alternative approach to the problems you mention in the introduction and use to motivate your contribution.
...what would be your reason for using DGC as opposed to just increasing the batch size?