Paper:
QUASI-HYPERBOLIC MOMENTUM AND ADAM FOR DEEP LEARNING
Momentum-based acceleration of stochastic gradient descent (SGD) is widely used in deep learning. We propose the quasi-hyperbolic momentum algorithm (QHM) as an extremely simple alteration of momentum SGD, averaging a plain SGD step with a momentum step. We describe numerous connections to and identities with other algorithms, and we characterize the set of two-state optimization algorithms that QHM can recover. Finally, we propose a QH variant of Adam called QHAdam, and we empirically demonstrate that our algorithms lead to significantly improved training in a variety of settings, including a new state-of-theart result on WMT16 EN-DE. We hope that these empirical results, combined with the conceptual and practical simplicity of QHM and QHAdam, will spur interest from both practitioners and researchers. Code is immediately available.

Review:
Edit: Following response, I have updated my score from 6 to 7.
I completed this review as an emergency reviewer - meaning that I had little time to complete the review. I did not have time to cover all of the material in the lengthy appendix but hope that I explored the parts most relevant to my comments below.
Paper summary: The paper introduces QHM, a simple variant of classical momentum which takes a weighted average of the momentum and gradient update. The authors comprehensively analyze the relationships between QHM and other momentum based optimization schemes. The authors present an empirical evaluation of QHM and QHAdam showing comparable performance with existing approaches.
Detailed comments:
I'll use CM to denote classical momentum, referred to as "momentum" in the paper.
1) In the introduction, you reference gradient variance reduction as a motivation for QHM. But in Section 3 you defer readers to the appendix for the motivation of QHM. I think that the main paper should include a brief explanation of this motivation.
2) The proposed QHM looks quite similar to a special case of Aggregated Momentum [1]. It seems that the key difference is with the use of damping but I suspect that this can be largely eliminated by using different learning rates for each velocity (as in Section 4 of [1]) and/or adopting damping in AggMo. In fact, Section 4.1 in your paper recovers Nesterov momentum in a very similar way. More generally, could one think of AggMo as a generalization of QHM? It averages plain SGD and several momentum steps on different time scales.
3) I thought that some of the surprising relations to other momentum based optimizers was the most interesting part of the paper. However, I found the presentation a little difficult. There are many algorithms presented but none are explored fully in the main paper. I had to flick between the main paper and appendix to uncover the information I wanted most from the paper.
Moreover, I found some of the arguments in the appendix a little tough to follow. For example, with AccSGD you should specify that epsilon is a constant typically chosen to be 0.7.  When the correspondence to QHM is presented it is not obvious that QHM -> AccSGD but not the other way around. I would suggest that you present a few algorithms in greater detail, and list the other algorithms you explore at the end of Section 4 with pointers to the appendix.
4) I am not sure that the QHAdam algorithm adds much to the paper. It is not explored theoretically and I found the empirical analysis fairly limited.
5) In general, the empirical results support QHM as an improvement on SGD/NAG. But I have some (fairly minor) concerns.
   a) For Figure 1, it looks like QHM beats QHAdam on MLP-EMNIST. Why not show these on the same plot? This goes back to my point 4 - it does not look like QHAdam improves on QHM and so I am not sure why it is included. The idea of averaging gradients and momentum is general - why explore QHAdam in particular?
   b) For Figure 2, while I certainly appreciate the inclusion of error bars, they suggest that the performance of all methods are very similar. In Table 3, QH and the baselines are often not just within a standard deviation of eachother but also have very close means (relatively).
6) I feel that some of the claims made in the paper are a little strong. E.g. "our algorithms lead to significantly improved training in a variety of settings". I felt that the evidence for this was lacking.
Overall, I felt that the paper offered many interesting results but clarity could be improved. I have some questions about the empirical results but felt that the overall story was strong. I hope that the issues I presented above can be easily addressed by the authors.
Minor comments:
- I thought the use of bold text in the introduction was unnecessary
- Some summary of the less common tasks in Table 2 should be given in the main paper
Clarity: I found the paper quite difficult to follow in places and found myself bouncing around the appendix frequently. While the writing is good I think that some light restructuring would improve the flow.
Significance: The paper presents a simple tweak to classical momentum but takes care to identify its relation to existing algorithms. The empirical results are not overwhelming but at least show QHM as competitive with CM on tasks and architecture where SGD is typically dominant.
Originality: To my knowledge, the paper presents original findings and places itself well amongst existing work.
References:
[1] Lucas et al. "Aggregated Momentum: Stability Through Passive Damping" https://arxiv.org/pdf/1804.00325.pdf