Paper:
We note that common implementations of adaptive gradient algorithms, such as Adam, limit the potential benefit of weight decay regularization, because the weights do not decay multiplicatively (as would be expected for standard weight decay) but by an additive constant factor. We propose a simple way to resolve this issue by decoupling weight decay and the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam, and (ii) substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter). We also demonstrate that longer optimization runs require smaller weight decay values for optimal results and introduce a normalized variant of weight decay to reduce this dependence. Finally, we propose a version of Adam with warm restarts (AdamWR) that has strong anytime performance while achieving state-ofthe-art results on CIFAR-10 and ImageNet32x32. Our source code will become available after the review process.

Review:
At the heart of the paper, there is a single idea: to decouple the weight decay from the number of steps taken by the optimization process (the paragraph at the end of page 2 is the key to the paper). This is an important and largely overlooked area of implementation and most off-the-shelf optimization algorithms, unfortunately, miss this point, too. I think that the proposed implementation should be taken seriously, especially in conjunction with the discussion that has been carried out with the work of Wilson et al., 2017 (https://arxiv.org/abs/1705.08292).
The introduction does a decent job explaining why it is necessary to pay attention to the norm of the weights as the training progresses within its scope. However, I would like to add a couple more points to the discussion: 
- "Optimal weight decay is a function (among other things) of the total number of epochs / batch passes." in principle, it is a function of weight updates. Clearly, it depends on the way the decay process is scheduled. However, there is a bad habit in DL where time is scaled by the number of epochs rather than the number of weight updates which sometimes lead to misleading plots (for instance, when comparing two algorithms with different batch sizes).
- Another ICLR 2018 submission has an interesting take on the norm of the weights and the algorithm (https://openreview.net/forum?id=HkmaTz-0W&noteId=HkmaTz-0W). Figure 3 shows the histograms of SGD/ADAM with and without WD (the *un-fixed* version), and it clearly shows how the landscape appear misleadingly different when one doesn't pay attention to the weight distribution in visualizations. 
- In figure 2, it appears that the training process has three phases, an initial decay, a steady progress, and a final decay that is more pronounced in AdamW. This final decay also correlates with the better test error of the proposed method. This third part also seems to correspond to the difference between Adam and AdamW through the way they branch out after following similar curves. One wonders what causes this branching and whether the key the desired effects are observed at the bottom of the landscape.
- The paper concludes with "Advani & Saxe (2017) analytically showed that in the limited data regime of deep networks the presence of eigenvalues that are zero forms a frozen subspace in which no learning occurs and thus smaller (e.g., zero) initial weight norms should be used to achieve best generalization results." Related to this there is another ICLR 2018 submission (https://openreview.net/forum?id=rJrTwxbCb), figure 1 shows that the eigenvalues of the Hessian of the loss have zero forms at the bottom of the landscape, not at the beginning. Back to the previous point, maybe that discussion should focus on the second and third phases of the training, not the beginning. 
- Finally, it would also be interesting to discuss the relation of the behavior of the weights at the last parts of the training and its connection to pruning. 
I'm aware that one can easily go beyond the scope of the paper by adding more material. Therefore, it is not completely reasonable to expect all such possible discussions to take place at once. The paper as it stands is reasonably self-contained and to the point. Just a minor last point that is irrelevant to the content of the work: The slash punctuation mark that is used to indicate 'or' should be used without spaces as in 'epochs/batch'.
Edit: Thanks very much for the updates and refinements. I stand by my original score and would like to indicate my support for this style of empirical work in scientific conferences.