Paper:
LOSS-AWARE WEIGHT QUANTIZATION OF DEEP NET- WORKS
The huge size of deep networks hinders their use in small computing devices. In this paper, we consider compressing the network by weight quantization. We extend a recently proposed loss-aware weight binarization scheme to ternarization, with possibly different scaling parameters for the positive and negative weights, and m-bit (where m > 2) quantization. Experiments on feedforward and recurrent neural networks show that the proposed scheme outperforms state-of-the-art weight quantization algorithms, and is as accurate (or even more accurate) than the full-precision network.

Review:
This paper proposes a new method to train DNNs with quantized weights, by including the quantization as a constraint in a proximal quasi-Newton algorithm, which simultaneously learns a scaling for the quantized values (possibly different for positive and negative weights). 
The paper is very clearly written, and the proposal is very well placed in the context of previous methods for the same purpose. The experiments are very clearly presented and solidly designed.
In fact, the paper is a somewhat simple extension of the method proposed by Hou, Yao, and Kwok (2017), which is where the novelty resides. Consequently, there is not a great degree of novelty in terms of the proposed method, and the results are only slightly better than those of previous methods.
Finally, in terms of analysis of the algorithm, the authors simply invoke a theorem from Hou, Yao, and Kwok (2017), which claims convergence of the proposed algorithm. However, what is shown in that paper is that the sequence of loss function values converges, which does not imply that the sequence of weight estimates also converges, because of the presence of a non-convex constraint ($b_j^t \in Q^{n_l}$). This may not be relevant for the practical results, but to be accurate, it can't be simply stated that the algorithm converges, without a more careful analysis.