Paper:
PROXQUANT: QUANTIZED NEURAL NETWORKS VIA PROXIMAL OPERATORS
To make deep neural networks feasible in resource-constrained environments (such as mobile devices), it is beneficial to quantize models by using low-precision weights. One common technique for quantizing neural networks is the straightthrough gradient method, which enables back-propagation through the quantization mapping. Despite its empirical success, little is understood about why the straight-through gradient method works. Building upon a novel observation that the straight-through gradient method is in fact identical to Nesterov's dual-averaging algorithm on a quantization constrained optimization problem, we propose a more principled alternative approach, called PROXQUANT, that formulates quantized network training as a regularized learning problem instead and optimizes it via the prox-gradient method. PROX-QUANT does back-propagation on the underlying full-precision vector and applies an efficient prox-operator in between stochastic gradient steps to encourage quantizedness. For quantizing ResNets and LSTMs, PROXQUANT outperforms state-of-the-art results on binary quantization and is on par with state-of-the-art on multi-bit quantization. We further perform theoretical analyses showing that PROXQUANT converges to stationary points under mild smoothness assumptions, whereas variants such as lazy prox-gradient method can fail to converge in the same setting.

Review:
This paper proposed ProxQuant method to train neural networks with quantized weights. ProxQuant relax the quantization constraint to a continuous regularizer and then solve the optimization problem with proximal gradient method. The authors argues that previous solvers straight through estimator (STE) in BinaryConnect (Courbariaux et al. 2015) may not converge, and the proposed ProxQuant is better.
 I have concerns about both theoretical and experimental contributions
1. The proposed regularizer for relaxing quantized constraint looks similar to BinaryRelax (Yin et al. 2018 BinaryRelax: A Relaxation Approach For Training Deep Neural Networks With Quantized Weights.), which is not cited. I hope the authors can discuss this work and clarify the novelty of the proposed method. One difference I noticed is that BinaryRelax use lazy prox-graident, while the proposed ProxQuant use non-lazy update. It is unclear which one is better.
2. On page 5, the authors claim ‘’Our proposed method can be viewed as … generalization ...’’ in page 5. It seems inaccurate because unlike proposed method, BinaryConnect use lazy prox-gradient.
3. What’s the purpose of equation (4)? I am confused and did not find it explained in the content.
4. The proposed method introduced more hyper-parameters, like the regularizer parameter \lambda, and the epoch to perform hard quantization. In section 4.2, it is indicated that parameter \lambda is tuned on validation set. I have doubts about the fairness comparing with baseline BinaryConnect. Though BC does not have this parameter, we can still tune learning rate.
5. ProxQuant is fine-tuned based on the pre-trained real-value weights. Is BinaryConnect also fine-tuned? For a CIFAR-10 experiments, 600 epochs are a lot for fine-tuning. As a comparison, training real-value weights usually use less than 300 epochs. BinaryConnect can be trained from scratch using same number of epochs. What does it mean to hard-quantize BinaryConnect? The weights are already quantized after projection step in BinaryConnect.  
6. The authors claim there are no reported results with ResNets on CIFAR-10 for BinaryConnect, which is not true. (Li et al. 2017 Training Quantized Nets: A Deeper Understanding) report results on ResNet-56, which I encourage authors to compare with. 
7. What is the benefit of ProxQuant? Is it faster than BinaryConnect? If yes, please show convergence curves. Does it generate better results? Table 1 and 2 does not look convincing, especially considering the fairness of comparison.
8. How to interpret Theorem 5.1? For example,  Li et al. 2017 show the real-value weights in BinaryConnect can converge for quadratic function, does it contradict with Theorem 5.1?
9. I would suggest authors to rephrase the last two paragraphs of section 5.2. It first states ‘’one needs to travel further to find a better net’’, and then state ProxQuant find good result nearby, which is confusing. 
10.  The theoretical benefit of ProxQuant is only intuitively explained, it looks to me there lacks a rigorous proof to show ProxQuant will converge to a solution of the original quantization constrained problem.
11. The draft is about 9 pages, which is longer than expected. Though the paper is well written and I generally enjoyed reading, I would appreciate it if the authors could shorten the content. 
My main concerns are novelty of the proposed method, and fairness of experiments. 

======================= after rebuttal =======================
I appreciate the authors' efforts and am generally satisfied with the revision. I raised my score. 
The authors show advantage of the proposed ProxQuant over previous BinaryConnect and BinaryRelax in both theory and practice. The analysis bring insights into training quantized neural networks and should be welcomed by the community. 
However, I still have concerns about novelty and experiments.
- The proposed ProxQuant is similar to BinaryRelax except for non-lazy vs. lazy updates. I personally like the theoretical analysis showing ProxQuant is better, although it is based on smooth assumptions. However, I am quite surprised BinaryRelax is so much worse than ProxQuant and BinaryConnect in practice (table 1). I would encourage the authors to give more unintuitive explanation.
-  The training time is still long, and the experimental setting seems uncommon. I appreciate the authors' efforts on shortening the finetuning time, and provide more parameter tuning.  However, 200 epochs training full precision network and 300 epochs for finetuning is still a long time, consider previous works like BinaryConnect can train from scratch without a full precision warm start. In this long-training setting, the empirical advantage of ProxQuant over baselines is not much (less than 0.3% for cifar-10 in table 1, and comparable with Xu 2018 in table 2).