Paper:
Recent work in network quantization has substantially reduced the time and space complexity of neural network inference, enabling their deployment on embedded and mobile devices with limited computational and memory resources. However, existing quantization methods often represent all weights and activations with the same precision (bit-width). In this paper, we explore a new dimension of the design space: quantizing different layers with different bit-widths. We formulate this problem as a neural architecture search problem and propose a novel differentiable neural architecture search (DNAS) framework to efficiently explore its exponential search space with gradient-based optimization. Experiments show we surpass the state-of-the-art compression of ResNet on CIFAR-10 and ImageNet. Our quantized models with 21.1x smaller model size or 103.9x lower computational cost can still outperform baseline quantized or even full precision models.

Review:
In this work the authors introduce a new method for neural architecture search (NAS) and use it in the context of network compression. Specifically, the NAS method is used to select the precision quantization of the weights at each layer of the neural network. Briefly, this is done by first defining a super network, which is a DAG where for each pair of nodes, the output node is the linear combination of the outputs of all possible operations (i.e., layers with different precision quantizations). Following [1], the weights of the linear combination are regarded as the probabilities of having certain operations (i.e., precision quantization), which allows for learning a probability distribution over the considered operations. Differently from [1], however, the authors bridge the soft sampling in [1] (where all operations are considered together but weighted accordingly to the corresponding probabilities) to a hard sampling (where a single operation is considered with the corresponding probability) through an annealing procedure based on the Gumbel Softmax technique. Through the proposed NAS algorithm, one can learn a probability distribution on the operations by minimizing a loss that accounts for both accuracy and model size. The final output of this search phase is a set of sampled architectures (containing a single operation at each connection between nodes), which are then retrained from scratch. In applications to CIFAR-10 and ImageNet, the authors achieve (and sometime surpass) state-of-the-art performance in model compression.
The two contributions of this work are
1)	A new approach to weight quantization using principles of NAS that is novel and promising;
2)	New insights/technical improvements in the broader field of NAS. While the utility of the method in the more general context of NAS has not been shown, this work will likely be of interest to the NAS community.
I only have one major concern. The architectures are sampled from the learnt probability distribution every certain number of epochs while training the supernet. Why? If we are learning the distribution, would not it make sense to sample all architectures only after training the supernet at our best?
This reasoning leads me to a second question. In the CIFAR-10 experiments, the authors sample 5 architecture every 10 epochs, which means 45 architectures (90 epochs were considered). This is a lot of architectures, which makes me wonder: how would a “cost-aware” random sampling perform with the same number of sampled architectures?
Also, I have some more questions/minor concerns:
1)	The authors say that the expectation of the loss function is not directly differentiable with respect to the architecture parameters because of the discrete random variable. For this reason, they introduce a Gumbel Softmax technique, which makes the mask soft, and thus the loss becomes differentiable with respect to the architecture parameters. However, subsequently in the manuscript, they write that Eq 6 provides an unbiased estimate for the gradients. Do they here refer to the gradients with respect to the weights ONLY? Could we say that the advantage of the Gumbel Softmax technique is two-fold? i) make the loss differentiable with respect to the arch parameters; ii) reduce the variance of the estimate of the loss gradients with respect to the network weights.
2)	Can the author discuss why the soft sampling procedure in [1] is not enough? I have an intuitive understanding of this, but I think this should be clearly discussed in the manuscript as this is a central aspect of the paper.
3)	The authors use a certain number of warmup steps to train the network weights without updating the architecture parameters to ensure that “the weights are sufficiently trained”. Can the authors discuss the choice on the number of warmup epochs?
I gave this paper a 5, but I am overall supportive. Happy to change my score if the authors can address my major concern.
[1] Liu H, Simonyan K, Yang Y. Darts: Differentiable architecture search. arXiv preprint arXiv:1806.09055. 2018 Jun 24.
-----------------------------------------------------------
Post-Rebuttal
---------------------------------------------------------
The authors have fully addressed my concerns. I changed the rating to a 7.