Efficient deep neural network (DNN) inference on mobile or embedded devices typically involves quantization of the network parameters and activations. In particular, mixed precision networks achieve better performance than networks with homogeneous bitwidth for the same size constraint. Since choosing the optimal bitwidths is not straight forward, training methods, which can learn them, are desirable. Differentiable quantization with straight-through gradients allows to learn the quantizer's parameters using gradient methods. We show that a suited parametrization of the quantizer is the key to achieve a stable training and a good final performance. Specifically, we propose to parametrize the quantizer with the step size and dynamic range. The bitwidth can then be inferred from them. Other parametrizations, which explicitly use the bitwidth, consistently perform worse. We confirm our findings with experiments on CIFAR-10 and ImageNet and we obtain mixed precision DNNs with learned quantization parameters, achieving state-of-the-art performance.
This paper considers the problem of training mixed-precision models. 
Since quantization involves non-differentiable operations, this paper discusses how to use the straight-through estimator to estimate the gradients,  and how different parameterizations of the quantized DNN affect the optimization process. The authors conclude that using the parameterization wrt the stepsize d and quantization range q_max has the best performance.
In the discussion for the three parameterization choices in section 2.1. 
It is not clear how the range for U2 is obtained. Given d an integer, the gradient wrt b is also bounded. In this case, why is case U3 better than U2? In Table 1, it is shown that U2 also has good performance for uniform quantization.
Indeed, the gradient of any of the three parameters (stepsize, bitwidth and quantization range) can be derived by using chain rule given the gradients of the other two. It is not clear to me why some of them can be unbounded while others do not. In addition, It is not clear to me why having different gradient scales is a big problem. Adaptive learning rate methods like Adam should be able to help deal with the different scale of the gradients for three parameters. Can the authors compare the three parameterizations using Adam and see if similar empirical results can still be observed.
At the end of Section 2.1, the authors said that "similar considerations can be made for power-of-two quantization".  However, from table 1, these three parameterizations indeed have quite different performances for uniform and power-of-two quantization.  E.g., for uniform quantization, U2 and U3 perform significantly better than U1, while for power-of-two quantization, U1 and U3 perform significantly better than U2. Can the authors elaborate more on the difference?
Is the proposed differential quantization method used for both weight and activation? If so, how are the gradients w.r.t. the weights propagated through the quantized activations?
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I thank the authors for their detailed response. It has solved most of my concerns and I accordingly raised my score.
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