Paper:
State-of-the-art quantization methods can compress deep neural networks down to 4 bits without losing accuracy. However, when it comes to 2 bits, the performance drop is still noticeable. One problem in these methods is that they assign equal bit rate to quantize weights and activations in all layers, which is not reasonable in the case of high rate compression (such as 2-bit quantization), as some of layers in deep neural networks are sensitive to quantization and performing coarse quantization on these layers can hurt the accuracy. In this paper, we address an important problem of how to optimize the bit allocation of weights and activations for deep CNNs compression. We first explore the additivity of output error caused by quantization and find that additivity property holds for deep neural networks which are continuously differentiable in the layers. Based on this observation, we formulate the optimal bit allocation problem of weights and activations in a joint framework and propose a very efficient method to solve the optimization problem via Lagrangian Formulation. Our method obtains excellent results on deep neural networks. It can compress deep CNN ResNet-50 down to 2 bits with only 0.7% accuracy loss. To the best of our knowledge, this is the first paper that reports 2-bit results on deep CNNs without hurting the accuracy.
In this supplementary material, we provide the mathematical analysis for the additivity of output error, and the additional results of Pareto-optimal bit allocation. Section 1 provides the proof of Proposition 1. Section 2 provides the results of Pareto-optimal bit allocation on ResNet-50  and MobileNet-v1 Howard et al. (2017) at different bitrates.

Review:
Very good paper that studies the error rate of low-bit quantized networks and uses Pareto condition for optimization to find the best allocation of weights over all layers of a network. The theoretical claims are strongly supported by experiments, and the experimental analysis covers state-of-the-art architectures and demonstrates competitive results. The paper in addition also analyzes the inference cost of their approach (in addition to the accuracy results), and shows positive results on ResNet and MobileNet architectures.
The paper primarily shows that the mean squared error of the final output of a quantized network has the additive property of being equal to the sum of squared errors of the outputs obtained by quantizing each layer individually. Although there is no reason why this should be case, experimental results from the authors on AlexNet and VGG-16 validate this. Based on this assumption, the authors then use a Lagrangian based constrained optimization to minimize the sum of squared errors of outputs when individual weights/activations are quantized, with the constraint being the total bit budget for weights and activations. The authors show that this can be optimized under the Pareto condition easily.
The experimental section is quite detailed and covers the popular architectures instead of toy ones. The accuracy results compared to other 2-bit and 4-bit approaches are competitive. It's also nice to see analysis of inference cost where unequal bitrate allocation performs better than other methods.
The authors show that given the constrained optimization, layers that have a large number of weights receive lower bitrates and vice-versa. While it makes sense that this would contribute to stronger inference speedup compared to methods with either equal bitrate allocation across layers or those that allocate higher bitrate to layers with large number of weights, it's not entirely clear why the optimization would produce this allocation in the first place. Do the authors mean to conclude that layers with large number of weights hold a lot of redundancy and don't have a significant impact on the overall accuracy of the model? This needs to be clarified further.