Abstract: Quantization is an effective method for reducing memory footprint and inference time of Neural Networks. However, ultra low precision quantization could lead to significant degradation in model accuracy. A promising method to address this is to perform mixed-precision quantization, where more sensitive layers are kept at higher precision. However, the search space for a mixed-precision quantization is exponential in the number of layers. Recent work has proposed a novel Hessian based framework, with the aim of reducing this exponential search space by using second-order information. While promising, this prior work has three major limitations: (i) they only use a heuristic metric based on top Hessian eigenvalue as a measure of sensitivity and do not consider the rest of the Hessian spectrum; (ii) their approach only provides relative sensitivity of different layers and therefore requires a manual selection of the mixed-precision setting; and (iii) they do not consider mixed-precision activation quantization. Here, we present HAWQ-V2 which addresses these shortcomings. For (i), we theoretically prove that the right sensitivity metric is the average Hessian trace, instead of just top Hessian eigenvalue. For (ii), we develop a Pareto frontier based method for automatic bit precision selection of different layers without any manual intervention. For (iii), we develop the first Hessian based analysis for mixed-precision activation quantization, which is very beneficial for object detection. We show that HAWQ-V2 achieves new state-of-the-art results for a wide range of tasks. In particular, we present quantization results for InceptionV3, ResNet50, and SqueezeNext, all without any manual bit selection. Furthermore, we present results for object detection on Microsoft COCO, where we achieve 2.6 higher mAP than direct uniform quantization and 1.6 higher mAP than the recently proposed method of FQN, with a smaller model size of 17.9MB.