The neural network quantization is highly desired procedure to perform before running the neural networks on mobile devices. Quantization without fine-tuning leads to accuracy drop of the model, whereas commonly used training with quantization is done on the full set of the labeled data and therefore is both time- and resource-consuming. Real life applications require simplification and acceleration of the quantization procedure that will maintain the accuracy of full-precision neural network, especially for modern mobile neural network architectures like Mobilenet-v1, MobileNet-v2 and MNAS. Here we present two methods to significantly optimize the training with the quantization procedure. The first one is introducing the trained scale factors for discretization thresholds that are separate for each filter. The second one is based on mutual rescaling of consequent depth-wise separable convolution and convolution layers. Using the proposed techniques, we quantize the modern mobile architectures of neural networks with the set of train data of only ∼ 10% of the total ImageNet 2012 sample. Such reduction of the train dataset size and a small number of trainable parameters allow to fine-tune the network for several hours while maintaining the high accuracy of the quantized model (the accuracy drop was less than 0.5%). The ready-for-use models and code are available at: https://github.com/agoncharenko1992/FAT-fast-adjustable-threshold.