  The quantized neural networks (QNNs) can be useful for neural network
acceleration and compression, but during the training process they pose a
challenge: how to propagate the gradient of loss function through the graph
flow with a derivative of 0 almost everywhere. In response to this
non-differentiable situation, we propose a novel Asymptotic-Quantized Estimator
(AQE) to estimate the gradient. In particular, during back-propagation, the
graph that relates inputs to output remains smoothness and differentiability.
At the end of training, the weights and activations have been quantized to
low-precision because of the asymptotic behaviour of AQE. Meanwhile, we propose
a M-bit Inputs and N-bit Weights Network (MINW-Net) trained by AQE, a quantized
neural network with 1-3 bits weights and activations. In the inference phase,
we can use XNOR or SHIFT operations instead of convolution operations to
accelerate the MINW-Net. Our experiments on CIFAR datasets demonstrate that our
AQE is well defined, and the QNNs with AQE perform better than that with
Straight-Through Estimator (STE). For example, in the case of the same ConvNet
that has 1-bit weights and activations, our MINW-Net with AQE can achieve a
prediction accuracy 1.5\% higher than the Binarized Neural Network (BNN) with
STE. The MINW-Net, which is trained from scratch by AQE, can achieve comparable
classification accuracy as 32-bit counterparts on CIFAR test sets. Extensive
experimental results on ImageNet dataset show great superiority of the proposed
AQE and our MINW-Net achieves comparable results with other state-of-the-art
QNNs.
