Discovering Low-Precision Networks Close to Full-Precision Networks for Efficient Embedded Inference

To realize the promise of ubiquitous embedded deep network inference, it is essential to seek limits of energy and area efficiency. To this end, low-precision networks offer tremendous promise because both energy and area scale down quadratically with the reduction in precision. Here we demonstrate ResNet-18, -34, -50, -152, Inception-v3, Densenet-161, and VGG-16bn networks on the ImageNet classification benchmark that, at 8-bit precision exceed the accuracy of the full-precision baseline networks after one epoch of finetuning, thereby leveraging the availability of pretrained models. We also demonstrate ResNet-18, -34, -50, -152, Densenet-161, and VGG-16bn 4-bit models that match the accuracy of the full-precision baseline networks -- the highest scores to date. Surprisingly, the weights of the low-precision networks are very close (in cosine similarity) to the weights of the corresponding baseline networks, making training from scratch unnecessary. We find that gradient noise due to quantization during training increases with reduced precision, and seek ways to overcome this noise. The number of iterations required by SGD to achieve a given training error is related to the square of (a) the distance of the initial solution from the final plus (b) the maximum variance of the gradient estimates. Therefore, we (a) reduce solution distance by starting with pretrained fp32 precision baseline networks and fine-tuning, and (b) combat gradient noise introduced by quantization by training longer and reducing learning rates. Sensitivity analysis indicates that these simple techniques, coupled with proper activation function range calibration to take full advantage of the limited precision, are sufficient to discover low-precision networks, if they exist, close to fp32 precision baseline networks. The results herein provide evidence that 4-bits suffice for classification.