  Adam is the important optimization algorithm to guarantee efficiency and
accuracy for training many important tasks such as BERT and ImageNet. However,
Adam is generally not compatible with information (gradient) compression
technology. Therefore, the communication usually becomes the bottleneck for
parallelizing Adam. In this paper, we propose a communication efficient {\bf
A}DAM {\bf p}reconditioned {\bf M}omentum SGD algorithm-- named APMSqueeze--
through an error compensated method compressing gradients. The proposed
algorithm achieves a similar convergence efficiency to Adam in term of epochs,
but significantly reduces the running time per epoch. In terms of end-to-end
performance (including the full-precision pre-condition step), APMSqueeze is
able to provide {sometimes by up to $2-10\times$ speed-up depending on network
bandwidth.} We also conduct theoretical analysis on the convergence and
efficiency.
