  Although the distributed machine learning methods can speed up the training
of large deep neural networks, the communication cost has become the
non-negligible bottleneck to constrain the performance. To address this
challenge, the gradient compression based communication-efficient distributed
learning methods were designed to reduce the communication cost, and more
recently the local error feedback was incorporated to compensate for the
corresponding performance loss. However, in this paper, we will show that a new
"gradient mismatch" problem is raised by the local error feedback in
centralized distributed training and can lead to degraded performance compared
with full-precision training. To solve this critical problem, we propose two
novel techniques, 1) step ahead and 2) error averaging, with rigorous
theoretical analysis. Both our theoretical and empirical results show that our
new methods can handle the "gradient mismatch" problem. The experimental
results show that we can even train faster with common gradient compression
schemes than both the full-precision training and local error feedback
regarding the training epochs and without performance loss.
