  Recently, researchers proposed various low-precision gradient compression,
for efficient communication in large-scale distributed optimization. Based on
these work, we try to reduce the communication complexity from a new direction.
We pursue an ideal bijective mapping between two spaces of gradient
distribution, so that the mapped gradient carries greater information entropy
after the compression. In our setting, all servers should share a reference
gradient in advance, and they communicate via the normalized gradients, which
are the subtraction or quotient, between current gradients and the reference.
To obtain a reference vector that yields a stronger signal-to-noise ratio,
dynamically in each iteration, we extract and fuse information from the past
trajectory in hindsight, and search for an optimal reference for compression.
We name this to be the trajectory-based normalized gradients (TNG). It bridges
the research from different societies, like coding, optimization, systems, and
learning. It is easy to implement and can universally combine with existing
algorithms. Our experiments on benchmarking hard non-convex functions, convex
problems like logistic regression demonstrate that TNG is more
compression-efficient for communication of distributed optimization of general
functions.
