  A very large number of communications are typically required to solve
distributed learning tasks, and this critically limits scalability and
convergence speed in wireless communications applications. In this paper, we
devise a Gradient Descent method with Sparsification and Error Correction
(GD-SEC) to improve the communications efficiency in a general worker-server
architecture. Motivated by a variety of wireless communications learning
scenarios, GD-SEC reduces the number of bits per communication from worker to
server with no degradation in the order of the convergence rate. This enables
larger-scale model learning without sacrificing convergence or accuracy. At
each iteration of GD-SEC, instead of directly transmitting the entire gradient
vector, each worker computes the difference between its current gradient and a
linear combination of its previously transmitted gradients, and then transmits
the sparsified gradient difference to the server. A key feature of GD-SEC is
that any given component of the gradient difference vector will not be
transmitted if its magnitude is not sufficiently large. An error correction
technique is used at each worker to compensate for the error resulting from
sparsification. We prove that GD-SEC is guaranteed to converge for strongly
convex, convex, and nonconvex optimization problems with the same order of
convergence rate as GD. Furthermore, if the objective function is strongly
convex, GD-SEC has a fast linear convergence rate. Numerical results not only
validate the convergence rate of GD-SEC but also explore the communication bit
savings it provides. Given a target accuracy, GD-SEC can significantly reduce
the communications load compared to the best existing algorithms without
slowing down the optimization process.
