  A stochastic gradient method for synchronous distributed optimization is
studied. For reducing communication cost, we particularly focus on utilization
of compression of communicated gradients. Several work has shown that
{\it{sparsified}} stochastic gradient descent method (SGD) with {\it{error
feedback}} asymptotically achieves the same rate as (non-sparsified) parallel
SGD. However, from a viewpoint of non-asymptotic behavior, the compression
error may cause slower convergence than non-sparsified SGD in early iterations.
This is problematic in practical situations since early stopping is often
adopted to maximize the generalization ability of learned models. For improving
the previous results, we propose and theoretically analyse a sparsified
stochastic gradient method with error feedback scheme combined with
{\it{Nesterov's acceleration}}. It is shown that the necessary per iteration
communication cost for maintaining the same rate as vanilla SGD can be smaller
than non-accelerated methods in convex and even in nonconvex optimization
problems. This indicates that our proposed method makes a better use of
compressed information than previous methods. Numerical experiments are
provided and empirically validates our theoretical findings.
