  Communication has been seen as a significant bottleneck in industrial
applications over large-scale networks. To alleviate the communication burden,
sign-based optimization algorithms have gained popularity recently in both
industrial and academic communities, which is shown to be closely related to
adaptive gradient methods, such as Adam. Along this line, this paper
investigates faster convergence for a variant of sign-based gradient descent,
called scaled signGD, in three cases: 1) the objective function is strongly
convex; 2) the objective function is nonconvex but satisfies the
Polyak-Lojasiewicz (PL) inequality; 3) the gradient is stochastic, called
scaled signGD in this case. For the first two cases, it can be shown that the
scaled signGD converges at a linear rate. For case 3), the algorithm is shown
to converge linearly to a neighborhood of the optimal value when a constant
learning rate is employed, and the algorithm converges at a rate of $O(1/k)$
when using a diminishing learning rate, where $k$ is the iteration number. The
results are also extended to the distributed setting by majority vote in a
parameter-server framework. Finally, numerical experiments on logistic
regression are performed to corroborate the theoretical findings.
