Sign-based methods reduce communication costs in distributed environments, but aggregating local signs can introduce bias when data are heterogeneous. As a result, existing sign-based variance reduction methods fail to obtain the optimal convergence rates. In this paper, we solve this problem and obtain optimal rates for both nonconvex stochastic and finite-sum optimization. We first give a counterexample showing that majority voting can fail to approach stationary points even with exact local gradients. Motivated by this limitation, we propose tracking the global gradient at the server through unbiased compression of recursive gradient increments. As a result, we can obtain the convergence rates of
O(d/K+d(a/(nK))1/3) for the
ℓ1-norm and
O(a/K+a/(nK)1/3) for the
ℓ2-norm. Here,
K is the iteration number,
n is the number of workers,
d is the dimension, and
a=1+ω, with
ω denoting the compressor's relative variance. For finite-sum problems with
M components, we combine periodic exact gradient refreshes with compressed component-gradient differences. The resulting total sample complexities are
O(M+daMε−2) and
O(M+aM epsilon−2) for
ℓ1 and
ℓ2 gradient norms at most
ε, matching the corresponding bounds in centralized settings.