  Sign-based stochastic methods have gained attention due to their ability to
achieve robust performance despite using only the sign information for
parameter updates. However, the current convergence analysis of sign-based
methods relies on the strong assumptions of first-order gradient Lipschitz and
second-order gradient Lipschitz, which may not hold in practical tasks like
deep neural network training that involve high non-smoothness. In this paper,
we revisit sign-based methods and analyze their convergence under more
realistic assumptions of first- and second-order smoothness. We first establish
the convergence of the sign-based method under weak first-order Lipschitz.
Motivated by the weak first-order Lipschitz, we propose a relaxed second-order
condition that still allows for nonconvex acceleration in sign-based methods.
Based on our theoretical results, we gain insights into the computational
advantages of the recently developed LION algorithm. In distributed settings,
we prove that this nonconvex acceleration persists with linear speedup in the
number of nodes, when utilizing fast communication compression gossip
protocols. The novelty of our theoretical results lies in that they are derived
under much weaker assumptions, thereby expanding the provable applicability of
sign-based algorithms to a wider range of problems.
