  Asynchronous Federated Learning (AFL) confronts inherent challenges arising
from the heterogeneity of devices (e.g., their computation capacities) and
low-bandwidth environments, both potentially causing stale model updates (e.g.,
local gradients) for global aggregation. Traditional approaches mitigating the
staleness of updates typically focus on either adjusting the local updating or
gradient compression, but not both. Recognizing this gap, we introduce a novel
approach that synergizes local updating with gradient compression. Our research
begins by examining the interplay between local updating frequency and gradient
compression rate, and their collective impact on convergence speed. The
theoretical upper bound shows that the local updating frequency and gradient
compression rate of each device are jointly determined by its computing power,
communication capabilities and other factors. Building on this foundation, we
propose an AFL framework called FedLuck that adaptively optimizes both local
update frequency and gradient compression rates. Experiments on image
classification and speech recognization show that FedLuck reduces communication
consumption by 56% and training time by 55% on average, achieving competitive
performance in heterogeneous and low-bandwidth scenarios compared to the
baselines.
