  Federated learning algorithms, such as FedAvg, are negatively affected by
data heterogeneity and partial client participation. To mitigate the latter
problem, global variance reduction methods, like FedVARP, leverage stale model
updates for non-participating clients. These methods are effective under
homogeneous client participation. Yet, this paper shows that, when some clients
participate much less than others, aggregating updates with different levels of
staleness can detrimentally affect the training process. Motivated by this
observation, we introduce FedStale, a novel algorithm that updates the global
model in each round through a convex combination of "fresh" updates from
participating clients and "stale" updates from non-participating ones. By
adjusting the weight in the convex combination, FedStale interpolates between
FedAvg, which only uses fresh updates, and FedVARP, which treats fresh and
stale updates equally. Our analysis of FedStale convergence yields the
following novel findings: i) it integrates and extends previous FedAvg and
FedVARP analyses to heterogeneous client participation; ii) it underscores how
the least participating client influences convergence error; iii) it provides
practical guidelines to best exploit stale updates, showing that their
usefulness diminishes as data heterogeneity decreases and participation
heterogeneity increases. Extensive experiments featuring diverse levels of
client data and participation heterogeneity not only confirm these findings but
also show that FedStale outperforms both FedAvg and FedVARP in many settings.
