  Existing theory predicts that data heterogeneity will degrade the performance
of the Federated Averaging (FedAvg) algorithm in federated learning. However,
in practice, the simple FedAvg algorithm converges very well. This paper
explains the seemingly unreasonable effectiveness of FedAvg that contradicts
the previous theoretical predictions. We find that the key assumption of
bounded gradient dissimilarity in previous theoretical analyses is too
pessimistic to characterize data heterogeneity in practical applications. For a
simple quadratic problem, we demonstrate there exist regimes where large
gradient dissimilarity does not have any negative impact on the convergence of
FedAvg. Motivated by this observation, we propose a new quantity, average drift
at optimum, to measure the effects of data heterogeneity, and explicitly use it
to present a new theoretical analysis of FedAvg. We show that the average drift
at optimum is nearly zero across many real-world federated training tasks,
whereas the gradient dissimilarity can be large. And our new analysis suggests
FedAvg can have identical convergence rates in homogeneous and heterogeneous
data settings, and hence, leads to better understanding of its empirical
success.
