  In federated optimization, heterogeneity in the clients' local datasets and
computation speeds results in large variations in the number of local updates
performed by each client in each communication round. Naive weighted
aggregation of such models causes objective inconsistency, that is, the global
model converges to a stationary point of a mismatched objective function which
can be arbitrarily different from the true objective. This paper provides a
general framework to analyze the convergence of federated heterogeneous
optimization algorithms. It subsumes previously proposed methods such as FedAvg
and FedProx and provides the first principled understanding of the solution
bias and the convergence slowdown due to objective inconsistency. Using
insights from this analysis, we propose FedNova, a normalized averaging method
that eliminates objective inconsistency while preserving fast error
convergence.
