Distributed Stochastic Optimization under a General Variance Condition

Distributed stochastic optimization has drawn great attention recently due to its effectiveness in solving large-scale machine learning problems. However, despite that numerous algorithms have been proposed with empirical successes, their theoretical guarantees are restrictive and rely on certain boundedness conditions on the stochastic gradients, varying from uniform boundedness to the relaxed growth condition. In addition, how to characterize the data heterogeneity among the agents and its impacts on the algorithmic performance remains challenging. In light of such motivations, we revisit the classical FedAvg algorithm for solving the distributed stochastic optimization problem and establish the convergence results under only a mild variance condition on the stochastic gradients for smooth nonconvex objective functions. Almost sure convergence to a stationary point is also established under the condition. Moreover, we discuss a more informative measurement for data heterogeneity as well as its implications.