Distributed Stochastic Optimization under a General Variance Condition

TL;DR

This paper establishes convergence guarantees for distributed stochastic optimization algorithms like FedAvg and SCAFFOLD under a mild variance condition, addressing data heterogeneity and extending theoretical understanding.

Contribution

It provides the first convergence analysis of these algorithms under a general variance condition, relaxing previous boundedness assumptions.

Findings

Convergence to stationary points under mild variance conditions

Almost sure convergence established for nonconvex objectives

Implications of data heterogeneity measurement discussed

Abstract

Distributed stochastic optimization has drawn great attention recently due to its effectiveness in solving large-scale machine learning problems. Though numerous algorithms have been proposed and successfully applied to general practical problems, their theoretical guarantees mainly 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 Federated Averaging (FedAvg) algorithm (McMahan et al., 2017) as well as the more recent SCAFFOLD method (Karimireddy et al., 2020) 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.