Revisiting LocalSGD and SCAFFOLD: Improved Rates and Missing Analysis

TL;DR

This paper provides a comprehensive theoretical analysis of LocalSGD and SCAFFOLD, demonstrating their improved convergence rates over minibatch SGD under weaker and more realistic conditions in distributed optimization.

Contribution

It offers new convergence guarantees for LocalSGD and SCAFFOLD under weaker assumptions, clarifying when these methods outperform simpler stochastic gradient methods.

Findings

LocalSGD converges faster than MbSGD for weakly convex functions.

Higher-order similarity and smoothness enhance LocalSGD performance.

SCAFFOLD outperforms MbSGD for a broader class of non-quadratic functions.

Abstract

LocalSGD and SCAFFOLD are widely used methods in distributed stochastic optimization, with numerous applications in machine learning, large-scale data processing, and federated learning. However, rigorously establishing their theoretical advantages over simpler methods, such as minibatch SGD (MbSGD), has proven challenging, as existing analyses often rely on strong assumptions, unrealistic premises, or overly restrictive scenarios. In this work, we revisit the convergence properties of LocalSGD and SCAFFOLD under a variety of existing or weaker conditions, including gradient similarity, Hessian similarity, weak convexity, and Lipschitz continuity of the Hessian. Our analysis shows that (i) LocalSGD achieves faster convergence compared to MbSGD for weakly convex functions without requiring stronger gradient similarity assumptions; (ii) LocalSGD benefits significantly from higher-order similarity and smoothness; and (iii) SCAFFOLD demonstrates faster convergence than MbSGD for a broader class of non-quadratic functions. These theoretical insights provide a clearer understanding of the conditions under which LocalSGD and SCAFFOLD outperform MbSGD.