Revisiting LocalSGD and SCAFFOLD: Improved Rates and Missing Analysis
This work clarifies theoretical advantages for distributed optimization methods, impacting machine learning and federated learning applications, but is incremental as it builds on existing analyses.
The paper revisits the convergence properties of LocalSGD and SCAFFOLD in distributed stochastic optimization, showing that LocalSGD achieves faster convergence than minibatch SGD for weakly convex functions without stronger assumptions, and SCAFFOLD outperforms it for a broader class of non-quadratic functions.
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.