Accelerating Distributed Optimization: A Primal-Dual Perspective on Local Steps

In distributed machine learning, efficient training across multiple agents with different data distributions poses significant challenges. Even with a centralized coordinator, current algorithms that achieve optimal communication complexity typically require either large minibatches or compromise on gradient complexity. In this work, we tackle both centralized and decentralized settings across strongly convex, convex, and nonconvex objectives. We first demonstrate that a basic primal-dual method, (Accelerated) Gradient Ascent Multiple Stochastic Gradient Descent (GA-MSGD), applied to the Lagrangian of distributed optimization inherently incorporates local updates, because the inner loops of running Stochastic Gradient Descent on the primal variable require no inter-agent communication. Notably, for strongly convex objectives, we show (Accelerated) GA-MSGD achieves linear convergence in communication rounds despite the Lagrangian being only linear in the dual variables. This is due to a unique structural property where the dual variable is confined to the span of the coupling matrix, rendering the dual problem strongly concave. When integrated with the Catalyst framework, our approach achieves nearly optimal communication complexity across various settings without the need for minibatches. Moreover, in stochastic decentralized problems, it attains communication complexities comparable to those in deterministic settings, improving over existing algorithms.