Acceleration in Distributed Optimization under Similarity

Abstract: We study distributed (strongly convex) optimization problems over a network of agents, with no centralized nodes. The loss functions of the agents are assumed to be \textit{similar}, due to statistical data similarity or otherwise. In order to reduce the number of communications to reach a solution accuracy, we proposed a {\it preconditioned, accelerated} distributed method. An $\varepsilon$-solution is achieved in $\tilde{\mathcal{O}}\big(\sqrt{\frac{\beta/\mu}{1-\rho}}\log1/\varepsilon\big)$ number of communications steps, where $\beta/\mu$ is the relative condition number between the global and local loss functions, and $\rho$ characterizes the connectivity of the network. This rate matches (up to poly-log factors) lower complexity communication bounds of distributed gossip-algorithms applied to the class of problems of interest. Numerical results show significant communication savings with respect to existing accelerated distributed schemes, especially when solving ill-conditioned problems.