Acceleration in Distributed Optimization under Similarity

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 similar, due to statistical data similarity or otherwise. In order to reduce the number of communications to reach a solution accuracy, we proposed a 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.