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We propose a novel distributed method for monotone variational inequalitiesand convex-concave saddle point problems arising in various machine learningapplications such as game theory and adversarial training. By exploiting\textit{similarity} our algorithm overcomes communication bottleneck which is amajor issue in distributed optimization. The proposed algorithm enjoys optimalcommunication complexity of $\delta/\epsilon$, where $\epsilon$ measures thenon-optimality gap function, and $\delta$ is a parameter of similarity. All theexisting distributed algorithms achieving this bound essentially utilize theEuclidean setup.  In contrast to them, our algorithm is built upon Bregman proximal maps and itis compatible with an arbitrary Bregman divergence. Thanks to this, it has moreflexibility to fit the problem geometry than algorithms with the Euclideansetup. Thereby the proposed method bridges the gap between the Euclidean andnon-Euclidean setting.  By using the restart technique, we extend our algorithm to variationalinequalities with $\mu$-strongly monotone operator, resulting in optimalcommunication complexity of $\delta/\mu$ (up to a logarithmic factor). Ourtheoretical results are confirmed by numerical experiments on a stochasticmatrix game.

Bregman Proximal Method for Efficient Communications under Similarity