We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. Our method and its accelerated extension also outperform existing distributed algorithms in terms of the rounds of communication as long as the condition number is not too large compared to the size of data in each machine. We also prove a lower bound for the number of rounds of communication for a broad class of distributed first-order methods including the proposed algorithms in this paper. We show that our accelerated distributed stochastic variance reduced gradient algorithm achieves this lower bound so that it uses the fewest rounds of communication among all distributed first-order algorithms.
Building on the work of @cite_19 , we prove a new lower bound showing that any distributed first-order algorithm requires @math rounds. This lower bound combined with the convergence analysis of DASVRG shows that DASVRG is optimal in the number of rounds of communication.
@cite_19
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We show that any randomised Monte Carlo distributed algorithm for the Lov 'asz local lemma requires @math communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of @math , where @math is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lov 'asz local lemma with a running time of @math rounds in bounded-degree graphs, and the best lower bound before our work was @math rounds [ 2014].