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.
We also make the connection to the recent lower bounds @cite_19 for the rounds of communication needed by distributed optimization. Arjevani and Shamir @cite_19 prove that, for a class of @math -related functions (see @cite_19 for the definition) and, for a class of algorithms, the rounds of communication achieved by DISCO is optimal. However, as mentioned above, DASVRG needs fewer rounds than DISCO. This is not a contradiction since DASVRG does not fall into the class of algorithms subject to the lower bound in @cite_19 . In particular, the algorithms concerned by @cite_19 can only use the @math local data points from the initial partition to update the local solutions while DASVRG samples and utilizes a second set of data points in each machine in addition to those @math data points.
@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].