text: 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.
  @cite_19: We study the fundamental limits to communication-efficient distributed methods for convex learning and optimization, under different assumptions on the information available to individual machines, and the types of functions considered. We identify cases where existing algorithms are already worst-case optimal, as well as cases where room for further improvement is still possible. Among other things, our results indicate that without similarity between the local objective functions (due to statistical data similarity or otherwise) many communication rounds may be required, even if the machines have unbounded computational power.
summary: 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.