A Decentralized Variance-Reduced Method for Stochastic Optimization Over Directed Graphs

In this paper, we propose a decentralized first-order stochastic optimization method Push-SAGA for finite-sum minimization over a strongly connected directed graph. This method features local variance reduction to remove the uncertainty caused by random sampling of the local gradients, global gradient tracking to address the distributed nature of the data, and push-sum consensus to handle the imbalance caused by the directed nature of the underlying graph. We show that, for a sufficiently small step-size, Push-SAGA linearly converges to the optimal solution for smooth and strongly convex problems, making it the first linearly-convergent stochastic algorithm over arbitrary strongly-connected directed graphs. We illustrate the behavior and convergence properties of Push-SAGA with the help of numerical experiments for strongly convex and non-convex problems.