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This paper proposes AB-SAGA, a first-order distributed stochasticoptimization method to minimize a finite-sum of smooth and strongly convexfunctions distributed over an arbitrary directed graph. AB-SAGA removes theuncertainty caused by the stochastic gradients using a node-level variancereduction and subsequently employs network-level gradient tracking to addressthe data dissimilarity across the nodes. Unlike existing methods that use thenonlinear push-sum correction to cancel the imbalance caused by the directedcommunication, the consensus updates in AB-SAGA are linear and uses both rowand column stochastic weights. We show that for a constant step-size, AB-SAGAconverges linearly to the global optimal. We quantify the directed nature ofthe underlying graph using an explicit directivity constant and characterizethe regimes in which AB-SAGA achieves a linear speed-up over its centralizedcounterpart. Numerical experiments illustrate the convergence of AB-SAGA forstrongly convex and nonconvex problems.

Variance reduced stochastic optimization over directed graphs with row and column stochastic weights