Push-SAGA: A decentralized stochastic algorithm with variance reduction over directed graphs

TL;DR

Push-SAGA is a novel decentralized stochastic optimization algorithm that combines variance reduction, gradient tracking, and push-sum consensus to achieve linear convergence over directed graphs, outperforming previous methods.

Contribution

It introduces Push-SAGA, the first linearly-convergent stochastic algorithm for strongly convex problems over arbitrary directed graphs.

Findings

Achieves linear convergence to the exact solution.

Attains linear speed-up over centralized algorithms.

Converges at a network-independent rate.

Abstract

In this paper, we propose Push-SAGA, a decentralized stochastic first-order method for finite-sum minimization over a directed network of nodes. Push-SAGA combines node-level variance reduction to remove the uncertainty caused by stochastic gradients, network-level gradient tracking to address the distributed nature of the data, and push-sum consensus to tackle the challenge of directed communication links. We show that Push-SAGA achieves linear convergence to the exact solution for smooth and strongly convex problems and is thus the first linearly-convergent stochastic algorithm over arbitrary strongly connected directed graphs. We also characterize the regimes in which Push-SAGA achieves a linear speed-up compared to its centralized counterpart and achieves a network-independent convergence rate. We illustrate the behavior and convergence properties of Push-SAGA with the help of numerical experiments on strongly convex and non-convex problems.