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We study distributed optimization to minimize a global objective that is asum of smooth and strongly-convex local cost functions. Recently, severalalgorithms over undirected and directed graphs have been proposed that use agradient tracking method to achieve linear convergence to the global minimizer.However, a connection between these different approaches has been unclear. Inthis paper, we first show that many of the existing first-order algorithms arein fact related with a simple state transformation, at the heart of which liesthe $\mathcal{AB}$ algorithm. We then describe \textit{distributed heavy-ball},denoted as $\mathcal{AB}m$, i.e., $\mathcal{AB}$ with momentum, that combinesgradient tracking with a momentum term and uses nonidentical local step-sizes.By simultaneously implementing both row- and column-stochastic weights,$\mathcal{AB}m$ removes the conservatism in the related work due todoubly-stochastic weights or eigenvector estimation. $\mathcal{AB}m$ thusnaturally leads to optimization and average-consensus over both undirected anddirected graphs, casting a unifying framework over several well-known consensusalgorithms over arbitrary strongly-connected graphs. We show that$\mathcal{AB}m$ has a global $R$-linear rate when the largest step-size ispositive and sufficiently small. Following the standard practice in theheavy-ball literature, we numerically show that $\mathcal{AB}m$ achievesaccelerated convergence especially when the objective function isill-conditioned.

Distributed heavy-ball: A generalization and acceleration of first-order methods with gradient tracking