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This paper considers the minimization of a sum of smooth and strongly convexfunctions dispatched over the nodes of a communication network. Previous workson the subject either focus on synchronous algorithms, which can be heavilyslowed down by a few slow nodes (the straggler problem), or consider a model ofasynchronous operation (Boyd et al., 2006) in which adjacent nodes communicateat the instants of Poisson point processes. We have two main contributions. 1)We propose CACDM (a Continuously Accelerated Coordinate Dual Method), and forthe Poisson model of asynchronous operation, we prove CACDM to converge tooptimality at an accelerated convergence rate in the sense of Nesterov etStich, 2017. In contrast, previously proposed asynchronous algorithms have notbeen proven to achieve such accelerated rate. While CACDM is based on discreteupdates, the proof of its convergence crucially depends on a continuous timeanalysis. 2) We introduce a new communication scheme based on Loss-Networks,that is programmable in a fully asynchronous and decentralized way, unlike thePoisson model of asynchronous operation that does not capture essential aspectsof asynchrony such as non-instantaneous communications and computations. Underthis Loss-Network model of asynchrony, we establish for CDM (a Coordinate DualMethod) a rate of convergence in terms of the eigengap of the Laplacian of thegraph weighted by local effective delays. We believe this eigengap to be afundamental bottleneck for convergence rates of asynchronous optimization.Finally, we verify empirically that CACDM enjoys an accelerated convergencerate in the Loss-Network model of asynchrony.

Asynchrony and Acceleration in Gossip Algorithms