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This paper studies Dictionary Learning problems wherein the learning task is distributed over a multi-agent network, modeled as a time-varying directed graph. This formulation is relevant, for instance, in Big Data scenarios where massive amounts of data are collected/stored in different locations (e.g., sensors, clouds) and aggregating and/or processing all data in a fusion center might be inefficient or unfeasible, due to resource limitations, communication overheads or privacy issues. We develop a unified decentralized algorithmic framework for this class of nonconvex problems, which is proved to converge to stationary solutions at a sublinear rate. The new method hinges on Successive Convex Approximation techniques, coupled with a decentralized tracking mechanism aiming at locally estimating the gradient of the smooth part of the sum-utility. To the best of our knowledge, this is the first provably convergent decentralized algorithm for Dictionary Learning and, more generally, bi-convex problems over (time-varying) (di)graphs 

This papers studies multi-agent (convex and nonconvex) optimization over static digraphs. We propose a general distributed asynchronous algorithmic framework whereby i) agents can update their local variables as well as communicate with their neighbors at any time, without any form of coordination; and ii) they can perform their local computations using (possibly) delayed, out-of-sync information from their neighbors. Delays need not be known to the agents or obey any specific profile, and can also be time-varying (but bounded). The algorithm builds on a tracking mechanism that is robust against asynchrony (in the above sense), whose goal is to estimate locally the sum of agents’ gradients. When applied to strongly convex functions, we prove that it converges at an R-linear (geometric) rate as long as the step-size is sufficiently small. A sublinear convergence rate is proved, when nonconvex problems and/or diminishing, uncoordinated step-sizes are employed. To the best of our knowledge, this is the first distributed algorithm with provable geometric convergence rate in such a general asynchonous setting.