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In decentralized optimization environments, each agent $i$ in a network of$n$ nodes has its own private function $f_i$, and nodes communicate with theirneighbors to cooperatively minimize the aggregate objective $\sum_{i=1}^n f_i$.In this setting, synchronizing the nodes' updates incurs significantcommunication overhead and computational costs, so much of the recentliterature has focused on the analysis and design of asynchronous optimizationalgorithms, where agents activate and communicate at arbitrary times withoutneeding a global synchronization enforcer. However, most works assume that whena node activates, it selects the neighbor to contact based on a fixedprobability (e.g., uniformly at random), a choice that ignores the optimizationlandscape at the moment of activation. Instead, in this work we introduce anoptimization-aware selection rule that chooses the neighbor providing thehighest dual cost improvement (a quantity related to a dualization of theproblem based on consensus). This scheme is related to the coordinate descent(CD) method with the Gauss-Southwell (GS) rule for coordinate updates; in oursetting however, only a subset of coordinates is accessible at each iteration(because each node can communicate only with its neighbors), so the existingliterature on GS methods does not apply. To overcome this difficulty, wedevelop a new analytical framework for smooth and strongly convex $f_i$ thatcovers the class of set-wise CD algorithms -- a class that directly applies todecentralized scenarios, but is not limited to them -- and we show that theproposed set-wise GS rule achieves a speedup factor of up to the maximum degreein the network (which is in the order of $\Theta(n)$ for highly connectedgraphs). The speedup predicted by our analysis is validated in numericalexperiments with synthetic data.

Pick your Neighbor: Local Gauss-Southwell Rule for Fast Asynchronous Decentralized Optimization