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Consider a set of agents collaboratively solving a distributed convexoptimization problem, asynchronously, under stringent communicationconstraints. In such situations, when an agent is activated and is allowed tocommunicate with only one of its neighbors, we would like to pick the oneholding the most informative local estimate. We propose new algorithms wherethe agents with maximal dissent average their estimates, leading to aninformation mixing mechanism that often displays faster convergence to anoptimal solution compared to randomized gossip. The core idea is that when twoneighboring agents whose distance between local estimates is the largest amongall neighboring agents in the network average their states, it leads to thelargest possible immediate reduction of the quadratic Lyapunov function used toestablish convergence to the set of optimal solutions. As a broadercontribution, we prove the convergence of max-dissent subgradient methods usinga unified framework that can be used for other state-dependent distributedoptimization algorithms. Our proof technique bypasses the need of establishingthe information flow between any two agents within a time interval of uniformlength by intelligently studying convergence properties of the Lyapunovfunction used in our analysis.

Maximal Dissent: a State-Dependent Way to Agree in Distributed Convex Optimization