Consider a set of agents collaboratively solving a distributed convex
optimization problem, asynchronously, under stringent communication
constraints. In such situations, when an agent is activated and is allowed to
communicate with only one of its neighbors, we would like to pick the one
holding the most informative local estimate. We propose new algorithms where
the agents with maximal dissent average their estimates, leading to an
information mixing mechanism that often displays faster convergence to an
optimal solution compared to randomized gossip. The core idea is that when two
neighboring agents whose distance between local estimates is the largest among
all neighboring agents in the network average their states, it leads to the
largest possible immediate reduction of the quadratic Lyapunov function used to
establish convergence to the set of optimal solutions. As a broader
contribution, we prove the convergence of max-dissent subgradient methods using
a unified framework that can be used for other state-dependent distributed
optimization algorithms. Our proof technique bypasses the need of establishing
the information flow between any two agents within a time interval of uniform
length by intelligently studying convergence properties of the Lyapunov
function used in our analysis