A Communication-Efficient Random-Walk Algorithm for Decentralized Optimization.

This paper addresses consensus optimization problem in a multi-agent network, where all the agents collaboratively find a common minimizer to the sum of their private functions. Our goal is to develop a decentralized algorithm in which there is no center agent and each agent only communicates with its neighbors.

State-of-the-art decentralized algorithms for consensus optimization with convex objectives use fixed step sizes but involve communications among either {\em all}, or a {\em random subset}, of the agents at each iteration. Another approach is to employ a \emph{random walk incremental} strategy, which activates a succession of nodes and their links, only one node and one link each time; since the existing algorithms in this approach require diminishing step sizes to converge to the optimal solution, its convergence is relatively slow.

In this work, we propose a random walk algorithm that uses a fixed step size and converges faster than the existing random walk incremental algorithms. It is also communication efficient. We derive our algorithm by modifying ADMM and analyze its convergence. We establish linear convergence for least squares problems, along with a state-of-the-art communication complexity. Numerical experiments verify our analyses.