This paper addresses consensus optimization problems in a multi-agent network, where all agents collaboratively find a minimizer for the sum of their private functions. We develop a new decentralized algorithm in which each agent communicates only with its neighbors. State-of-the-art decentralized algorithms use communications between either all pairs of adjacent agents or a random subset of them at each iteration. Another class of algorithms uses a random walk incremental strategy, which sequentially activates a succession of nodes; these incremental algorithms require diminishing step sizes to converge to the solution, so their 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. Our algorithm is also communication efficient. Each iteration uses only one link to communicate the latest information for an agent to another. Since this communication rule mimics a man walking around the network, we call our new algorithm Walkman. We establish convergence for convex and nonconvex objectives. For decentralized least squares, we derive a linear rate of convergence and obtain a better communication complexity than those of other decentralized algorithms. Numerical experiments verify our analysis results.
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