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This paper introduces a new algorithm for consensus optimization in a multi-agent network, where all agents collaboratively find a minimizer for the sum of their private functions. All decentralized algorithms rely on communications between adjacent nodes. One class of algorithms use communications between some or all pairs of adjacent agents at each iteration. Another class of algorithms uses a random walk incremental strategy, which sequentially activates a succession of agents. Existing incremental algorithms require diminishing step sizes to converge to the solution, and their convergence is slow. In this work, we propose a random walk algorithm that uses a fixed step size and converges faster to the solution than the existing random walk incremental algorithms. Our algorithm uses only one link to communicate the latest information from an agent to another. Since this style of communication mimics a man walking in a network, we call our 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.

Walkman: A Communication-Efficient Random-Walk Algorithm for Decentralized Optimization