Distributed Multiagent Optimization: Linear Convergence Rate of ADMM

We propose a distributed algorithm based on Alternating Direction Method of Multipliers (ADMM) to minimize the sum of locally known convex functions. This optimization problem captures many applications in distributed machine learning and statistical estimation. We provide a novel analysis that shows if the functions are strongly convex and have Lipschitz gradients, then an -optimal solution can be computed with O(κf log(1/ )) iterations, where κf is the condition number of the problem. This is the first paper in the literature that establishes this rate of convergence for distributed ADMM, matching the best known iteration complexity for centralized ADMM. Our analysis also highlights the effect of network structure on the convergence rate.