Convergence Rate of Distributed ADMM over Networks

We propose a distributed algorithm based on Alternating Direction Method of Multipliers (ADMM) to minimize the sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications in distributed machine learning and statistical estimation. We show that when functions are convex, both the objective function values and the feasibility violation converge with rate $O(\frac{1}{T})$, where $T$ is the number of iterations. We then show that if the functions are strongly convex and have Lipschitz continuous gradients, the sequence generated by our algorithm converges linearly to the optimal solution. In particular, an $\epsilon$-optimal solution can be computed with $O(\sqrt{\kappa_f} \log (1/\epsilon))$ iterations, where $\kappa_f$ is the condition number of the problem. Our analysis also highlights the effect of network structure on the convergence rate through maximum and minimum degree of nodes as well as the algebraic connectivity of the network.