An Asynchronous Approximate Distributed Alternating Direction Method of Multipliers in Digraphs

TL;DR

This paper introduces AsyAD-ADMM, an asynchronous distributed optimization algorithm for directed graphs, which achieves convergence using approximate consensus and is applicable to convex, possibly non-differentiable functions.

Contribution

It proposes a novel asynchronous ADMM-based algorithm leveraging finite-time approximate consensus for directed network topologies.

Findings

Converges at a rate of O(1/k) for convex, non-differentiable functions.

Effective in distributed least-squares optimization.

Demonstrates robustness to asynchronous communication delays.

Abstract

In this work, we consider the asynchronous distributed optimization problem in which each node has its own convex cost function and can communicate directly only with its neighbors, as determined by a directed communication topology (directed graph or digraph). First, we reformulate the optimization problem so that Alternating Direction Method of Multipliers (ADMM) can be utilized. Then, we propose an algorithm, herein called Asynchronous Approximate Distributed Alternating Direction Method of Multipliers (AsyAD-ADMM), using finite-time asynchronous approximate ratio consensus, to solve the multi-node convex optimization problem, in which every node performs iterative computations and exchanges information with its neighbors asynchronously. More specifically, at every iteration of AsyAD-ADMM, each node solves a local convex optimization problem for one of the primal variables and utilizes a finite-time asynchronous approximate consensus protocol to obtain the value of the other variable which is close to the optimal value, since the cost function for the second primal variable is not decomposable. If the individual cost functions are convex but not necessarily differentiable, the proposed algorithm converges at a rate of $\mathcal{O}(1/k)$, where $k$ is the iteration counter. The efficacy of AsyAD-ADMM is exemplified via a proof-of-concept distributed least-square optimization problem with different performance-influencing factors investigated.