D-ADMM: A Communication-Efficient Distributed Algorithm for Separable Optimization

TLDR

Separable optimization problems involve private cost functions and constraint sets at each node, and low communication is critical in sensor networks where energy consumption is dominated by data exchange. The authors propose D‑ADMM to minimize the sum of node‑specific cost functions while ensuring the solution lies in the intersection of all constraint sets. D‑ADMM is a distributed algorithm that iteratively updates local variables to solve problems such as average consensus, compressed sensing, and support vector machines. The method converges under bipartite networks or strongly convex functions, often converges beyond these conditions, and outperforms state‑of‑the‑art algorithms by requiring fewer communications to reach a given accuracy.

Abstract

We propose a distributed algorithm, named Distributed Alternating Direction Method of Multipliers (D-ADMM), for solving separable optimization problems in networks of interconnected nodes or agents. In a separable optimization problem there is a private cost function and a private constraint set at each node. The goal is to minimize the sum of all the cost functions, constraining the solution to be in the intersection of all the constraint sets. D-ADMM is proven to converge when the network is bipartite or when all the functions are strongly convex, although in practice, convergence is observed even when these conditions are not met. We use D-ADMM to solve the following problems from signal processing and control: average consensus, compressed sensing, and support vector machines. Our simulations show that D-ADMM requires less communications than state-of-the-art algorithms to achieve a given accuracy level. Algorithms with low communication requirements are important, for example, in sensor networks, where sensors are typically battery-operated and communicating is the most energy consuming operation.