Accelerated Alternating Direction Method of Multipliers Gradient Tracking for Distributed Optimization

TL;DR

This paper introduces an accelerated distributed optimization algorithm that combines momentum, ADMM, and gradient tracking to achieve faster convergence over static networks, outperforming existing methods.

Contribution

It develops a novel accelerated ADMM gradient tracking algorithm with a rigorous convergence analysis and demonstrates superior performance through simulations.

Findings

Achieves faster convergence than non-accelerated ADMM gradient tracking.

Proves the algorithm's convergence rate surpasses existing methods.

Numerical results confirm improved convergence speed in simulations.

Abstract

This paper presents a novel accelerated distributed algorithm for unconstrained consensus optimization over static undirected networks. The proposed algorithm combines the benefits of acceleration from momentum, the robustness of the alternating direction method of multipliers, and the computational efficiency of gradient tracking to surpass existing state-of-the-art methods in convergence speed, while preserving their computational and communication cost. First, we prove that, by applying momentum on the average dynamic consensus protocol over the estimates and gradient, we can study the algorithm as an interconnection of two singularly perturbed systems: the outer system connects the consensus variables and the optimization variables, and the inner system connects the estimates of the optimum and the auxiliary optimization variables. Next, we prove that, by adding momentum to the auxiliary dynamics, our algorithm always achieves faster convergence than the achievable linear convergence rate for the non-accelerated alternating direction method of multipliers gradient tracking algorithm case. Through simulations, we numerically show that our accelerated algorithm surpasses the existing accelerated and non-accelerated distributed consensus first-order optimization protocols in convergence speed.