Multi-Agent Distributed Optimization via Inexact Consensus ADMM (1402.6065v2)

Abstract: Multi-agent distributed consensus optimization problems arise in many signal processing applications. Recently, the alternating direction method of multipliers (ADMM) has been used for solving this family of problems. ADMM based distributed optimization method is shown to have faster convergence rate compared with classic methods based on consensus subgradient, but can be computationally expensive, especially for problems with complicated structures or large dimensions. In this paper, we propose low-complexity algorithms that can reduce the overall computational cost of consensus ADMM by an order of magnitude for certain large-scale problems. Central to the proposed algorithms is the use of an inexact step for each ADMM update, which enables the agents to perform cheap computation at each iteration. Our convergence analyses show that the proposed methods converge well under some convexity assumptions. Numerical results show that the proposed algorithms offer considerably lower computational complexity than the standard ADMM based distributed optimization methods.

• The paper introduces inexact update methods that significantly reduce computational complexity in multi-agent consensus optimization.
• It employs proximal gradient steps for ADMM updates, achieving linear convergence under convexity conditions with validated numerical experiments.
• The developed approaches demonstrate practical efficiency in large-scale machine learning and decentralized sensor network applications.

Exploring Multi-Agent Distributed Optimization via Inexact Consensus ADMM

The paper "Multi-Agent Distributed Optimization via Inexact Consensus ADMM" by Tsung-Hui Chang, Mingyi Hong, and Xiangfeng Wang introduces novel methodologies for enhancing multi-agent distributed consensus optimization frameworks using inexact Consensus Alternating Direction Method of Multipliers (ADMM). This paper presents a significant exploration into reducing computational cost in distributed systems commonly observed in contemporary signal processing applications.

The research targets two principal distributed consensus optimization problem formulations, (P1) and (P2), characterized by their multi-agent collaborative decision-making nature. Traditional ADMM-based approaches are leveraged for their facilitation of faster convergence rates concerning problem dimensions or complication levels. However, the proposed methodology intends to further mitigate computational costs through the adoption of inexact updates in the ADMM process.

In the domain of distributed optimization, the traditional Consensus ADMM (C-ADMM) iteratively solves subproblems until global convergence conditions are satisfied. While effective, it necessitates high computational accuracy — an aspect that the authors identify as a bottleneck in scenarios involving large-scale and structurally complex functions. The presented Inexact Consensus ADMM (IC-ADMM) and Inexact Dual Consensus ADMM (IDC-ADMM) strategically introduce low-complexity operations by employing inexact proximal gradient steps for ADMM updates. This modification permits an order of magnitude reduction in computational complexity by replacing costly iterations with more computationally feasible approximations.

The paper rigorously details the mathematical foundation of the proposed methods. A focal point of the research is analyzing the global convergence properties of IC-ADMM, demonstrating that under specific convexity conditions, the convergence is linear — a claim supported by theoretical derivations and numerical experiments. Similarly, the IDC-ADMM method is designed for dual forms by addressing primal congruence through an inexact solution strategy that provides a substantial computational efficiency benefit.

The implications of this research extend beyond theoretical considerations into practical applications. For instance, the proposed methodologies have distinct advantages in applications involving large-scale machine learning, where distributed architectures are leveraged for tasks such as parameter learning and data mining. The paper's numerical simulations on sparse logistic regression underscore the potential for practical deployment, showcasing considerable speed-ups compared to traditional Consensus Subgradient methods without compromising solution accuracy.

In future explorations, refining and adapting these inexact ADMM techniques could catalyze enhanced performance across various facets of distributed optimization scenarios, extending their utility in vast computational environments including sensor networks, data clouds, and decentralized communication systems.

In conclusion, the paper makes substantial contributions to the field of distributed optimization by presenting IC-ADMM and IDC-ADMM as computationally efficient alternatives to conventional methods. These advancements not only make a theoretical impact on how distributed systems can be optimized but also provide practical frameworks for enhancing performance in expansive, real-world applications. Future research may consider further strengthening these techniques in asynchronous settings or other non-standard distributed environments, thus continuing to push boundaries in distributed computational methodologies.