ADMM-Tracking Gradient for Distributed Optimization over Asynchronous and Unreliable Networks

TL;DR

This paper introduces a robust distributed optimization algorithm based on ADMM that ensures linear convergence even with asynchronous agents and packet losses, validated through theoretical proofs and numerical simulations.

Contribution

It presents a novel ADMM-based distributed algorithm with robustness to asynchrony and packet loss, extending convergence guarantees to unreliable network conditions.

Findings

Proves linear convergence for ideal networks with strongly convex costs.

Extends convergence results to asynchronous agents and packet losses.

Demonstrates robustness and efficiency through numerical simulations.

Abstract

In this paper, we propose a novel distributed algorithm for consensus optimization over networks and a robust extension tailored to deal with asynchronous agents and packet losses. Indeed, to robustly achieve dynamic consensus on the solution estimates and the global descent direction, we embed in our algorithms a distributed implementation of the Alternating Direction Method of Multipliers (ADMM). Such a mechanism is suitably interlaced with a local proportional action steering each agent estimate to the solution of the original consensus optimization problem. First, in the case of ideal networks, by using tools from system theory, we prove the linear convergence of the scheme with strongly convex costs. Then, by exploiting the averaging theory, we extend such a first result to prove that the robust extension of our method preserves linear convergence in the case of asynchronous agents and packet losses. Further, by using the notion of Input-to-State Stability, we also guarantee the robustness of the schemes with respect to additional, generic errors affecting the agents' updates. Finally, some numerical simulations confirm our theoretical findings and compare our algorithms with other distributed schemes in terms of speed and robustness.