Proximal-Free ADMM for Decentralized Composite Optimization via Graph Simplification

TL;DR

This paper introduces a proximal-free ADMM algorithm for decentralized composite optimization that leverages graph simplification to enhance convergence speed over existing methods.

Contribution

The paper proposes a novel ADMM approach that eliminates the need for proximal terms by using graph simplification techniques, improving convergence in decentralized optimization.

Findings

Achieves faster convergence than existing algorithms

Uses graph simplification to remove proximal terms in ADMM

Demonstrates effectiveness through simulation results

Abstract

We consider the problem of decentralized composite optimization over a symmetric connected graph, in which each node holds its own agent-specific private convex functions, and communications are only allowed between nodes with direct links. A variety of algorithms have been proposed to solve such a problem in an alternating direction method of multiplier (ADMM) framework. Many of these algorithms, however, need to include some proximal term in the augmented Lagrangian function such that the resulting algorithm can be implemented in a decentralized manner. The use of the proximal term slows down the convergence speed because it forces the current solution to stay close to the solution obtained in the previous iteration. To address this issue, in this paper, we first introduce the notion of simplest bipartite graph, which is defined as a bipartite graph that has a minimum number of edges to keep the graph connected. A simple two-step message passing-based procedure is proposed to find a simplest bipartite graph associated with the original graph. We show that the simplest bipartite graph has some interesting properties. By utilizing these properties, an ADMM without involving any proximal terms can be developed to perform decentralized composite optimization over the simplest bipartite graph. Simulation results show that our proposed method achieves a much faster convergence speed than existing state-of-the-art decentralized algorithms.