On the Linear Convergence of the ADMM in Decentralized Consensus Optimization

TL;DR

This paper proves that the ADMM algorithm converges linearly in decentralized consensus optimization with strongly convex local functions, providing explicit rates based on network and function properties.

Contribution

It establishes the first explicit linear convergence rate for ADMM in decentralized consensus optimization with strongly convex objectives.

Findings

Linear convergence rate depends on network topology and function properties

Explicit convergence rate formula provided

Guidelines for accelerating ADMM convergence derived

Abstract

In decentralized consensus optimization, a connected network of agents collaboratively minimize the sum of their local objective functions over a common decision variable, where their information exchange is restricted between the neighbors. To this end, one can first obtain a problem reformulation and then apply the alternating direction method of multipliers (ADMM). The method applies iterative computation at the individual agents and information exchange between the neighbors. This approach has been observed to converge quickly and deemed powerful. This paper establishes its linear convergence rate for decentralized consensus optimization problem with strongly convex local objective functions. The theoretical convergence rate is explicitly given in terms of the network topology, the properties of local objective functions, and the algorithm parameter. This result is not only a performance guarantee but also a guideline toward accelerating the ADMM convergence.