Optimizing ADMM and Over-Relaxed ADMM Parameters for Linear Quadratic Problems

TL;DR

This paper develops a method to optimally select ADMM and over-relaxation parameters specifically for linear quadratic problems, improving convergence in various imaging applications.

Contribution

It introduces a general approach for penalty parameter optimization and a closed-form formula for the relaxation parameter in LQPs, validated through experiments.

Findings

Optimized parameters enhance ADMM convergence rate.

Validated on diverse imaging tasks like registration and MRI reconstruction.

Closed-form formula simplifies parameter selection process.

Abstract

The Alternating Direction Method of Multipliers (ADMM) has gained significant attention across a broad spectrum of machine learning applications. Incorporating the over-relaxation technique shows potential for enhancing the convergence rate of ADMM. However, determining optimal algorithmic parameters, including both the associated penalty and relaxation parameters, often relies on empirical approaches tailored to specific problem domains and contextual scenarios. Incorrect parameter selection can significantly hinder ADMM's convergence rate. To address this challenge, in this paper we first propose a general approach to optimize the value of penalty parameter, followed by a novel closed-form formula to compute the optimal relaxation parameter in the context of linear quadratic problems (LQPs). We then experimentally validate our parameter selection methods through random instantiations and diverse imaging applications, encompassing diffeomorphic image registration, image deblurring, and MRI reconstruction.