Extending the ergodic convergence rate of the proximal ADMM

TL;DR

This paper extends the ergodic convergence rate analysis of the proximal ADMM to include the critical stepsize (1+√5)/2, providing the first such result at this stepsize in the literature, using a non-Euclidean hybrid proximal extragradient framework.

Contribution

It introduces the first ergodic iteration-complexity result for the proximal ADMM at the stepsize (1+√5)/2, expanding the theoretical understanding of its convergence behavior.

Findings

Extended ergodic convergence rate to stepsize (1+√5)/2

Provided alternative proofs for existing results

Connected proximal ADMM to a non-Euclidean hybrid framework

Abstract

Pointwise and ergodic iteration-complexity results for the proximal alternating direction method of multipliers (ADMM) for any stepsize in(0,(1+\sqrt{5})/2) have been recently established in the literature. In addition to giving alternative proofs of these results, this paper also extends the ergodic iteration-complexity result to include the case in which the stepsize is equal to (1+\sqrt{5})/$. As far as we know, this is the first ergodic iteration-complexity for the stepsize (1+\sqrt{5})/2 obtained in the ADMM literature. These results are obtained by showing that the proximal ADMM is an instance of a non-Euclidean hybrid proximal extragradient framework whose pointwise and ergodic convergence rate are also studied.