On the (linear) convergence of Generalized Newton Inexact ADMM

Zachary Frangella, Theo Diamandis, Bartolomeo Stellato, Madeleine Udell

Abstract: This paper presents GeNI-ADMM, a framework for large-scale composite convex optimization that facilitates theoretical analysis of both existing and new approximate ADMM schemes. GeNI-ADMM encompasses any ADMM algorithm that solves a first- or second-order approximation to the ADMM subproblem inexactly. GeNI-ADMM exhibits the usual O(1/t)-convergence rate under standard hypotheses and converges linearly under additional hypotheses such as strong convexity. Further, the GeNI-ADMM framework provides explicit convergence rates for ADMM variants accelerated with randomized linear algebra, such as NysADMM and sketch-and-solve ADMM, resolving an important open question on the convergence of these methods. This analysis quantifies the benefit of improved approximations and can aid in the design of new ADMM variants with faster convergence.

Submission Length: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=ofAXexUGLO&referrer=%5BAuthor%20Console%5D(%2Fgroup%3Fid%3DTMLR%2FAuthors%23your-submissions)

Changes Since Last Submission: A revised manuscript that takes into account the comments and suggestions of the Associate Editor.

Code: https://github.com/tjdiamandis/GeNIADMM.jl

Assigned Action Editor: ~Ahmet_Alacaoglu2

Submission Number: 5874