On the (linear) convergence of Generalized Newton Inexact ADMM

TL;DR

This paper introduces GeNI-ADMM, a comprehensive framework for analyzing the convergence of approximate ADMM algorithms, including new variants accelerated with randomized linear algebra, establishing their convergence rates.

Contribution

The paper develops GeNI-ADMM, enabling unified convergence analysis of approximate ADMM schemes, including novel accelerated variants with explicit rates and open questions addressed.

Findings

Standard ADMM exhibits $ ext{O}(1/t)$ convergence.

Strong convexity leads to linear convergence.

Accelerated ADMM variants with randomized linear algebra also converge linearly.

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 $\mathcal{O} (1/t)$-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.