A General Continuous-Time Formulation of Stochastic ADMM and Its Variants

TL;DR

This paper introduces a unified continuous-time framework for stochastic ADMM and its variants, providing new theoretical insights into their convergence behavior and parameter selection.

Contribution

It develops a generalized stochastic ADMM framework and analyzes its continuous-time limit, explaining convergence and parameter choices.

Findings

Stochastic ADMM trajectories converge to stochastic differential equations.

Proper scaling ensures weak convergence of the algorithm.

The relaxation parameter should be between 0 and 2 for optimal performance.

Abstract

Stochastic versions of the alternating direction method of multiplier (ADMM) and its variants play a key role in many modern large-scale machine learning problems. In this work, we introduce a unified algorithmic framework called generalized stochastic ADMM and investigate their continuous-time analysis. The generalized framework widely includes many stochastic ADMM variants such as standard, linearized and gradient-based ADMM. Our continuous-time analysis provides us with new insights into stochastic ADMM and variants, and we rigorously prove that under some proper scaling, the trajectory of stochastic ADMM weakly converges to the solution of a stochastic differential equation with small noise. Our analysis also provides a theoretical explanation of why the relaxation parameter should be chosen between 0 and 2.