# Online Alternating Direction Method of Multipliers for Online Composite Optimization

## TL;DR

This paper introduces an online semi-proximal ADMM algorithm with proven sublinear regret bounds for solving linearly constrained convex composite problems, demonstrating its effectiveness through theoretical analysis and numerical experiments.

## Contribution

It develops an online ADMM method with regret bounds and analyzes its parameter settings, extending the applicability of ADMM to online convex optimization.

## Key findings

- Achieves ${
m O}(
oot{N}{})$ regret bounds for objective and constraint violations.
- Provides guidelines for parameter selection in online ADMM.
- Validates theoretical results with numerical experiments.

## Abstract

In this paper, we investigate regrets of an online semi-proximal alternating direction method of multiplier (Online-spADMM) for solving online linearly constrained convex composite optimization problems. Under mild conditions, we establish ${\rm O}(\sqrt{N})$ objective regret and ${\rm O}(\sqrt{N})$ constraint violation regret at round $N$ when the dual step-length is taken in $(0,(1 +\sqrt{5})/2)$ and penalty parameter $\sigma$ is taken as $\sqrt{N}$. We explain that the optimal value of parameter $\sigma$ is of order ${\rm O}(\sqrt{N})$. Like the semi-proximal alternating direction method of multiplier (spADMM), Online-spADMM has the advantage to resolve the potentially non-solvability issue of the subproblems efficiently. We show the usefulness of the obtained results when applied to different types of online optimization problems and verify the theoretical result by numerical experiments}. The inequalities established for Online-spADMM are also used to develop iteration complexity of the average update of spADMM for solving linearly constrained convex composite optimization problems.