# A Control-Theoretic Approach to Analysis and Parameter Selection of Douglas-Rachford Splitting

## TL;DR

This paper introduces a control-theoretic framework for analyzing and selecting parameters in Douglas-Rachford splitting and ADMM algorithms, providing convergence guarantees and simplifying rate proofs.

## Contribution

It offers a unified, dynamical system-based analysis method for convergence and parameter tuning of Douglas-Rachford splitting, applicable under various convexity and smoothness conditions.

## Key findings

- Derived a dimensionally independent matrix inequality for convergence analysis.
- Provided performance guarantees and optimal parameter settings.
- Simplified proof of O(1/k) convergence rate without strong convexity.

## Abstract

Douglas-Rachford splitting and its equivalent dual formulation ADMM are widely used iterative methods in composite optimization problems arising in control and machine learning applications. The performance of these algorithms depends on the choice of step size parameters, for which the optimal values are known in some specific cases, and otherwise are set heuristically. We provide a new unified method of convergence analysis and parameter selection by interpreting the algorithm as a linear dynamical system with nonlinear feedback. This approach allows us to derive a dimensionally independent matrix inequality whose feasibility is sufficient for the algorithm to converge at a specified rate. By analyzing this inequality, we are able to give performance guarantees and parameter settings of the algorithm under a variety of assumptions regarding the convexity and smoothness of the objective function. In particular, our framework enables us to obtain a new and simple proof of the O(1/k) convergence rate of the algorithm when the objective function is not strongly convex.