Proximal gradient flow and Douglas-Rachford splitting dynamics: global exponential stability via integral quadratic constraints

TL;DR

This paper models proximal gradient and Douglas-Rachford splitting algorithms as dynamical systems and uses control theory to prove their global exponential stability and convergence properties, even without strong convexity.

Contribution

It introduces a control-theoretic framework using integral quadratic constraints to analyze the stability of these algorithms as dynamical systems.

Findings

Proximal algorithms are globally exponentially stable for strongly convex problems.

The analysis applies to envelopes derived from the augmented Lagrangian.

Conditions for exponential convergence are established without requiring strong convexity.

Abstract

Many large-scale and distributed optimization problems can be brought into a composite form in which the objective function is given by the sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal gradient method and its variants, thereby generalizing gradient descent to a nonsmooth setup. In this paper, we view proximal algorithms as dynamical systems and leverage techniques from control theory to study their global properties. In particular, for problems with strongly convex objective functions, we utilize the theory of integral quadratic constraints to prove the global exponential stability of the equilibrium points of the differential equations that govern the evolution of proximal gradient and Douglas-Rachford splitting flows. In our analysis, we use the fact that these algorithms can be interpreted as variable-metric gradient methods on the suitable envelopes and exploit structural properties of the nonlinear terms that arise from the gradient of the smooth part of the objective function and the proximal operator associated with the nonsmooth regularizer. We also demonstrate that these envelopes can be obtained from the augmented Lagrangian associated with the original nonsmooth problem and establish conditions for global exponential convergence even in the absence of strong convexity.