Accelerated forward-backward and Douglas-Rachford splitting dynamics

TL;DR

This paper analyzes continuous-time accelerated splitting algorithms for nonsmooth optimization, establishing accelerated convergence rates for specific problem classes through Lyapunov-based methods and computational validation.

Contribution

It introduces a Lyapunov-based analysis framework for continuous-time accelerated FB and DR splitting dynamics, proving new convergence rates for quadratic and strongly convex problems.

Findings

Accelerated sublinear convergence for convex problems.

Exponential convergence for strongly convex problems.

Validation through computational experiments.

Abstract

We examine convergence properties of continuous-time variants of accelerated Forward-Backward (FB) and Douglas-Rachford (DR) splitting algorithms for nonsmooth composite optimization problems. When the objective function is given by the sum of a quadratic and a nonsmooth term, we establish accelerated sublinear and exponential convergence rates for convex and strongly convex problems, respectively. Moreover, for FB splitting dynamics, we demonstrate that accelerated exponential convergence rate carries over to general strongly convex problems. In our Lyapunov-based analysis we exploit the variable-metric gradient interpretations of FB and DR splittings to obtain smooth Lyapunov functions that allow us to establish accelerated convergence rates. We provide computational experiments to demonstrate the merits and the effectiveness of our analysis.