Accelerated Gradient Methods Through Variable and Operator Splitting

TL;DR

This paper presents a unified framework for accelerated gradient methods using variable and operator splitting, enabling efficient solutions for various convex and saddle point optimization problems.

Contribution

It introduces a novel VOS framework with Lyapunov analysis and advanced discretization techniques, enhancing acceleration and stability in gradient methods.

Findings

Developed strong Lyapunov functions for convergence analysis.

Implemented advanced discretization methods like AOR and EPC.

Demonstrated effectiveness on convex and saddle point problems.

Abstract

This paper introduces a unified framework for accelerated gradient methods through the variable and operator splitting (VOS). The operator splitting decouples the optimization process into simpler subproblems, and more importantly, the variable splitting leads to acceleration. The key contributions include the development of strong Lyapunov functions to analyze stability and convergence rates, as well as advanced discretization techniques like Accelerated Over-Relaxation (AOR) and extrapolation by the predictor-corrector methods (EPC). For convex case, we introduce a dynamic updating parameter and a perturbed VOS flow. The framework effectively handles a wide range of optimization problems, including convex optimization, composite convex optimization, and saddle point systems with bilinear coupling.