A continuous perspective on the inertial corrected primal-dual proximal splitting

TL;DR

This paper offers a continuous-time analysis of the inertial corrected primal-dual proximal splitting algorithm, revealing new ODE models and convergence insights for saddle-point problem solutions.

Contribution

It introduces a semi-implicit Euler scheme perspective and two novel second-order ODE models, enhancing understanding of the algorithm's dynamics and convergence.

Findings

Reconstruction of IC-PDPS as a semi-implicit Euler scheme.

Derivation of two equivalent second-order ODE models.

Convergence analysis of the Lagrangian gap using Lyapunov functions.

Abstract

We give a continuous perspective on the Inertial Corrected Primal-Dual Proximal Splitting (IC-PDPS) proposed by Valkonen ({\it SIAM J. Optim.}, 30(2): 1391--1420, 2020) for solving saddle-point problems. The algorithm possesses nonergodic convergence rate and admits a tight preconditioned proximal point formulation which involves both inertia and additional correction. Based on new understandings on the relation between the discrete step size and rescaling effect, we rebuild IC-PDPS as a semi-implicit Euler scheme with respect to its iterative sequences and integrated parameters. This leads to two novel second-order ordinary differential equation (ODE) models that are equivalent under proper time transformation, and also provides an alternative interpretation from the continuous point of view. Besides, we present the convergence analysis of the Lagrangian gap along the continuous trajectory by using proper Lyapunov functions.