Automated tight Lyapunov analysis for first-order methods

TL;DR

This paper introduces an automated methodology to verify quadratic Lyapunov inequalities for various first-order convex optimization methods, enabling broader convergence analysis through semidefinite programming.

Contribution

It provides a necessary and sufficient condition for the existence of quadratic Lyapunov inequalities, facilitating automated convergence proofs for a wide class of methods.

Findings

Extended the convergence region for Chambolle-Pock method with identity operator.

Automated semidefinite programming approach for Lyapunov inequality verification.

Applicable to a broad class of first-order methods in convex optimization.

Abstract

We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, ii) first-order methods that can be written as a linear system in state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We present a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality within a predefined class of Lyapunov inequalities, which amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality-gap convergence in the Chambolle-Pock method when the linear operator is the identity mapping.