Convergence analysis of dynamical systems for optimization by an improved Lyapunov framework

TL;DR

This paper introduces an improved Lyapunov framework for analyzing the convergence of continuous-time dynamical systems in optimization, achieving better or matching convergence rates for strongly convex functions.

Contribution

It combines a recent Lyapunov reorganization technique with a computer-assisted discovery framework to enhance convergence analysis methods.

Findings

Reproduces existing convergence rates

Achieves improved convergence rates in some cases

Provides a systematic approach for Lyapunov function construction

Abstract

We study the convergence analysis of continuous-time dynamical systems associated with optimization methods for strongly convex functions. Recent works have proposed systematic constructions of Lyapunov functions for such analysis, while also revealing limitations of the Lyapunov analysis. Aujol--Dossal--Rondepierre (2023) have proposed a technique to address this issue by reorganizing Lyapunov functions so as to evaluate a quantity $f(x(t)) - f_* - g(t)\|x(t)-x_*\|^2$ rather than $f(x(t)) - f_*$. By combining this technique with our computer-assisted framework to discover Lyapunov functions, we develop an improved method that reproduces an existing convergence rate or yields better rates than previous studies.