Convergence analysis of dynamical systems for optimization by an improved Lyapunov framework
This work addresses convergence analysis for optimization methods, but it is incremental as it builds on prior techniques to refine rates.
The paper tackles the problem of analyzing convergence in continuous-time dynamical systems for optimization by improving Lyapunov function techniques, resulting in a method that reproduces or enhances existing convergence rates.
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.