A systematic approach to Lyapunov analyses of continuous-time models in convex optimization

TL;DR

This paper develops a systematic method to find and verify Lyapunov functions for continuous-time models in convex optimization, extending existing frameworks and providing new insights into stochastic accelerated gradient flows.

Contribution

It extends the performance estimation framework to continuous-time models, enabling more efficient Lyapunov function construction and analysis for both deterministic and stochastic differential equations.

Findings

Retrieved convergence results similar to discrete methods with fewer assumptions.

Provided new results for stochastic accelerated gradient flows.

Enhanced understanding of Lyapunov functions in continuous-time convex optimization.

Abstract

First-order methods are often analyzed via their continuous-time models, where their worst-case convergence properties are usually approached via Lyapunov functions. In this work, we provide a systematic and principled approach to find and verify Lyapunov functions for classes of ordinary and stochastic differential equations. More precisely, we extend the performance estimation framework, originally proposed by Drori and Teboulle [10], to continuous-time models. We retrieve convergence results comparable to those of discrete methods using fewer assumptions and convexity inequalities, and provide new results for stochastic accelerated gradient flows.