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.

翻译：一阶方法通常通过其连续时间模型进行分析，其最坏情况收敛性质通常借助Lyapunov函数进行研究。本文提出了一种系统且规范的方法，用于寻找并验证常微分方程和随机微分方程类别的Lyapunov函数。具体而言，我们将Drori与Teboulle [10]最初提出的性能估计框架扩展至连续时间模型。通过减少假设条件与凸性不等式的使用，我们获得了与离散方法相媲美的收敛结果，并为随机加速梯度流提供了新的理论发现。