A Systematic Approach to Lyapunov Analyses of Continuous-Time Models in Convex Optimization
Céline Moucer, Adrien Taylor, Francis Bach · SIAM Journal on Optimization · 2023
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 finding and verifying Lyapunov functions for classes of ordinary and stochastic differential equations. More precisely, we extend the performance estimation framework, originally proposed by Drori and Teboulle [ Math. Program., 145 (2014), pp. 451–482], to continuous-time models. We retrieve convergence results comparable to those of discrete-time methods using fewer assumptions and inequalities and provide new results for a family of stochastic accelerated gradient flows.