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We revisit the general framework introduced by Fazylab et al. (SIAM J. Optim.28, 2018) to construct Lyapunov functions for optimization algorithms indiscrete and continuous time. For smooth, strongly convex objective functions,we relax the requirements necessary for such a construction. As a result we areable to prove for Polyak's ordinary differential equations and for atwo-parameter family of Nesterov algorithms rates of convergence that improveon those available in the literature. We analyse the interpretation of Nesterovalgorithms as discretizations of the Polyak equation. We show that thealgorithms are instances of Additive Runge-Kutta integrators and discuss thereasons why most discretizations of the differential equation do not result inoptimization algorithms with acceleration. We also introduce a modification ofPolyak's equation and study its convergence properties. Finally we extend thegeneral framework to the stochastic scenario and consider an application torandom algorithms with acceleration for overparameterized models; again we areable to prove convergence rates that improve on those in the literature.

On the connections between optimization algorithms, Lyapunov functions, and differential equations: theory and insights