A Lyapunov Analysis of Momentum Methods in Optimization

Abstract

Momentum methods play a significant role in optimization. Examples include Nesterov's accelerated gradient method and the conditional gradient algorithm. Several momentum methods are provably optimal under standard oracle models, and all use a technique called estimate sequences to analyze their convergence properties. The technique of estimate sequences has long been considered difficult to understand, leading many researchers to generate alternative, "more intuitive" methods and analyses. We show there is an equivalence between the technique of estimate sequences and a family of Lyapunov functions in both continuous and discrete time. This connection allows us to develop a simple and unified analysis of many existing momentum algorithms, introduce several new algorithms, and strengthen the connection between algorithms and continuous-time dynamical systems.