Lyapunov Functions for First-Order Methods: Tight Automated Convergence Guarantees

Authors

Adrien Taylor, Bryan Van Scoy, Laurent Lessard

Year

2018

Preprint

Algorithm

Lyapunov Functions for First-Order Methods

Source

arXiv

Identifiers

arXiv 1803.06073

Abstract¶

We present a novel way of generating Lyapunov functions for proving linear convergence rates of first-order optimization methods. Our approach provably obtains the fastest linear convergence rate that can be verified by a quadratic Lyapunov function (with given states), and only relies on solving a small-sized semidefinite program. Our approach combines the advantages of performance estimation problems (PEP, due to Drori & Teboulle (2014)) and integral quadratic constraints (IQC, due to Lessard et al. (2016)), and relies on convex interpolation (due to Taylor et al. (2017c;b)).