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

Understand

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.