A Unified Convergence Analysis of First Order Convex Optimization Methods via Strong Lyapunov Functions

TL;DR

This paper introduces a unified convergence analysis framework for various first order convex optimization methods using strong Lyapunov functions, applicable to both convex and strongly convex cases, and derives new algorithms from dynamical systems.

Contribution

It develops a systematic Lyapunov-based analysis for first order methods and introduces a novel Hessian-driven Nesterov flow for accelerated optimization.

Findings

Unified convergence analysis for multiple methods

Derivation of existing algorithms from dynamical systems

Introduction of Hessian-driven Nesterov flow

Abstract

We present a unified convergence analysis for first order convex optimization methods using the concept of strong Lyapunov conditions. Combining this with suitable time scaling factors, we are able to handle both convex and strong convex cases, and establish contraction properties of Lyapunov functions for many existing ordinary differential equation models. Then we derive prevailing first order optimization algorithms, such as proximal gradient methods, heavy ball methods (also known as momentum methods), Nesterov accelerated gradient methods, and accelerated proximal gradient methods from numerical discretizations of corresponding dynamical systems. We also apply strong Lyapunov conditions to the discrete level and provide a more systematical analysis framework. Another contribution is a novel second order dynamical system called Hessian-driven Nesterov accelerated gradient flow which can be used to design and analyze accelerated first order methods for smooth and non-smooth convex optimizations.