On the convergence analysis of the optimized gradient method

TL;DR

This paper analyzes the convergence properties of the optimized gradient method (OGM), establishing new bounds and properties that deepen understanding of its optimality in smooth convex minimization.

Contribution

It provides an analytic convergence bound for OGM's primary sequence and explores its worst-case functions, advancing the theoretical understanding of optimal first-order methods.

Findings

OGM has a proven convergence bound for the primary sequence.

OGM exhibits two types of worst-case functions: piecewise affine-quadratic and quadratic.

The results complete the theoretical framework of optimal first-order methods.

Abstract

This paper considers the problem of unconstrained minimization of smooth convex functions having Lipschitz continuous gradients with known Lipschitz constant. We recently proposed an optimized gradient method (OGM) for this problem and showed that it has a worst-case convergence bound for the cost function decrease that is twice as small as that of Nesterov's fast gradient method (FGM), yet has a similarly efficient practical implementation. Drori showed recently that OGM has optimal complexity over the general class of first-order methods. This optimality makes it important to study fully the convergence properties of OGM. The previous worst-case convergence bound for OGM was derived for only the last iterate of a secondary sequence. This paper provides an analytic convergence bound for the primary sequence generated by OGM. We then discuss additional convergence properties of OGM, including the interesting fact that OGM has two types of worst-case functions: a piecewise affine-quadratic function and a quadratic function. These results help complete the theory of optimal first-order methods for smooth convex minimization.