An optimal lower bound for smooth convex functions

TL;DR

This paper introduces an optimal global lower bound for smooth convex functions, enhancing the performance of gradient methods with memory and providing improved convergence guarantees validated by simulations.

Contribution

It defines a globally optimal lower bound for smooth functions and develops an improved gradient method with memory and adaptive guarantees.

Findings

The optimal lower bound improves theoretical convergence guarantees.

The proposed method outperforms existing algorithms on synthetic problems.

Adaptive adjustment enhances convergence without line-search.

Abstract

First order methods endowed with global convergence guarantees operate using global lower bounds on the objective. The tightening of the bounds has been shown to increase both the theoretical guarantees and the practical performance. In this work, we define a global lower bound for smooth differentiable objectives that is optimal with respect to the collected oracle information. The bound can be readily employed by the Gradient Method with Memory to improve its performance. Further using the machinery underlying the optimal bounds, we introduce a modified version of the estimate sequence that we use to construct an Optimized Gradient Method with Memory possessing the best known convergence guarantees for its class of algorithms, even in terms of the proportionality constant. We additionally equip the method with an adaptive convergence guarantee adjustment procedure that is an effective replacement for line-search. Simulation results on synthetic but otherwise difficult smooth problems validate the theoretical properties of the bound and proposed methods.