Adaptive first-order methods with enhanced worst-case rates

TL;DR

This paper introduces an adaptive framework for first-order methods that improves convergence guarantees and extends to composite problems, combining memory and dynamic adjustment capabilities.

Contribution

A novel estimate sequence framework that unifies and enhances existing methods like OGM, ITEM, and TMM with adaptivity and applicability to composite optimization.

Findings

Developed a unified adaptive framework for first-order methods.

Extended the framework to composite optimization problems.

Constructed an enhanced accelerated composite gradient method with line-search.

Abstract

The Optimized Gradient Method (OGM), its strongly convex extension, the Information Theoretical Exact Method (ITEM), as well as the related Triple Momentum Method (TMM) have superior convergence guarantees when compared to the Fast Gradient Method but lack adaptivity and their derivation is incompatible with composite problems. In this work we introduce a slightly modified version of the estimate sequence that can be used to simultaneously derive OGM, ITEM and TMM while adding memory along with the ability to dynamically adjust the convergence guarantees at runtime. Our framework can be extended to the composite setup and we use it to construct an Enhanced Accelerated Composite Gradient Method equipped with fully-adaptive line-search.