Efficient First-order Methods for Convex Minimization: a Constructive Approach

TL;DR

This paper introduces a new constructive approach for designing efficient first-order methods for various convex minimization problems, achieving optimal or improved worst-case guarantees across smooth, non-smooth, and strongly convex cases.

Contribution

It presents a novel technique based on a variant of conjugate gradient to create a family of methods, including fixed-step and universal methods, with optimal worst-case guarantees.

Findings

Derived optimal methods for smooth and non-smooth convex problems.

Developed a universal method requiring a three-dimensional search.

Achieved improved worst-case bounds over Nesterov's fast gradient method.

Abstract

We describe a novel constructive technique for devising efficient first-order methods for a wide range of large-scale convex minimization settings, including smooth, non-smooth, and strongly convex minimization. The technique builds upon a certain variant of the conjugate gradient method to construct a family of methods such that a) all methods in the family share the same worst-case guarantee as the base conjugate gradient method, and b) the family includes a fixed-step first-order method. We demonstrate the effectiveness of the approach by deriving optimal methods for the smooth and non-smooth cases, including new methods that forego knowledge of the problem parameters at the cost of a one-dimensional line search per iteration, and a universal method for the union of these classes that requires a three-dimensional search per iteration. In the strongly convex case, we show how numerical tools can be used to perform the construction, and show that the resulting method offers an improved worst-case bound compared to Nesterov's celebrated fast gradient method.