First-Order Methods for Convex Optimization

TL;DR

This survey reviews the evolution and key advancements of first-order convex optimization methods, emphasizing their applications, efficiency, and recent algorithmic developments like non-Euclidean and projection-free techniques.

Contribution

It provides a comprehensive overview of recent developments in gradient-based convex optimization, including proofs and unifying perspectives on various algorithms.

Findings

Non-Euclidean extensions improve flexibility of proximal methods

Accelerated methods enhance convergence rates

Projection-free algorithms reduce computational complexity

Abstract

First-order methods for solving convex optimization problems have been at the forefront of mathematical optimization in the last 20 years. The rapid development of this important class of algorithms is motivated by the success stories reported in various applications, including most importantly machine learning, signal processing, imaging and control theory. First-order methods have the potential to provide low accuracy solutions at low computational complexity which makes them an attractive set of tools in large-scale optimization problems. In this survey we cover a number of key developments in gradient-based optimization methods. This includes non-Euclidean extensions of the classical proximal gradient method, and its accelerated versions. Additionally we survey recent developments within the class of projection-free methods, and proximal versions of primal-dual schemes. We give complete proofs for various key results, and highlight the unifying aspects of several optimization algorithms.