Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent

TL;DR

This paper introduces linear coupling, a unifying framework that combines gradient and mirror descent to create faster optimization algorithms, including a new interpretation of Nesterov's accelerated methods and extensions to broader settings.

Contribution

The paper presents linear coupling as a novel unification of gradient and mirror descent, enabling the design of improved algorithms and new insights into existing methods.

Findings

Reconstructed Nesterov's accelerated methods using linear coupling.

Provided a cleaner interpretation of Nesterov's proofs.

Extended linear coupling to settings beyond Nesterov's methods.

Abstract

First-order methods play a central role in large-scale machine learning. Even though many variations exist, each suited to a particular problem, almost all such methods fundamentally rely on two types of algorithmic steps: gradient descent, which yields primal progress, and mirror descent, which yields dual progress. We observe that the performances of gradient and mirror descent are complementary, so that faster algorithms can be designed by LINEARLY COUPLING the two. We show how to reconstruct Nesterov's accelerated gradient methods using linear coupling, which gives a cleaner interpretation than Nesterov's original proofs. We also discuss the power of linear coupling by extending it to many other settings that Nesterov's methods cannot apply to.