A Variational Perspective on Accelerated Methods in Optimization

TL;DR

This paper introduces a unified continuous-time framework using the Bregman Lagrangian to understand and generate a broad class of accelerated optimization methods, revealing their common geometric structure.

Contribution

It presents the Bregman Lagrangian as a unifying continuous-time perspective for various accelerated methods, including their derivation and generalizations.

Findings

All accelerated methods follow the same continuous-time curve at different speeds.

Nesterov's acceleration can be viewed as discretizing a continuous-time trajectory.

The framework encompasses Euclidean and non-Euclidean accelerated methods.

Abstract

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. While many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the \emph{Bregman Lagrangian} which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods correspond to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov's technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.