A Dynamical Systems Perspective on Nesterov Acceleration

TL;DR

This paper offers a dynamical systems perspective on Nesterov acceleration, deriving it from discretizing an ODE without vanishing step size, and highlights the role of curvature-dependent damping in the acceleration phenomenon.

Contribution

It introduces a novel dynamical systems framework for Nesterov acceleration that does not rely on traditional vanishing step size assumptions.

Findings

Nesterov acceleration arises from discretizing a specific ODE.

A curvature-dependent damping term is central to acceleration.

Connections between continuous and discretized dynamics are established.

Abstract

We present a dynamical system framework for understanding Nesterov's accelerated gradient method. In contrast to earlier work, our derivation does not rely on a vanishing step size argument. We show that Nesterov acceleration arises from discretizing an ordinary differential equation with a semi-implicit Euler integration scheme. We analyze both the underlying differential equation as well as the discretization to obtain insights into the phenomenon of acceleration. The analysis suggests that a curvature-dependent damping term lies at the heart of the phenomenon. We further establish connections between the discretized and the continuous-time dynamics.