Nesterov Flow May Travel Infinitely Long to Converge to a Minimizer

TL;DR

This paper demonstrates that the continuous-time Nesterov flow can converge to a minimizer while still accumulating infinite path length, challenging assumptions about convergence behavior.

Contribution

It constructs a differentiable convex potential where the Nesterov flow converges but has infinite path length, showing point convergence does not imply rectifiability.

Findings

Nesterov flow can have infinite path length despite convergence

Constructs a specific convex potential with this property

Challenges previous assumptions about convergence and path length

Abstract

Recent work has established that the trajectory of the Nesterov ODE, a the continuous-time model of Nesterov's accelerated gradient method, exhibits point convergence towards a minimizer of a convex potential. A natural next question is whether this point convergence can be upgraded to rectifiability, namely whether the convergent orbit has finite path length. This work provides the answer in the negative by constructing a differentiable convex potential in $\mathbb{R}^2$ for which the flow converges to its minimizer but still accumulates infinite path length. All proofs of this work are due entirely to an internal model at OpenAI.