Open Problem: Two Riddles in Heavy-Ball Dynamics

TL;DR

This paper discusses two open problems related to Polyak's Heavy-Ball algorithm, focusing on its acceleration in one dimension and its behavior with certain parameters, providing detailed descriptions and partial insights.

Contribution

It introduces two open problems in Heavy-Ball dynamics, offering detailed problem descriptions and initial analysis for each, advancing understanding of the algorithm's behavior.

Findings

Identification of open problems in Heavy-Ball acceleration

Partial analysis and description of problem behaviors

Insights into parameter regimes where behavior is unclear

Abstract

This short paper presents two open problems on the widely used Polyak's Heavy-Ball algorithm. The first problem is the method's ability to exactly \textit{accelerate} in dimension one exactly. The second question regards the behavior of the method for parameters for which it seems that neither a Lyapunov nor a cycle exists. For both problems, we provide a detailed description of the problem and elements of an answer.