Convergence rates for the Heavy-Ball continuous dynamics for non-convex optimization, under Polyak-\L ojasiewicz condition

TL;DR

This paper analyzes the convergence behavior of the Heavy Ball dynamical system in non-convex optimization, establishing new linear convergence rates under the Polyak-ojasiewicz condition without requiring a unique minimizer.

Contribution

It provides novel linear convergence rate results for Heavy Ball dynamics in non-convex settings under the Polyak-ojasiewicz condition, expanding understanding beyond convex cases.

Findings

Derived new linear convergence rates for Heavy Ball trajectories

Applicable to non-convex smooth optimization without minimizer uniqueness

Results depend on the Polyak-ojasiewicz condition

Abstract

We study convergence of the trajectories of the Heavy Ball dynamical system, with constant damping coefficient, in the framework of convex and non-convex smooth optimization. By using the Polyak-{\L}ojasiewicz condition, we derive new linear convergence rates for the associated trajectory, in terms of objective function values, without assuming uniqueness of the minimizer.