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

arXiv: https://arxiv.org/abs/2107.10123 · HTML: https://mlsanity.com/paper/2107.10123/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2021.

## Built on by (most cited first, 11 of 11)

- [Provable Acceleration of Heavy Ball beyond Quadratics for a Class of Polyak-Lojasiewicz Functions when the Non-Convexity is Averaged-Out](https://mlsanity.com/paper/2206.11872/) (2022)
- [Leveraging Continuous Time to Understand Momentum When Training Diagonal Linear Networks](https://mlsanity.com/paper/2403.05293/) (2024)
- [Exponential convergence rates for momentum stochastic gradient descent in the overparametrized setting](https://mlsanity.com/paper/2302.03550/) (2023)
- [Nesterov acceleration in benignly non-convex landscapes](https://mlsanity.com/paper/2410.08395/) (2024)
- [Variance reduction techniques for stochastic proximal point algorithms](https://mlsanity.com/paper/2308.09310/) (2023)
- [Polyak's Heavy Ball Method Achieves Accelerated Local Rate of Convergence under Polyak-Lojasiewicz Inequality](https://mlsanity.com/paper/2410.16849/) (2024)
- [Convergence of the Stochastic Heavy Ball Method With Approximate Gradients and/or Block Updating](https://mlsanity.com/paper/2303.16241/) (2023)
- [A Moreau Envelope Approach for LQR Meta-Policy Estimation](https://mlsanity.com/paper/2403.17364/) (2024)
- [Convergence of Momentum-Based Optimization Algorithms with Time-Varying Parameters](https://mlsanity.com/paper/2506.11904/) (2025)
- [DRSOM: A Dimension Reduced Second-Order Method](https://mlsanity.com/paper/2208.00208/) (2022)
- [Iterative regularization in classification via hinge loss diagonal descent](https://mlsanity.com/paper/2212.12675/) (2022)

## Abstract (arXiv)

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-Lojasiewicz condition, we derive new linear convergence rates for the associated trajectory, in terms of objective function values, without assuming uniqueness of the minimizer.