# Local Convergence of the Heavy-ball Method and iPiano for Non-convex Optimization

arXiv: https://arxiv.org/abs/1606.09070 · HTML: https://mlsanity.com/paper/1606.09070/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jun 2016.

## Built on by (most cited first, 9 of 9)

- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Restarted Nonconvex Accelerated Gradient Descent: No More Polylogarithmic Factor in the O(ε-7/4) Complexity](https://mlsanity.com/paper/2201.11411/) (2022)
- [Unified Group Fairness on Federated Learning](https://mlsanity.com/paper/2111.04986/) (2021)
- [Accelerated Proximal Alternating Gradient-Descent-Ascent for Nonconvex Minimax Machine Learning](https://mlsanity.com/paper/2112.11663/) (2021)
- [Polyak's Heavy Ball Method Achieves Accelerated Local Rate of Convergence under Polyak-Lojasiewicz Inequality](https://mlsanity.com/paper/2410.16849/) (2024)
- [Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games](https://mlsanity.com/paper/2505.19537/) (2025)
- [Inertial Newton Algorithms Avoiding Strict Saddle Points](https://mlsanity.com/paper/2111.04596/) (2021)
- [Who breaks early, looses: goal oriented training of deep neural networks based on port Hamiltonian dynamics](https://mlsanity.com/paper/2304.07070/) (2023)
- [Optimal Asymptotic Rates for (Stochastic) Gradient Descent under the Local PL-Condition: A Geometric Approach](https://mlsanity.com/paper/2605.14663/) (2026)

## Abstract (arXiv)

A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges within this neighborhood. This result allows algorithms to exploit local properties of the objective function. In particular, the abstract theory in this paper applies to the inertial forward--backward splitting method: iPiano---a generalization of the Heavy-ball method. Moreover, it reveals an equivalence between iPiano and inertial averaged/alternating proximal minimization and projection methods. Key for this equivalence is the attraction to a local minimum within a neighborhood and the fact that, for a prox-regular function, the gradient of the Moreau envelope is locally Lipschitz continuous and expressible in terms of the proximal mapping. In a numerical feasibility problem, the inertial alternating projection method significantly outperforms its non-inertial variants.