# Non-ergodic Convergence Analysis of Heavy-Ball Algorithms

arXiv: https://arxiv.org/abs/1811.01777 · HTML: https://mlsanity.com/paper/1811.01777/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2018.

## Built on by (most cited first, 14 of 14)

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- [Unified convergence analysis for gradient descent optimization methods in the training of deep neural networks](https://mlsanity.com/paper/2607.04233/) (2026)
- [DRSOM: A Dimension Reduced Second-Order Method](https://mlsanity.com/paper/2208.00208/) (2022)
- [A Lower Bound for the Heavy-Ball Method on Smooth Convex Functions](https://mlsanity.com/paper/2609.08656/) (2026)
- [Continuous Optimization for p-adic Models](https://mlsanity.com/paper/2609.25501/) (2026)
- [Stochastic Heavy Ball with Polyak Step Size and Armijo Line Search: A General Convergence Analysis](https://mlsanity.com/paper/2609.36668/) (2026)

## Abstract (arXiv)

In this paper, we revisit the convergence of the Heavy-ball method, and present improved convergence complexity results in the convex setting. We provide the first non-ergodic O(1/k) rate result of the Heavy-ball algorithm with constant step size for coercive objective functions. For objective functions satisfying a relaxed strongly convex condition, the linear convergence is established under weaker assumptions on the step size and inertial parameter than made in the existing literature. We extend our results to multi-block version of the algorithm with both the cyclic and stochastic update rules. In addition, our results can also be extended to decentralized optimization, where the ergodic analysis is not applicable.