# Understanding the Role of Momentum in Stochastic Gradient Methods

arXiv: https://arxiv.org/abs/1910.13962 · HTML: https://mlsanity.com/paper/1910.13962/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2019.

## Built on by (most cited first, 30 of 44)

- [Quasi-Global Momentum: Accelerating Decentralized Deep Learning on Heterogeneous Data](https://mlsanity.com/paper/2102.04761/) (2021)
- [DecentLaM: Decentralized Momentum SGD for Large-batch Deep Training](https://mlsanity.com/paper/2104.11981/) (2021)
- [Strength of Minibatch Noise in SGD](https://mlsanity.com/paper/2102.05375/) (2021)
- [The Marginal Value of Momentum for Small Learning Rate SGD](https://mlsanity.com/paper/2307.15196/) (2023)
- [On the fast convergence of minibatch heavy ball momentum](https://mlsanity.com/paper/2206.07553/) (2022)
- [Stochastic Polyak Step-sizes and Momentum: Convergence Guarantees and Practical Performance](https://mlsanity.com/paper/2406.04142/) (2024)
- [Minibatch and Momentum Model-based Methods for Stochastic Weakly Convex Optimization](https://mlsanity.com/paper/2106.03034/) (2021)
- [Information-Theoretic Generalization Bounds for Deep Neural Networks](https://mlsanity.com/paper/2404.03176/) (2024)
- [Momentum Further Constrains Sharpness at the Edge of Stochastic Stability](https://mlsanity.com/paper/2604.14108/) (2026)
- [Acceleration of stochastic gradient descent with momentum by averaging: finite-sample rates and asymptotic normality](https://mlsanity.com/paper/2305.17665/) (2023)
- [Accelerated Convergence of Stochastic Heavy Ball Method under Anisotropic Gradient Noise](https://mlsanity.com/paper/2312.14567/) (2023)
- [Training Deep Neural Networks with Adaptive Momentum Inspired by the Quadratic Optimization](https://mlsanity.com/paper/2110.09057/) (2021)
- [The Role of Momentum Parameters in the Optimal Convergence of Adaptive Polyak's Heavy-ball Methods](https://mlsanity.com/paper/2102.07314/) (2021)
- [On the Convergence of mSGD and AdaGrad for Stochastic Optimization](https://mlsanity.com/paper/2201.11204/) (2022)
- [Revisiting the Role of Euler Numerical Integration on Acceleration and Stability in Convex Optimization](https://mlsanity.com/paper/2102.11537/) (2021)
- [Analytical Study of Momentum-Based Acceleration Methods in Paradigmatic High-Dimensional Non-Convex Problems](https://mlsanity.com/paper/2102.11755/) (2021)
- [Enhance Diffusion to Improve Robust Generalization](https://mlsanity.com/paper/2306.02618/) (2023)
- [Bandwidth-based Step-Sizes for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/2106.02888/) (2021)
- [Gradient Temporal Difference with Momentum: Stability and Convergence](https://mlsanity.com/paper/2111.11004/) (2021)
- [Lean and Mean Adaptive Optimization via Subset-Norm and Subspace-Momentum with Convergence Guarantees](https://mlsanity.com/paper/2411.07120/) (2024)
- [Explicit and Implicit Graduated Optimization in Deep Neural Networks](https://mlsanity.com/paper/2412.11501/) (2024)
- [Increasing Batch Size Improves Convergence of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2501.08883/) (2025)
- [AdamNX: An Adam improvement algorithm based on a novel exponential decay mechanism for the second-order moment estimate](https://mlsanity.com/paper/2511.13465/) (2025)
- [Scaling transition from momentum stochastic gradient descent to plain stochastic gradient descent](https://mlsanity.com/paper/2106.06753/) (2021)
- [Computing the Variance of Shuffling Stochastic Gradient Algorithms via Power Spectral Density Analysis](https://mlsanity.com/paper/2206.00632/) (2022)
- [Tradeoffs between convergence rate and noise amplification for momentum-based accelerated optimization algorithms](https://mlsanity.com/paper/2209.11920/) (2022)
- [Effectively Leveraging Momentum Terms in Stochastic Line Search Frameworks for Fast Optimization of Finite-Sum Problems](https://mlsanity.com/paper/2411.07102/) (2024)
- [Sharpness-Aware Minimization with Adaptive Regularization for Training Deep Neural Networks](https://mlsanity.com/paper/2412.16854/) (2024)
- [Both Asymptotic and Non-Asymptotic Convergence of Quasi-Hyperbolic Momentum using Increasing Batch Size](https://mlsanity.com/paper/2506.23544/) (2025)
- [Accelerating SGDM via Learning Rate and Batch Size Schedules: A Lyapunov-Based Analysis](https://mlsanity.com/paper/2508.03105/) (2025)

## Abstract (arXiv)

The use of momentum in stochastic gradient methods has become a widespread practice in machine learning. Different variants of momentum, including heavy-ball momentum, Nesterov's accelerated gradient (NAG), and quasi-hyperbolic momentum (QHM), have demonstrated success on various tasks. Despite these empirical successes, there is a lack of clear understanding of how the momentum parameters affect convergence and various performance measures of different algorithms. In this paper, we use the general formulation of QHM to give a unified analysis of several popular algorithms, covering their asymptotic convergence conditions, stability regions, and properties of their stationary distributions. In addition, by combining the results on convergence rates and stationary distributions, we obtain sometimes counter-intuitive practical guidelines for setting the learning rate and momentum parameters.