mlsanityData to 14 Oct 2026

Stochastic Heavy Ball

arXiv 1609.04228 ↗First posted 14 Sep 2016math.ST math.PR stat.MLcited by 42

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Abstract (arXiv)

This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f . The family of second-order methods recently received a large amount of attention, until the famous contribution of Nesterov [Nes83], leading to the explosion of large-scale optimization problems. This work provides an in-depth description of the stochastic heavy-ball method, which is an adaptation of the deterministic one when only unbiased evalutions of the gradient are available and used throughout the iterations of the algorithm. We first describe some almost sure convergence results in the case of general non-convex coercive functions f . We then examine the situation of convex and strongly convex potentials and derive some non-asymptotic results about the stochastic heavy-ball method. We end our study with limit theorems on several rescaled algorithms.

Key facts. Cited by 42 papers in the map (data through 14 Oct 2026). First posted on arXiv 14 Sep 2016.

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Citing papers per year: 2021: 11; 2022: 5; 2023: 10; 2024: 3; 2025: 6; 2026: 7.

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  2. 2023Stochastic Policy Gradient Methods: Improved Sample Complexity for Fisher-non-degenerate Policiescited by 58
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  4. 2021A Statistical Analysis of Polyak-Ruppert Averaged Q-learningcited by 29
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  8. 2023Reinforcement Learning with General Utilities: Simpler Variance Reduction and Large State-Action Spacecited by 18
  9. 2025Can SGD Handle Heavy-Tailed Noise?cited by 17
  10. 2021Dynamics of Stochastic Momentum Methods on Large-scale, Quadratic Modelscited by 16
  11. 2022On the fast convergence of minibatch heavy ball momentumcited by 13
  12. 2022Accelerated SGD for Non-Strongly-Convex Least Squarescited by 9
30 more
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  4. 2023Self-Repellent Random Walks on General Graphs -- Achieving Minimal Sampling Variance via Nonlinear Markov Chainscited by 8
  5. 2024Central Limit Theorem for Two-Timescale Stochastic Approximation with Markovian Noise: Theory and Applicationscited by 7
  6. 2021Analytical Study of Momentum-Based Acceleration Methods in Paradigmatic High-Dimensional Non-Convex Problemscited by 5
  7. 2024Almost sure convergence rates of stochastic gradient methods under gradient dominationcited by 5
  8. 2021SGD with Coordinate Sampling: Theory and Practicecited by 4
  9. 2021Randomized Stochastic Gradient Descent Ascentcited by 3
  10. 2025Randomised Splitting Methods and Stochastic Gradient Descentcited by 3
  11. 2021Bandwidth-based Step-Sizes for Non-Convex Stochastic Optimizationcited by 2
  12. 2021Gradient Temporal Difference with Momentum: Stability and Convergencecited by 2
  13. 2023Convergence of the Stochastic Heavy Ball Method With Approximate Gradients and/or Block Updatingcited by 2
  14. 2024Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walkscited by 2
  15. 2025Random feature approximation for general spectral methodscited by 2
  16. 2026A Polyak-Ruppert Central Limit Theorem for SA-Adam with Momentum and Non-Convergent Adaptive Preconditioningcited by 2
  17. 2021Improving Adversarial Robustness for Free with Snapshot Ensemblecited by 1
  18. 2023Stochastic Ratios Tracking Algorithm for Large Scale Machine Learning Problemscited by 1
  19. 2025Convergence of Momentum-Based Optimization Algorithms with Time-Varying Parameterscited by 1
  20. 2025Accelerating SGDM via Learning Rate and Batch Size Schedules: A Lyapunov-Based Analysiscited by 1
  21. 2026Vanilla SGD with Momentum Survives Heavy-Tailed Noise: Convergence Analysis without Gradient Clipping or Normalizationcited by 1
  22. 2021On the Generalization of Stochastic Gradient Descent with Momentumcited by 0
  23. 2023Stability and Convergence of Distributed Stochastic Approximations with large Unbounded Stochastic Information Delayscited by 0
  24. 2023Random feature approximation for general spectral methodscited by 0
  25. 2025Convergence Rate for the Last Iterate of Stochastic Gradient Descent Schemescited by 0
  26. 2026Perfect Parallelization in Mini-Batch SGD with Classical Momentum Accelerationcited by 0
  27. 2026Dynamics of Stochastic Momentum with Sparse Updates in High Dimensionscited by 0
  28. 2026Nonsmooth Optimization via Orthogonalized Momentumcited by 0
  29. 2026Stochastic Heavy Ball with Polyak Step Size and Armijo Line Search: A General Convergence Analysiscited by 0
  30. 2026A Nonlinear Time-Series Theory for Constant-step SGD with Momentumcited by 0