# Complexity of normalized stochastic first-order methods with momentum under heavy-tailed noise

arXiv: https://arxiv.org/abs/2506.11214 · HTML: https://mlsanity.com/paper/2506.11214/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 37 references that are in the map.
- First posted on arXiv 12 Jun 2025.

## Builds on (most cited first, 29 of 29)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter](https://mlsanity.com/paper/1910.01108/) (2019)
- [Flickr30k Entities: Collecting Region-to-Phrase Correspondences for Richer Image-to-Sentence Models](https://mlsanity.com/paper/1505.04870/) (2015)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [The Heavy-Tail Phenomenon in SGD](https://mlsanity.com/paper/2006.04740/) (2020)
- [PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization](https://mlsanity.com/paper/2008.10898/) (2020)
- [High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails](https://mlsanity.com/paper/2106.14343/) (2021)
- [Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming](https://mlsanity.com/paper/1310.3787/) (2013)
- [Non-convex Finite-Sum Optimization Via SCSG Methods](https://mlsanity.com/paper/1706.09156/) (2017)
- [High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance](https://mlsanity.com/paper/2302.00999/) (2023)
- [From Gradient Clipping to Normalization for Heavy Tailed SGD](https://mlsanity.com/paper/2410.13849/) (2024)
- ["Convex Until Proven Guilty": Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions](https://mlsanity.com/paper/1705.02766/) (2017)
- [On the Heavy-Tailed Theory of Stochastic Gradient Descent for Deep Neural Networks](https://mlsanity.com/paper/1912.00018/) (2019)
- [Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping](https://mlsanity.com/paper/2412.19529/) (2024)
- [Breaking the Lower Bound with (Little) Structure: Acceleration in Non-Convex Stochastic Optimization with Heavy-Tailed Noise](https://mlsanity.com/paper/2302.06763/) (2023)
- [Revisiting Gradient Normalization and Clipping for Nonconvex SGD under Heavy-Tailed Noise: Necessity, Sufficiency, and Acceleration](https://mlsanity.com/paper/2410.16561/) (2024)
- [Non-convex Stochastic Composite Optimization with Polyak Momentum](https://mlsanity.com/paper/2403.02967/) (2024)
- [Smoothness parameter of power of Euclidean norm](https://mlsanity.com/paper/1907.12346/) (2019)

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

- [Low-rank Orthogonalization for Large-scale Matrix Optimization with Applications to Foundation Model Training](https://mlsanity.com/paper/2509.11983/) (2025)
- [Can SGD Handle Heavy-Tailed Noise?](https://mlsanity.com/paper/2508.04860/) (2025)
- [Stochastic Bilevel Optimization with Heavy-Tailed Noise](https://mlsanity.com/paper/2509.14952/) (2025)
- [DeMuon: A Decentralized Muon for Matrix Optimization over Graphs](https://mlsanity.com/paper/2510.01377/) (2025)
- [Accelerated stochastic first-order method for convex optimization under heavy-tailed noise](https://mlsanity.com/paper/2510.11676/) (2025)
- [Second-order Optimization under Heavy-Tailed Noise: Hessian Clipping and Sample Complexity Limits](https://mlsanity.com/paper/2510.10690/) (2025)
- [Optimal Asynchronous Stochastic Nonconvex Optimization under Heavy-Tailed Noise](https://mlsanity.com/paper/2601.19379/) (2026)
- [The Convergence Behavior of Adam under Heavy-Tailed Noise](https://mlsanity.com/paper/2607.27383/) (2026)
- [Near-Optimal Decentralized Stochastic Nonconvex Optimization with Heavy-Tailed Noise](https://mlsanity.com/paper/2601.11435/) (2026)
- [Stochastic Weakly Convex Optimization Under Heavy-Tailed Noises](https://mlsanity.com/paper/2507.13283/) (2025)
- [Bias-Variance Trade-off for Clipped Stochastic First-Order Methods: From Bounded Variance to Infinite Mean](https://mlsanity.com/paper/2512.14686/) (2025)

## Abstract (arXiv)

In this paper, we propose practical normalized stochastic first-order methods with Polyak momentum, multi-extrapolated momentum, and recursive momentum for solving unconstrained optimization problems. These methods employ dynamically updated algorithmic parameters and do not require explicit knowledge of problem-dependent quantities such as the Lipschitz constant or noise bound. We establish first-order oracle complexity results for finding approximate stochastic stationary points under heavy-tailed noise and weakly average smoothness conditions -- both of which are weaker than the commonly used bounded variance and mean-squared smoothness assumptions. Our complexity bounds either improve upon or match the best-known results in the literature. Numerical experiments are presented to demonstrate the practical effectiveness of the proposed methods.