# Vanilla SGD with Momentum Survives Heavy-Tailed Noise: Convergence Analysis without Gradient Clipping or Normalization

arXiv: https://arxiv.org/abs/2607.08104 · HTML: https://mlsanity.com/paper/2607.08104/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 48 references that are in the map.
- First posted on arXiv 9 Jul 2026.

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

- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [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)
- [An Improved Analysis of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2007.07989/) (2020)
- [Noise Is Not the Main Factor Behind the Gap Between SGD and Adam on Transformers, but Sign Descent Might Be](https://mlsanity.com/paper/2304.13960/) (2023)
- [Linear attention is (maybe) all you need (to understand transformer optimization)](https://mlsanity.com/paper/2310.01082/) (2023)
- [Unified Optimal Analysis of the (Stochastic) Gradient Method](https://mlsanity.com/paper/1907.04232/) (2019)
- [High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails](https://mlsanity.com/paper/2106.14343/) (2021)
- [SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation](https://mlsanity.com/paper/2006.10311/) (2020)
- [Global convergence of the Heavy-ball method for convex optimization](https://mlsanity.com/paper/1412.7457/) (2014)
- [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)
- [Stochastic Heavy Ball](https://mlsanity.com/paper/1609.04228/) (2016)
- [Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping](https://mlsanity.com/paper/2412.19529/) (2024)
- [Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimization](https://mlsanity.com/paper/1604.03257/) (2016)
- [Revisiting the Noise Model of Stochastic Gradient Descent](https://mlsanity.com/paper/2303.02749/) (2023)
- [Convergence of a Stochastic Gradient Method with Momentum for Non-Smooth Non-Convex Optimization](https://mlsanity.com/paper/2002.05466/) (2020)
- [On Proximal Policy Optimization's Heavy-tailed Gradients](https://mlsanity.com/paper/2102.10264/) (2021)
- [Momentum via Primal Averaging: Theoretical Insights and Learning Rate Schedules for Non-Convex Optimization](https://mlsanity.com/paper/2010.00406/) (2020)
- [High Probability Convergence of Clipped-SGD Under Heavy-tailed Noise](https://mlsanity.com/paper/2302.05437/) (2023)
- [Clipping Improves Adam-Norm and AdaGrad-Norm when the Noise Is Heavy-Tailed](https://mlsanity.com/paper/2406.04443/) (2024)
- [Can SGD Handle Heavy-Tailed Noise?](https://mlsanity.com/paper/2508.04860/) (2025)
- [Smoothness parameter of power of Euclidean norm](https://mlsanity.com/paper/1907.12346/) (2019)
- [Convergence of SGD with momentum in the nonconvex case: A time window-based analysis](https://mlsanity.com/paper/2405.16954/) (2024)
- [Accelerated stochastic first-order method for convex optimization under heavy-tailed noise](https://mlsanity.com/paper/2510.11676/) (2025)
- [Accelerating SGDM via Learning Rate and Batch Size Schedules: A Lyapunov-Based Analysis](https://mlsanity.com/paper/2508.03105/) (2025)

## Built on by (most cited first, 1 of 1)

- [The Role of Gradient Modification in Heavy-Tailed Nonconvex Stochastic Min-Max Optimization](https://mlsanity.com/paper/2609.06064/) (2026)

## Abstract (arXiv)

Stochastic gradient descent (SGD) is a cornerstone of modern optimization. While its performance under heavy-tailed noise is often addressed through specialized modifications such as gradient clipping or normalization, we investigate a more fundamental question: how does vanilla SGD, particularly with momentum, perform in the presence of heavy-tailed noise? In this paper, we refine existing convergence results for vanilla SGD and, more importantly, provide the first comprehensive convergence analysis of vanilla SGD with momentum for strongly convex, convex, and nonconvex objectives, without employing any gradient control mechanisms. Our results demonstrate that the obtained convergence rates are inferior to the optimal rates achieved by clipped or normalized variants of SGD, thereby revealing inherent limitations of vanilla methods under heavy-tailed noise. The theoretical findings are supported by experiments on synthetic functions.