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In this work we study high probability bounds for stochastic subgradient methods under heavy tailed noise. In this setting the noise is only assumed to have finite variance as opposed to a sub-Gaussian distribution for which it is known…

Optimization and Control · Mathematics 2024-04-16 Daniela A. Parletta , Andrea Paudice , Massimiliano Pontil , Saverio Salzo

The empirical evidence indicates that stochastic optimization with heavy-tailed gradient noise is more appropriate to characterize the training of machine learning models than that with standard bounded gradient variance noise. Most…

Machine Learning · Computer Science 2026-01-28 Hongxu Chen , Ke Wei , Xiaoming Yuan , Luo Luo

Recent empirical evidence indicates that many machine learning applications involve heavy-tailed gradient noise, which challenges the standard assumptions of bounded variance in stochastic optimization. Gradient clipping has emerged as a…

Optimization and Control · Mathematics 2025-07-10 Florian Hübler , Ilyas Fatkhullin , Niao He

Stochastic optimization is fundamental to modern machine learning. Recent research has extended the study of stochastic first-order methods (SFOMs) from light-tailed to heavy-tailed noise, which frequently arises in practice, with clipping…

Machine Learning · Computer Science 2025-12-17 Chuan He

High-probability analysis of stochastic first-order optimization methods under mild assumptions on the noise has been gaining a lot of attention in recent years. Typically, gradient clipping is one of the key algorithmic ingredients to…

During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity…

We consider non-convex stochastic optimization using first-order algorithms for which the gradient estimates may have heavy tails. We show that a combination of gradient clipping, momentum, and normalized gradient descent yields convergence…

Machine Learning · Computer Science 2021-11-10 Ashok Cutkosky , Harsh Mehta

Heavy-tailed noise in nonconvex stochastic optimization has garnered increasing research interest, as empirical studies, including those on training attention models, suggest it is a more realistic gradient noise condition. This paper…

Optimization and Control · Mathematics 2026-04-17 Shuhua Yu , Dusan Jakovetic , Soummya Kar

In this work, we study the convergence \emph{in high probability} of clipped gradient methods when the noise distribution has heavy tails, ie., with bounded $p$th moments, for some $1<p\le2$. Prior works in this setting follow the same…

Optimization and Control · Mathematics 2023-04-05 Ta Duy Nguyen , Alina Ene , Huy L. Nguyen

We study the distributed stochastic optimization (DSO) problem under a heavy-tailed noise condition by utilizing a multi-agent system. Despite the extensive research on DSO algorithms used to solve DSO problems under light-tailed noise…

Optimization and Control · Mathematics 2025-09-23 Zhan Yu , Lan Liao , Deming Yuan , Daniel W. C. Ho , Ding-Xuan Zhou

We study high-probability convergence in online learning, in the presence of heavy-tailed noise. To combat the heavy tails, a general framework of nonlinear SGD methods is considered, subsuming several popular nonlinearities like sign,…

Machine Learning · Computer Science 2025-03-24 Aleksandar Armacki , Shuhua Yu , Pranay Sharma , Gauri Joshi , Dragana Bajovic , Dusan Jakovetic , Soummya Kar

We study the performance of stochastic first-order methods for finding saddle points of convex-concave functions. A notorious challenge faced by such methods is that the gradients can grow arbitrarily large during optimization, which may…

Machine Learning · Computer Science 2024-06-10 Gergely Neu , Nneka Okolo

In this paper, the problem of distributed optimization is studied via a network of agents. Each agent only has access to a noisy gradient of its own objective function, and can communicate with its neighbors via a network. To handle this…

Optimization and Control · Mathematics 2025-06-19 Yuchen Yang , Kaihong Lu , Long Wang

We consider a first order stochastic optimization framework where, at each iteration, $K$ independent identically distributed (i.i.d.) data point samples are drawn, based on which stochastic gradients can be queried. We allow gradient noise…

Optimization and Control · Mathematics 2026-05-11 Manojlo Vukovic , Dusan Jakovetic

Gradient clipping is a widely used technique in Machine Learning and Deep Learning (DL), known for its effectiveness in mitigating the impact of heavy-tailed noise, which frequently arises in the training of large language models.…

Optimization and Control · Mathematics 2025-09-30 Savelii Chezhegov , Aleksandr Beznosikov , Samuel Horváth , Eduard Gorbunov

This paper studies decentralized stochastic nonconvex optimization problem over row-stochastic networks. We consider the heavy-tailed gradient noise which is empirically observed in many popular real-world applications. Specifically, we…

Optimization and Control · Mathematics 2026-01-19 Menglian Wang , Zhuanghua Liu , Luo Luo

We develop a worst-case complexity theory for stochastically preconditioned stochastic gradient descent (SPSGD) and its accelerated variants under heavy-tailed noise, a setting that encompasses widely used adaptive methods such as Adam,…

Machine Learning · Computer Science 2026-02-17 Yuchen Fang , James Demmel , Javad Lavaei

Existing decentralized stochastic optimization methods assume the lower-level loss function is strongly convex and the stochastic gradient noise has finite variance. These strong assumptions typically are not satisfied in real-world machine…

Machine Learning · Computer Science 2026-05-26 Xinwen Zhang , Yihan Zhang , Heng Liang , Hongchang Gao

We provide tight finite-time convergence bounds for gradient descent and stochastic gradient descent on quadratic functions, when the gradients are delayed and reflect iterates from $\tau$ rounds ago. First, we show that without stochastic…

Optimization and Control · Mathematics 2018-06-28 Yossi Arjevani , Ohad Shamir , Nathan Srebro

Heavy-tailed noise has attracted growing attention in nonconvex stochastic optimization, as numerous empirical studies suggest it offers a more realistic assumption than standard bounded variance assumption. In this work, we investigate…

Machine Learning · Computer Science 2025-11-07 Xinwen Zhang , Hongchang Gao