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Stochastic gradient descent (SGD) is a simple and popular method to solve stochastic optimization problems which arise in machine learning. For strongly convex problems, its convergence rate was known to be O(\log(T)/T), by running SGD for…

Machine Learning · Computer Science 2015-03-19 Alexander Rakhlin , Ohad Shamir , Karthik Sridharan

We consider a class of stochastic smooth convex optimization problems under rather general assumptions on the noise in the stochastic gradient observation. As opposed to the classical problem setting in which the variance of noise is…

Optimization and Control · Mathematics 2024-08-23 Sasila Ilandarideva , Anatoli Juditsky , Guanghui Lan , Tianjiao Li

There are several applications of stochastic optimization where one can benefit from a robust estimate of the gradient. For example, domains such as distributed learning with corrupted nodes, the presence of large outliers in the training…

Machine Learning · Statistics 2025-10-30 Fabian Schaipp , Guillaume Garrigos , Umut Simsekli , Robert Gower

The injection of heavy-tailed noise into the iterates of stochastic gradient descent (SGD) has garnered growing interest in recent years due to its theoretical and empirical benefits for optimization and generalization. However, its…

Machine Learning · Statistics 2025-05-13 Umut Şimşekli , Mert Gürbüzbalaban , Sinan Yıldırım , Lingjiong Zhu

Differentially Private Stochastic Gradient Descent (DPSGD) is widely utilized to preserve training data privacy in deep learning, which first clips the gradients to a predefined norm and then injects calibrated noise into the training…

Machine Learning · Computer Science 2024-05-29 Haichao Sha , Yang Cao , Yong Liu , Yuncheng Wu , Ruixuan Liu , Hong Chen

Non-convex optimization problems are ubiquitous in machine learning, especially in Deep Learning. While such complex problems can often be successfully optimized in practice by using stochastic gradient descent (SGD), theoretical analysis…

Machine Learning · Computer Science 2022-02-21 Harsh Vardhan , Sebastian U. Stich

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

This paper studies the distributed optimization problem under the influence of heavy-tailed gradient noises. Here, a heavy-tailed noise means that the noise does not necessarily satisfy the bounded variance assumption. Instead, it satisfies…

Optimization and Control · Mathematics 2025-05-12 Chao Sun , Huiming Zhang , Bo Chen , Li Yu

In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only…

Optimization and Control · Mathematics 2023-03-01 Zijian Liu , Ta Duy Nguyen , Thien Hang Nguyen , Alina Ene , Huy Lê Nguyen

Learning with a {\it convex loss} function has been a dominating paradigm for many years. It remains an interesting question how non-convex loss functions help improve the generalization of learning with broad applicability. In this paper,…

Machine Learning · Computer Science 2018-05-22 Yi Xu , Shenghuo Zhu , Sen Yang , Chi Zhang , Rong Jin , Tianbao Yang

Stochastic gradient descent (SGD) with mini-batching is a standard tool in large-scale optimization, yet its theoretical properties under heavy-tailed gradient noise remain largely unexplored. In this paper we study SGD with increasing…

Probability · Mathematics 2026-05-11 Bartosz Glowacki , Rafal Kulik , Philippe Soulier

Stochastic gradient descent (SGD) is a popular and efficient method with wide applications in training deep neural nets and other nonconvex models. While the behavior of SGD is well understood in the convex learning setting, the existing…

Machine Learning · Computer Science 2019-12-16 Yunwen Lei , Ting Hu , Guiying Li , Ke Tang

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

Recent empirical work on stochastic gradient descent (SGD) applied to over-parameterized deep learning has shown that most gradient components over epochs are quite small. Inspired by such observations, we rigorously study properties of…

Machine Learning · Computer Science 2021-10-19 Yingxue Zhou , Xinyan Li , Arindam Banerjee

Gradient compression has surfaced as a key technique to address the challenge of communication efficiency in distributed learning. In distributed deep learning, however, it is observed that gradient distributions are heavy-tailed, with…

Machine Learning · Computer Science 2024-02-07 Guangfeng Yan , Tan Li , Yuanzhang Xiao , Hanxu Hou , Linqi Song

Recent works have shown that high probability metrics with stochastic gradient descent (SGD) exhibit informativeness and in some cases advantage over the commonly adopted mean-square error-based ones. In this work we provide a formal…

Machine Learning · Computer Science 2022-11-03 Dragana Bajovic , Dusan Jakovetic , Soummya Kar

We study a scalable alternative to robust gradient descent (RGD) techniques that can be used when the gradients can be heavy-tailed, though this will be unknown to the learner. The core technique is simple: instead of trying to robustly…

Machine Learning · Statistics 2020-12-16 Matthew J. Holland

This paper studies the convergence of clipped stochastic gradient descent (SGD) algorithms with decision-dependent data distribution. Our setting is motivated by privacy preserving optimization algorithms that interact with performative…

Optimization and Control · Mathematics 2025-01-31 Qiang Li , Michal Yemini , Hoi-To Wai

The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the \emph{classical} central limit theorem (CLT) kicks in. This assumption is often made…

Machine Learning · Statistics 2019-12-03 Umut Şimşekli , Mert Gürbüzbalaban , Thanh Huy Nguyen , Gaël Richard , Levent Sagun

State-of-the-art training algorithms for deep learning models are based on stochastic gradient descent (SGD). Recently, many variations have been explored: perturbing parameters for better accuracy (such as in Extragradient), limiting SGD…

Machine Learning · Computer Science 2022-03-23 Amirkeivan Mohtashami , Martin Jaggi , Sebastian U. Stich