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In this paper, we propose a new, simplified high probability analysis of AdaGrad for smooth, non-convex problems. More specifically, we focus on a particular accelerated gradient (AGD) template (Lan, 2020), through which we recover the…

Optimization and Control · Mathematics 2022-04-07 Ali Kavis , Kfir Yehuda Levy , Volkan Cevher

In this study, we revisit the convergence of AdaGrad with momentum (covering AdaGrad as a special case) on non-convex smooth optimization problems. We consider a general noise model where the noise magnitude is controlled by the function…

Optimization and Control · Mathematics 2024-09-16 Yusu Hong , Junhong Lin

We investigate the Randomized Stochastic Accelerated Gradient (RSAG) method, utilizing either constant or adaptive step sizes, for stochastic optimization problems with generalized smooth objective functions. Under relaxed affine variance…

Optimization and Control · Mathematics 2025-02-25 Chenhao Yu , Yusu Hong , Junhong Lin

In this work, we describe a generic approach to show convergence with high probability for stochastic convex optimization. In previous works, either the convergence is only in expectation or the bound depends on the diameter of the domain.…

Optimization and Control · Mathematics 2022-10-04 Alina Ene , Huy L. Nguyen

Stochastic gradient descent is one of the most common iterative algorithms used in machine learning and its convergence analysis is a rich area of research. Understanding its convergence properties can help inform what modifications of it…

Optimization and Control · Mathematics 2025-11-25 Liam Madden , Emiliano Dall'Anese , Stephen Becker

Adaptive gradient methods, such as AdaGrad, are among the most successful optimization algorithms for neural network training. While these methods are known to achieve better dimensional dependence than stochastic gradient descent (SGD) for…

Optimization and Control · Mathematics 2025-06-09 Ruichen Jiang , Devyani Maladkar , Aryan Mokhtari

We study Stochastic Gradient Descent with AdaGrad stepsizes: a popular adaptive (self-tuning) method for first-order stochastic optimization. Despite being well studied, existing analyses of this method suffer from various shortcomings:…

Machine Learning · Computer Science 2023-06-13 Amit Attia , Tomer Koren

We prove novel convergence results for a stochastic proximal gradient algorithm suitable for solving a large class of convex optimization problems, where a convex objective function is given by the sum of a smooth and a possibly non-smooth…

Optimization and Control · Mathematics 2016-08-11 Lorenzo Rosasco , Silvia Villa , Bang Công Vũ

Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a…

Machine Learning · Statistics 2019-02-28 Xiaoyu Li , Francesco Orabona

Adaptive gradient methods such as AdaGrad and its variants update the stepsize in stochastic gradient descent on the fly according to the gradients received along the way; such methods have gained widespread use in large-scale optimization…

Machine Learning · Statistics 2021-04-20 Rachel Ward , Xiaoxia Wu , Leon Bottou

Stochastic first-order methods are standard for training large-scale machine learning models. Random behavior may cause a particular run of an algorithm to result in a highly suboptimal objective value, whereas theoretical guarantees are…

Optimization and Control · Mathematics 2024-09-02 Eduard Gorbunov , Marina Danilova , Innokentiy Shibaev , Pavel Dvurechensky , Alexander Gasnikov

Existing analysis of AdaGrad and other adaptive methods for smooth convex optimization is typically for functions with bounded domain diameter. In unconstrained problems, previous works guarantee an asymptotic convergence rate without an…

Machine Learning · Computer Science 2023-10-05 Zijian Liu , Ta Duy Nguyen , Alina Ene , Huy L. Nguyen

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 study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. Despite their…

Under mild assumptions stochastic gradient methods asymptotically achieve an optimal rate of convergence if the arithmetic mean of all iterates is returned as an approximate optimal solution. However, in the absence of stochastic noise, the…

Optimization and Control · Mathematics 2022-10-06 Melinda Hagedorn , Florian Jarre

Recently, Stochastic Gradient Descent (SGD) and its variants have become the dominant methods in the large-scale optimization of machine learning (ML) problems. A variety of strategies have been proposed for tuning the step sizes, ranging…

Machine Learning · Computer Science 2022-08-02 Xiaoyu Li

We consider stochastic gradient descent algorithms for minimizing a non-smooth, strongly-convex function. Several forms of this algorithm, including suffix averaging, are known to achieve the optimal $O(1/T)$ convergence rate in…

Machine Learning · Computer Science 2019-09-04 Nicholas J. A. Harvey , Christopher Liaw , Sikander Randhawa

Adaptive gradient methods, such as AdaGrad, have become fundamental tools in deep learning. Despite their widespread use, the asymptotic convergence of AdaGrad remains poorly understood in non-convex scenarios. In this work, we present the…

Optimization and Control · Mathematics 2026-01-06 Ruinan Jin , Xiaoyu Wang

We establish the O($\frac{1}{k}$) convergence rate for distributed stochastic gradient methods that operate over strongly convex costs and random networks. The considered class of methods is standard each node performs a weighted average of…

Optimization and Control · Mathematics 2018-03-22 Dusan Jakovetic , Dragana Bajovic , Anit Kumar Sahu , Soummya Kar

Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper…

Optimization and Control · Mathematics 2023-05-23 Zijian Liu , Zhengyuan Zhou
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