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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…

Optimization and Control · Mathematics 2026-02-12 Chuan He , Zhaosong Lu , Defeng Sun , Zhanwang Deng

We introduce a detailed analysis of the convergence of first-order methods with composite noise (sum of relative and absolute) in gradient for convex and smooth function minimization. This paper illustrates instances of practical problems…

Optimization and Control · Mathematics 2026-03-16 Artem Vasin , Alexander Gasnikov

We study the robustness of accelerated first-order algorithms to stochastic uncertainties in gradient evaluation. Specifically, for unconstrained, smooth, strongly convex optimization problems, we examine the mean-squared error in the…

Optimization and Control · Mathematics 2020-02-21 Hesameddin Mohammadi , Meisam Razaviyayn , Mihailo R. Jovanović

Boosting algorithms have been widely used to tackle a plethora of problems. In the last few years, a lot of approaches have been proposed to provide standard AdaBoost with cost-sensitive capabilities, each with a different focus. However,…

Computer Vision and Pattern Recognition · Computer Science 2016-07-25 Iago Landesa-Vázquez , José Luis Alba-Castro

Using an optimization algorithm to solve a machine learning problem is one of mainstreams in the field of science. In this work, we demonstrate a comprehensive comparison of some state-of-the-art first-order optimization algorithms for…

Machine Learning · Computer Science 2014-04-29 Yu Wei , Pock Thomas

There has been much recent interest in finding unconstrained local minima of smooth functions, due in part of the prevalence of such problems in machine learning and robust statistics. A particular focus is algorithms with good complexity…

Optimization and Control · Mathematics 2017-12-12 Clément W. Royer , Stephen J. Wright

We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has…

Machine Learning · Computer Science 2017-01-31 Diederik P. Kingma , Jimmy Ba

Numerous empirical evidence has corroborated that the noise plays a crucial rule in effective and efficient training of neural networks. The theory behind, however, is still largely unknown. This paper studies this fundamental problem…

Machine Learning · Computer Science 2019-09-10 Mo Zhou , Tianyi Liu , Yan Li , Dachao Lin , Enlu Zhou , Tuo Zhao

We introduce a class of first-order methods for smooth constrained optimization that are based on an analogy to non-smooth dynamical systems. Two distinctive features of our approach are that (i) projections or optimizations over the entire…

Optimization and Control · Mathematics 2025-04-15 Michael Muehlebach , Michael I. Jordan

A new algorithm for smooth constrained optimization is proposed that never computes the value of the problem's objective function and that handles both equality and inequality constraints. The algorithm uses an adaptive switching strategy…

Optimization and Control · Mathematics 2026-02-13 S. Bellavia , S. Gratton , B. Morini , Ph. L. Toint

We consider in this paper a class of composite optimization problems whose objective function is given by the summation of a general smooth and nonsmooth component, together with a relatively simple nonsmooth term. We present a new class of…

Optimization and Control · Mathematics 2015-10-27 Guanghui Lan

We consider sum-type strongly convex optimization problem (first term) with smooth convex not proximal friendly composite (second term). We show that the complexity of this problem can be split into optimal number of incremental oracle…

Optimization and Control · Mathematics 2020-03-12 Darina Dvinskikh , Sergey Omelchenko , Alexander Tyurin , Alexander Gasnikov

Gradient-free/zeroth-order methods for black-box convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration…

The speed-up provided by quantum algorithms with respect to their classical counterparts is at the origin of scientific interest in quantum computation. However, the fundamental reasons for such a speed-up are not yet completely understood…

Quantum Physics · Physics 2007-05-23 C. Di Franco , M. Paternostro , M. S. Kim

We tackle the problem of learning linear classifiers from noisy datasets in a multiclass setting. The two-class version of this problem was studied a few years ago where the proposed approaches to combat the noise revolve around a…

Machine Learning · Computer Science 2015-06-25 Ugo Louche , Liva Ralaivola

We study the trade-off between convergence rate and sensitivity to stochastic additive gradient noise for first-order optimization methods. Ordinary Gradient Descent (GD) can be made fast-and-sensitive or slow-and-robust by increasing or…

Optimization and Control · Mathematics 2025-11-07 Bryan Van Scoy , Laurent Lessard

Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…

Optimization and Control · Mathematics 2025-12-03 Stephen J. Wright

The development of nonlinear optimization algorithms capable of performing reliably in the presence of noise has garnered considerable attention lately. This paper advocates for strategies to create noise-tolerant nonlinear optimization…

Optimization and Control · Mathematics 2024-10-04 Yuchen Lou , Shigeng Sun , Jorge Nocedal

We investigate the convergence properties of a class of iterative algorithms designed to minimize a potentially non-smooth and noisy objective function, which may be algebraically intractable and whose values may be obtained as the output…

Computation · Statistics 2025-12-04 Christophe Andrieu , Nicolas Chopin , Ettore Fincato , Mathieu Gerber

Second-order optimization methods are computationally expensive for large-scale problems. Recently, Doikov, Chayti, and Jaggi (ICML 2023) proposed the LazyCRN method that reduces computation by studying the gradient complexity of…

Optimization and Control · Mathematics 2026-01-26 Lesi Chen , Chengchang Liu , Luo Luo , Jingzhao Zhang