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This paper studies a class of adaptive gradient based momentum algorithms that update the search directions and learning rates simultaneously using past gradients. This class, which we refer to as the "Adam-type", includes the popular…

Machine Learning · Computer Science 2019-03-12 Xiangyi Chen , Sijia Liu , Ruoyu Sun , Mingyi Hong

Federated learning allows loads of edge computing devices to collaboratively learn a global model without data sharing. The analysis with partial device participation under non-IID and unbalanced data reflects more reality. In this work, we…

Machine Learning · Computer Science 2020-12-23 Qianqian Tong , Guannan Liang , Jinbo Bi

We provide a simple proof of convergence covering both the Adam and Adagrad adaptive optimization algorithms when applied to smooth (possibly non-convex) objective functions with bounded gradients. We show that in expectation, the squared…

Machine Learning · Statistics 2022-10-18 Alexandre Défossez , Léon Bottou , Francis Bach , Nicolas Usunier

Adam has achieved strong empirical success, but its theory remains incomplete even in the deterministic full-batch setting, largely because adaptive preconditioning and momentum are tightly coupled. In this work, a convergent reformulation…

Optimization and Control · Mathematics 2026-04-13 Yaxin Yu , Long Chen , Zeyi Xu

In this paper, we present a comprehensive study on the convergence properties of Adam-family methods for nonsmooth optimization, especially in the training of nonsmooth neural networks. We introduce a novel two-timescale framework that…

Optimization and Control · Mathematics 2024-02-20 Nachuan Xiao , Xiaoyin Hu , Xin Liu , Kim-Chuan Toh

Understanding the remarkable efficacy of Adam when training transformer-based language models has become a central research topic within the optimization community. To gain deeper insights, several simplifications of Adam have been…

Machine Learning · Computer Science 2025-12-02 Antonio Orvieto , Robert M. Gower

Learning from a sequence of interactions, as soon as observations are perceived and acted upon, without explicitly storing them, holds the promise of simpler, more efficient and adaptive algorithms. For over a decade, however, deep…

Machine Learning · Computer Science 2026-05-11 Florin Gogianu , Adrian Catalin Lutu , Razvan Pascanu

Adam and RMSProp are two of the most influential adaptive stochastic algorithms for training deep neural networks, which have been pointed out to be divergent even in the convex setting via a few simple counterexamples. Many attempts, such…

Machine Learning · Computer Science 2019-06-26 Fangyu Zou , Li Shen , Zequn Jie , Weizhong Zhang , Wei Liu

Stochastic gradient descent (SGD) optimization methods are nowadays the method of choice for the training of deep neural networks (DNNs) in artificial intelligence systems. In practically relevant training problems, usually not the plain…

Optimization and Control · Mathematics 2024-08-01 Steffen Dereich , Arnulf Jentzen

Adam-type methods, the extension of adaptive gradient methods, have shown great performance in the training of both supervised and unsupervised machine learning models. In particular, Adam-type optimizers have been widely used empirically…

Machine Learning · Computer Science 2021-09-30 Zehao Dou , Yuanzhi Li

We propose a stochastic optimization method for minimizing loss functions, expressed as an expected value, that adaptively controls the batch size used in the computation of gradient approximations and the step size used to move along such…

Machine Learning · Computer Science 2020-03-04 Achraf Bahamou , Donald Goldfarb

Optimizer is an essential component for the success of deep learning, which guides the neural network to update the parameters according to the loss on the training set. SGD and Adam are two classical and effective optimizers on which…

Machine Learning · Computer Science 2023-07-04 Yineng Chen , Zuchao Li , Lefei Zhang , Bo Du , Hai Zhao

In this paper, we consider the integrating factor midpoint method for wave-type equations and derive optimal order a posteriori error estimates. We first introduce an integrating factor midpoint approximation defined by the piecewise linear…

Numerical Analysis · Mathematics 2026-03-03 Xianfa Hu , Fazhan Geng , Wansheng Wang

In deep learning, different kinds of deep networks typically need different optimizers, which have to be chosen after multiple trials, making the training process inefficient. To relieve this issue and consistently improve the model…

Machine Learning · Computer Science 2024-12-02 Xingyu Xie , Pan Zhou , Huan Li , Zhouchen Lin , Shuicheng Yan

Despite superior training outcomes, adaptive optimization methods such as Adam, Adagrad or RMSprop have been found to generalize poorly compared to Stochastic gradient descent (SGD). These methods tend to perform well in the initial portion…

Machine Learning · Computer Science 2017-12-21 Nitish Shirish Keskar , Richard Socher

In this paper, we propose a new method called Self-Training with Dynamic Weighting (STDW), which aims to enhance robustness in Gradual Domain Adaptation (GDA) by addressing the challenge of smooth knowledge migration from the source to the…

Machine Learning · Computer Science 2025-10-17 Zixi Wang , Yushe Cao , Yubo Huang , Jinzhu Wei , Jingzehua Xu , Shuai Zhang , Xin Lai

In reinforcement learning (RL), it is common to apply techniques used broadly in machine learning such as neural network function approximators and momentum-based optimizers. However, such tools were largely developed for supervised…

Optimization algorithms with momentum, e.g., (ADAM), have been widely used for building deep learning models due to the faster convergence rates compared with stochastic gradient descent (SGD). Momentum helps accelerate SGD in the relevant…

Machine Learning · Computer Science 2020-01-24 Jiyang Bai , Yuxiang Ren , Jiawei Zhang

AdamW has been the default optimizer for transformer pretraining. For many years, our community searched for faster and more stable optimizers with only constrained positive outcomes. In this work, we propose a \textbf{one-line modification…

Machine Learning · Computer Science 2026-02-17 Kaizhao Liang , Lizhang Chen , Bo Liu , Qiang Liu

The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial…

Machine Learning · Computer Science 2021-10-01 Chao Ma , Lei Wu , Weinan E