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A central question in modern deep learning is how to design optimizers whose behavior remains stable as the network width $w$ increases. We address this question by interpreting several widely used neural-network optimizers, including…

Machine Learning · Computer Science 2026-03-11 Ruihan Xu , Jiajin Li , Yiping Lu

Spectral gradient methods, such as the recently popularized Muon optimizer, are a promising alternative to standard Euclidean gradient descent for training deep neural networks and transformers, but it is still unclear in which regimes they…

Machine Learning · Computer Science 2026-01-15 Damek Davis , Dmitriy Drusvyatskiy

The best performing Binary Neural Networks (BNNs) are usually attained using Adam optimization and its multi-step training variants. However, to the best of our knowledge, few studies explore the fundamental reasons why Adam is superior to…

Machine Learning · Computer Science 2021-06-22 Zechun Liu , Zhiqiang Shen , Shichao Li , Koen Helwegen , Dong Huang , Kwang-Ting Cheng

Matrix-based optimizers have attracted growing interest for improving LLM training efficiency, with significant progress centered on orthogonalization/whitening based methods. While yielding substantial performance gains, a fundamental…

Machine Learning · Computer Science 2026-02-10 Wenbo Gong , Javier Zazo , Qijun Luo , Puqian Wang , James Hensman , Chao Ma

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods…

Optimization and Control · Mathematics 2023-01-31 Nikita Doikov , Anton Rodomanov

Robustness evaluation against adversarial examples has become increasingly important to unveil the trustworthiness of the prevailing deep models in natural language processing (NLP). However, in contrast to the computer vision domain where…

Computation and Language · Computer Science 2022-12-20 Bairu Hou , Jinghan Jia , Yihua Zhang , Guanhua Zhang , Yang Zhang , Sijia Liu , Shiyu Chang

Fast gradient-based optimization algorithms have become increasingly essential for the computationally efficient training of machine learning models. One technique is to multiply the gradient by a preconditioner matrix to produce a step,…

Machine Learning · Computer Science 2023-09-12 Isaac Liao , Rumen R. Dangovski , Jakob N. Foerster , Marin Soljačić

Primal-Dual Hybrid Gradient (PDHG) and Alternating Direction Method of Multipliers (ADMM) are two widely-used first-order optimization methods. They reduce a difficult problem to simple subproblems, so they are easy to implement and have…

Optimization and Control · Mathematics 2019-09-10 Yanli Liu , Yunbei Xu , Wotao Yin

Efficient stochastic optimization typically integrates an update direction that performs well in the deterministic regime with a mechanism adapting to stochastic perturbations. While Adam uses adaptive moment estimates to promote stability,…

Machine Learning · Computer Science 2026-02-23 Minxin Zhang , Yuxuan Liu , Hayden Schaeffer

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

The paper presents the formulation, implementation, and evaluation of the ArcGD optimiser. The evaluation is conducted initially on a non-convex benchmark function and subsequently on a real-world ML dataset. The initial comparative study…

Machine Learning · Computer Science 2026-03-25 Nikhil Verma , Joonas Linnosmaa , Leonardo Espinosa-Leal , Napat Vajragupta

We present a class of novel optimisers for training neural networks that makes use of the Riemannian metric naturally induced when the loss landscape is embedded in higher-dimensional space. This is the same metric that underlies common…

Machine Learning · Computer Science 2025-09-05 Thomas R. Harvey

The concept of learning to optimize involves utilizing a trainable optimization strategy rather than relying on manually defined full gradient estimations such as ADAM. We present a framework that jointly trains the full gradient estimator…

Machine Learning · Computer Science 2026-01-30 Ruiqi Wang , Diego Klabjan

Modern optimizers like Adam and Muon are central to training large language models, but their reliance on first- and second-order momenta introduces significant memory overhead, which constrains scalability and computational efficiency. In…

Machine Learning · Computer Science 2026-03-02 Zhengbo Wang , Jian Liang , Ran He , Zilei Wang , Tieniu Tan

The Muon optimizer, based on matrix orthogonalization, has recently shown faster convergence and better computational efficiency over AdamW in LLM pre-training. However, the memory overhead of maintaining high-precision optimizer states…

We introduce a new method inspired by Adam that enhances convergence speed and achieves better loss function minima. Traditional optimizers, including Adam, apply uniform or globally adjusted learning rates across neural networks without…

Machine Learning · Computer Science 2024-11-01 Remi Genet , Hugo Inzirillo

Dictionary learning aims at seeking a dictionary under which the training data can be sparsely represented. Methods in the literature typically formulate the dictionary learning problem as an optimization w.r.t. two variables, i.e.,…

Signal Processing · Electrical Eng. & Systems 2021-10-27 Cheng Cheng , Wei Dai

Learning an efficient update rule from data that promotes rapid learning of new tasks from the same distribution remains an open problem in meta-learning. Typically, previous works have approached this issue either by attempting to train a…

Machine Learning · Computer Science 2020-02-19 Sebastian Flennerhag , Andrei A. Rusu , Razvan Pascanu , Francesco Visin , Hujun Yin , Raia Hadsell

This paper proposes a conjugate-gradient-based Adam algorithm blending Adam with nonlinear conjugate gradient methods and shows its convergence analysis. Numerical experiments on text classification and image classification show that the…

Optimization and Control · Mathematics 2020-03-04 Yu Kobayashi , Hideaki Iiduka

The $\mu$-parameterization ($\mu$P) provides a principled foundation for large language model (LLM) training by prescribing width-independent learning dynamics, which in turn enables predictable scaling behavior and robust hyperparameter…

Machine Learning · Computer Science 2026-01-08 John Zhao