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Orthonormalized updates accelerate training, improve stability, and enable robust hyperparameter transfer, but existing methods like Muon rely on dense matrix operations that clash with sharded weights in large-scale LLM training, causing…

Machine Learning · Computer Science 2025-09-16 Kwangjun Ahn , Byron Xu , Natalie Abreu , Ying Fan , Gagik Magakyan , Pratyusha Sharma , Zheng Zhan , John Langford

To enable the pre-trained models to be fine-tuned with local data on edge devices without sharing data with the cloud, we design an efficient split fine-tuning (SFT) framework for edge and cloud collaborative learning. We propose three…

Distributed, Parallel, and Cluster Computing · Computer Science 2022-12-01 Shaohuai Shi , Qing Yang , Yang Xiang , Shuhan Qi , Xuan Wang

Stochastic Frank-Wolfe is a classical optimization method for solving constrained optimization problems. On the other hand, recent optimizers such as Lion and Muon have gained quite significant popularity in deep learning. In this work,…

Optimization and Control · Mathematics 2026-02-03 Maria-Eleni Sfyraki , Jun-Kun Wang

This paper proposes and analyzes a communication-efficient distributed optimization framework for general nonconvex nonsmooth signal processing and machine learning problems under an asynchronous protocol. At each iteration, worker machines…

Optimization and Control · Mathematics 2020-07-15 Jineng Ren , Jarvis Haupt

This work introduces a hybrid non-Euclidean optimization method which generalizes gradient norm clipping by combining steepest descent and conditional gradient approaches. The method achieves the best of both worlds by establishing a…

Machine Learning · Computer Science 2026-02-05 Thomas Pethick , Wanyun Xie , Mete Erdogan , Kimon Antonakopoulos , Antonio Silveti-Falls , Volkan Cevher

Motivated by the increasing popularity and importance of large-scale training under differential privacy (DP) constraints, we study distributed gradient methods with gradient clipping, i.e., clipping applied to the gradients computed from…

Machine Learning · Computer Science 2023-05-31 Sarit Khirirat , Eduard Gorbunov , Samuel Horváth , Rustem Islamov , Fakhri Karray , Peter Richtárik

The Muon optimizer has emerged as a compelling alternative to Adam for training large language models, achieving remarkable computational savings through gradient orthogonalization. However, Muon's optimizer state is more sensitive to…

Machine Learning · Computer Science 2026-05-13 Yupeng Su , Ruijie Zhang , Ziyue Liu , Yequan Zhao , Zheng Zhang

In distributed or federated optimization and learning, communication between the different computing units is often the bottleneck and gradient compression is widely used to reduce the number of bits sent within each communication round of…

Machine Learning · Computer Science 2023-03-07 Laurent Condat , Kai Yi , Peter Richtárik

In large-scale optimization, the cheapness and effectiveness of update steps are the most crucial factors for a successful optimizer. Sign-based optimizers like Lion or Signum produce cheap per-step updates, whereas Muon's spectral…

A central challenge in continual learning for large language models (LLMs) is catastrophic forgetting, where adapting to new tasks can substantially degrade performance on previously learned ones. Existing projection-based methods mitigate…

Machine Learning · Computer Science 2026-05-18 Binghang Lu , Zheyuan Deng , Runyu Zhang , Bing Hu , Yunhan Zhao , Yuan Tian , Changhong Mou , Guang Lin , Xiaomin Li

The majority of parameters in neural networks are naturally represented as matrices. However, most commonly used optimizers treat these matrix parameters as flattened vectors during optimization, potentially overlooking their inherent…

Machine Learning · Statistics 2026-04-15 Wei Shen , Ruichuan Huang , Minhui Huang , Cong Shen , Jiawei Zhang

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

Distributed optimization methods are often applied to solving huge-scale problems like training neural networks with millions and even billions of parameters. In such applications, communicating full vectors, e.g., (stochastic) gradients,…

Optimization and Control · Mathematics 2022-05-31 Marina Danilova , Eduard Gorbunov

Recent optimizers such as Lion and Muon have demonstrated strong empirical performance by normalizing gradient momentum via linear minimization oracles (LMOs). While variance reduction has been explored to accelerate LMO-based methods, it…

Machine Learning · Computer Science 2026-05-08 Won-Jun Jang , Si-Hyeon Lee

Muon, a recently proposed optimizer that leverages the inherent matrix structure of neural network parameters, has demonstrated strong empirical performance, indicating its potential as a successor to standard optimizers such as AdamW. This…

Machine Learning · Computer Science 2025-11-24 Naoki Sato , Hiroki Naganuma , Hideaki Iiduka

In the last few years, various communication compression techniques have emerged as an indispensable tool helping to alleviate the communication bottleneck in distributed learning. However, despite the fact biased compressors often show…

Machine Learning · Computer Science 2024-01-17 Aleksandr Beznosikov , Samuel Horváth , Peter Richtárik , Mher Safaryan

The Muon optimizer has recently attracted considerable attention for its strong empirical performance and use of orthogonalized updates on matrix-shaped parameters, yet its underlying mechanisms and relationship to adaptive optimizers such…

Machine Learning · Computer Science 2026-02-05 Xianbiao Qi , Marco Chen , Jiaquan Ye , Yelin He , Rong Xiao

In this work, we develop proximal preconditioned gradient methods with a focus on spectral gradient methods providing a proximal extension to the Muon and Scion optimizers. We introduce a family of stochastic algorithms that can handle a…

We present a comprehensive theoretical and empirical study of the Muon optimizer for training transformers only with a small to medium decoder (30M - 200M parameters), with an emphasis on its mathematical foundations, convergence properties…

Machine Learning · Computer Science 2025-09-30 Sushant Mehta , Raj Dandekar , Rajat Dandekar , Sreedath Panat

We provide the first proof of convergence for normalized error feedback algorithms across a wide range of machine learning problems. Despite their popularity and efficiency in training deep neural networks, traditional analyses of error…

Machine Learning · Computer Science 2024-10-23 Sarit Khirirat , Abdurakhmon Sadiev , Artem Riabinin , Eduard Gorbunov , Peter Richtárik