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This paper studies the AdamW-style Shampoo optimizer, an effective implementation of classical Shampoo that notably won the external tuning track of the AlgoPerf neural network training algorithm competition. Our analysis unifies one-sided…

Optimization and Control · Mathematics 2026-05-26 Huan Li , Yiming Dong , Zhouchen Lin

The rise of large language models (LLMs) has created a significant disparity: industrial research labs with their computational resources, expert teams, and advanced infrastructures, can effectively fine-tune LLMs, while individual…

Normalization techniques are a boon for modern deep learning. They let weights converge more quickly with often better generalization performances. It has been argued that the normalization-induced scale invariance among the weights…

Machine Learning · Computer Science 2021-01-19 Byeongho Heo , Sanghyuk Chun , Seong Joon Oh , Dongyoon Han , Sangdoo Yun , Gyuwan Kim , Youngjung Uh , Jung-Woo Ha

DiLoCo is a powerful framework for training large language models (LLMs), enabling larger optimal batch sizes and increased accelerator utilization under networking constraints. However, DiLoCo's performance has been shown to degrade as the…

Machine Learning · Computer Science 2026-02-26 Benjamin Thérien , Xiaolong Huang , Aaron Defazio , Irina Rish , Eugene Belilovsky

Large Language Models (LLMs) have seen great advance in both academia and industry, and their popularity results in numerous open-source frameworks and techniques in accelerating LLM pre-training, fine-tuning, and inference. Training and…

Performance · Computer Science 2023-12-04 Longteng Zhang , Xiang Liu , Zeyu Li , Xinglin Pan , Peijie Dong , Ruibo Fan , Rui Guo , Xin Wang , Qiong Luo , Shaohuai Shi , Xiaowen Chu

The Muon optimizer has recently offered a promising alternative to AdamW for large language model training, leveraging matrix orthogonalization to produce geometry-aware updates. However, like all first-order methods, Muon can become…

Machine Learning · Computer Science 2026-05-27 Jiacheng Li , Jianchao Tan , Hongtao Xu , Jiaqi Zhang , Yifan Lu , Yerui Sun , Yuchen Xie , Xunliang Cai

Large language models (LLMs) have demonstrated impressive generalization and emergent capabilities, yet their pre-training remains computationally expensive and sensitive to optimization dynamics. While Adam-based optimizers offer fast…

Machine Learning · Computer Science 2025-10-01 Junjie Wang , Pan Zhou , Yiming Dong , Huan Li , Jia Li , Xun Zhou , Qicheng Lao , Cong Fang , Zhouchen Lin

Optimizers leveraging the matrix structure in neural networks, such as Shampoo and Muon, are more data-efficient than element-wise algorithms like Adam and Signum. While in specific settings, Shampoo and Muon reduce to spectral descent…

Machine Learning · Computer Science 2026-02-11 Runa Eschenhagen , Anna Cai , Tsung-Hsien Lee , Hao-Jun Michael Shi

Existing works show that augmenting the training data of pre-trained language models (PLMs) for classification tasks fine-tuned via parameter-efficient fine-tuning methods (PEFT) using both clean and adversarial examples can enhance their…

Computation and Language · Computer Science 2024-06-18 Tuc Nguyen , Thai Le

Uncovering early-stage metrics that reflect final model performance is one core principle for large-scale pretraining. The existing scaling law demonstrates the power-law correlation between pretraining loss and training flops, which serves…

The foundational capabilities of large language models are acquired during pretraining on internet-scale, highly heterogeneous data mixtures. In this work, we investigate an interesting geometric question regarding the converged state of…

Machine Learning · Computer Science 2026-05-28 Huanran Chen , Huaqing Zhang , Xiao Li , Yinpeng Dong , Ke Shen , Jun Zhu

How does the choice of optimization algorithm shape a model's ability to learn features? To address this question for steepest descent methods --including sign descent, which is closely related to Adam --we introduce steepest mirror flows…

Machine Learning · Computer Science 2026-03-03 Tom Jacobs , Chao Zhou , Rebekka Burkholz

Large-batch training has become a cornerstone in accelerating the training of deep neural networks, yet it poses challenges in optimization and generalization. Existing optimizers like AdamW present performance degradation during language…

Machine Learning · Computer Science 2025-08-29 Yang Luo , Zangwei Zheng , Ziheng Qin , Zirui Zhu , Yong Liu , Yang You

AdamZ is an advanced variant of the Adam optimiser, developed to enhance convergence efficiency in neural network training. This optimiser dynamically adjusts the learning rate by incorporating mechanisms to address overshooting and…

Machine Learning · Computer Science 2024-11-26 Ilia Zaznov , Atta Badii , Alfonso Dufour , Julian Kunkel

Hyper-parameters optimization (HPO) is vital for machine learning models. Besides model accuracy, other tuning intentions such as model training time and energy consumption are also worthy of attention from data analytic service providers.…

Machine Learning · Computer Science 2023-04-21 Hui Dou , Shanshan Zhu , Yiwen Zhang , Pengfei Chen , Zibin Zheng

While adaptive gradient methods are the workhorse of modern machine learning, sign-based optimization algorithms such as Lion and Muon have recently demonstrated superior empirical performance over AdamW in training large language models…

Machine Learning · Computer Science 2026-05-11 Dingzhi Yu , Hongyi Tao , Yuanyu Wan , Luo Luo , Lijun Zhang

Determining the optimal data mixture for large language model training remains a challenging problem with an outsized impact on performance. In practice, language model developers continue to rely on heuristic exploration since no…

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 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

Optimizers are crucial for the efficient training of Large Language Models (LLMs). While AdamW is the de facto standard, recent structure-aware optimizers like Muon have emerged, which regularize gradient updates by operating on entire…

Machine Learning · Computer Science 2025-10-07 Zehua Liu , Han Wu , Xiaojin Fu , Shuqi Liu , Xiongwei Han , Tao Zhong , Mingxuan Yuan