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Geometry-aware optimization algorithms, such as Muon, have achieved remarkable success in training deep neural networks (DNNs). These methods leverage the underlying geometry of DNNs by selecting appropriate norms for different layers and…

Machine Learning · Computer Science 2026-02-04 Jie Hao , Xiaochuan Gong , Jie Xu , Zhengdao Wang , Mingrui Liu

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions typically either rescale updates after orthogonalization or use heavier whitening-based preconditioners before it. We…

Machine Learning · Computer Science 2026-05-12 Da Chang , Qiankun Shi , Lvgang Zhang , Yu Li , Ruijie Zhang , Yao Lu , Yongxiang Liu , Ganzhao Yuan

Large language models (LLMs) suffer from catastrophic forgetting in sequential multi-task learning. Existing parameter regularization methods (e.g., O-LoRA, N-LoRA) mitigate interference via low-rank subspace orthogonality, but additive…

Computation and Language · Computer Science 2026-01-26 Kefan Cao , Shuaicheng Wu

Memory-efficient optimization methods have recently gained increasing attention for scaling full-parameter training of large language models under the GPU-memory bottleneck. Existing approaches either lack clear convergence guarantees, or…

Machine Learning · Computer Science 2026-03-11 Hui Yang , Tao Ren , Jinyang Jiang , Wan Tian , Yijie Peng

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even…

Artificial Intelligence · Computer Science 2025-12-05 Thibaut Boissin , Thomas Massena , Franck Mamalet , Mathieu Serrurier

Recent empirical research has demonstrated that deep learning optimizers based on the linear minimization oracle (LMO) over specifically chosen Non-Euclidean norm balls, such as Muon and Scion, outperform Adam-type methods in the training…

Optimization and Control · Mathematics 2025-12-19 Xun Qian , Hussein Rammal , Dmitry Kovalev , Peter Richtárik

In neural networks, continual learning results in gradient interference among sequential tasks, leading to catastrophic forgetting of old tasks while learning new ones. This issue is addressed in recent methods by storing the important…

Machine Learning · Computer Science 2023-02-06 Gobinda Saha , Kaushik Roy

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

To define a steepest descent method over a neural network, we need to choose a norm for each layer, a way to aggregate these norms across layers, and whether to use normalization. We systematically explore different alternatives for…

Machine Learning · Computer Science 2025-10-14 Michael Crawshaw , Chirag Modi , Mingrui Liu , Robert M. Gower

Drawing inspiration from gradient-based meta-learning methods with infinitely small gradient steps, we introduce Continuous-Time Meta-Learning (COMLN), a meta-learning algorithm where adaptation follows the dynamics of a gradient vector…

Machine Learning · Computer Science 2022-03-04 Tristan Deleu , David Kanaa , Leo Feng , Giancarlo Kerg , Yoshua Bengio , Guillaume Lajoie , Pierre-Luc Bacon

Muon has emerged as a strong competitor to AdamW for language model pre-training, yet its behavior at scale is sensitive to weight decay. Recent work has observed that, for Muon without decoupled weight decay, the spectral norm of weight…

Machine Learning · Computer Science 2026-05-12 Kai Lion , Florian Hübler , Bingcong Li , Antonio Orvieto , Niao He

The Muon optimizer has recently demonstrated remarkable empirical success in training large language models. However, the theoretical understanding of its mechanisms remains limited. Current convergence guarantees for Muon rely heavily on…

Machine Learning · Computer Science 2026-05-27 Yixuan Yang , Yuqing He , Song Li

Recent developments in deep learning optimization have brought about radically new algorithms based on the Linear Minimization Oracle (LMO) framework, such as $\sf Muon$ and $\sf Scion$. After over a decade of $\sf Adam$'s dominance, these…

Machine Learning · Computer Science 2025-05-20 Artem Riabinin , Egor Shulgin , Kaja Gruntkowska , Peter Richtárik

Large models recently are widely applied in artificial intelligence, so efficient training of large models has received widespread attention. More recently, a useful Muon optimizer is specifically designed for matrix-structured parameters…

Machine Learning · Computer Science 2025-09-22 Feihu Huang , Yuning Luo , Songcan Chen

Recent MLLMs have shown emerging visual understanding and reasoning abilities after being pre-trained on large-scale multimodal datasets. Unlike pre-training, where MLLMs receive rich visual-text alignment, instruction-tuning is often…

Machine Learning · Computer Science 2026-01-27 Junda Wu , Yuxin Xiong , Xintong Li , Yu Xia , Ruoyu Wang , Yu Wang , Tong Yu , Sungchul Kim , Ryan A. Rossi , Lina Yao , Jingbo Shang , Julian McAuley

Low-rank adaptation (LoRA) has become the default approach to fine-tune large language models (LLMs) due to its significant reduction in trainable parameters. However, trainable parameter demand for LoRA increases with increasing model…

Computation and Language · Computer Science 2024-06-19 Seyedarmin Azizi , Souvik Kundu , Massoud Pedram

Despite the significant progress in multimodal large language models (MLLMs), their high computational cost remains a barrier to real-world deployment. Inspired by the mixture of depths (MoDs) in natural language processing, we aim to…

Computer Vision and Pattern Recognition · Computer Science 2024-10-18 Yaxin Luo , Gen Luo , Jiayi Ji , Yiyi Zhou , Xiaoshuai Sun , Zhiqiang Shen , Rongrong Ji

Optimization lies at the core of modern deep learning, yet existing methods often face a fundamental trade-off between adapting to problem geometry and leveraging curvature utilization. Steepest descent algorithms adapt to different…

Low-rank gradient compression reduces communication in distributed training by representing updates with rank-$r$ factors. Dion is a recent method that approximates Muon, a spectral optimizer that orthogonalizes momentum, using one step of…

Existing prompt-tuning methods have demonstrated impressive performances in continual learning (CL), by selecting and updating relevant prompts in the vision-transformer models. On the contrary, this paper aims to learn each task by tuning…

Computer Vision and Pattern Recognition · Computer Science 2024-10-29 Yue Lu , Shizhou Zhang , De Cheng , Yinghui Xing , Nannan Wang , Peng Wang , Yanning Zhang