Related papers: Muon-OGD: Muon-based Spectral Orthogonal Gradient …

Muon and related norm-constrained matrix optimizers have become central to large-scale learning problems. They are formulated as a linear maximization oracle (LMO) over an ambient matrix-norm ball in unconstrained Euclidean space. However,…

Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this…

Orthogonal Gradient Descent (OGD) has emerged as a powerful method for continual learning. However, its Euclidean projections do not leverage the underlying information-geometric structure of the problem, which can lead to suboptimal…

Muon is a matrix-aware optimizer that leverages Newton-Schulz (NS) iterations to enforce spectral gradient orthogonalization by driving all singular values of the momentum matrix toward 1. While this uniform spectral whitening enhances…

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…

Memory-efficient optimization is critical for training increasingly large language models (LLMs). A popular strategy involves gradient low-rank projection, storing only the projected optimizer states, with GaLore being a representative…

Fine-tuning large language models (LLMs) for specialized domains often necessitates a trade-off between acquiring domain expertise and retaining general reasoning capabilities, a phenomenon known as catastrophic forgetting. Existing…

Large language models (LLMs) often suffer from catastrophic forgetting in continual learning (CL) scenarios, where performance on previously learned tasks degrades severely while training on sequentially arriving tasks. Although pioneering…

Modern optimizers, like Muon, impose matrix-wise geometry constraints on their updates. These matrix-wise constraints can be unified under Linear Minimization Oracle (LMO) theory. However, all current methods impose fixed LMO geometries for…

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…

Muon has recently emerged as a strong alternative to AdamW for training neural networks, with encouraging large-scale pretraining results and growing evidence that matrix-structured updates can be faster in practice. Yet Muon, and more…

The core bottleneck of Federated Learning (FL) lies in the communication rounds. That is, how to achieve more effective local updates is crucial for reducing communication rounds. Existing FL methods still primarily use element-wise local…

Muon updates matrix parameters via the matrix sign of the gradient and has shown strong empirical gains, yet its dynamics and scaling behavior remain unclear in theory. We study Muon in a linear associative memory model with softmax…

Continual learning in multimodal large language models (MLLMs) aims to sequentially acquire knowledge while mitigating catastrophic forgetting, yet existing methods face inherent limitations: architecture-based approaches incur additional…

Zeroth-order (ZO) optimization provides a gradient-free alternative to first-order (FO) methods by estimating gradients via finite differences of function evaluations, and has recently emerged as a memory-efficient paradigm for fine-tuning…

Language models (LMs) exhibit impressive performance and generalization capabilities. However, LMs struggle with the persistent challenge of catastrophic forgetting, which undermines their long-term sustainability in continual learning…

Adversarial training (AT) remains one of the most reliable empirical defenses against adversarial attacks. Its robustness critically depends on how the underlying min-max objective is optimized. In practice, Stochastic Gradient Descent…

Orthogonalized-momentum optimizers such as Muon improve transformer training by approximately whitening/orthogonalizing matrix-valued momentum updates via a short polar-decomposition iteration. However, polar-factor approximations typically…

Large Vision-Language Models (LVLMs) have advanced multimodal understanding, yet their reliability is limited by hallucination, where generated content conflicts with visual facts. Existing mitigation methods either rely on costly external…

The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon's efficiency…