Related papers: Adam Improves Muon: Adaptive Moment Estimation wit…

Optimization algorithms with momentum, e.g., (ADAM), have been widely used for building deep learning models due to the faster convergence rates compared with stochastic gradient descent (SGD). Momentum helps accelerate SGD in the relevant…

Recent work by Xia et al. leveraged the continuous-limit of the classical momentum accelerated gradient descent and proposed heavy-ball neural ODEs. While this model offers computational efficiency and high utility over vanilla neural ODEs,…

The Adaptive Momentum Estimation (Adam) algorithm is highly effective in training various deep learning tasks. Despite this, there's limited theoretical understanding for Adam, especially when focusing on its vanilla form in non-convex…

Modern optimization algorithms that incorporate momentum and adaptive step-size offer improved performance in numerous challenging deep learning tasks. However, their effectiveness is often highly sensitive to the choice of hyperparameters,…

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…

Lowering the memory requirement in full-parameter training on large models has become a hot research area. MeZO fine-tunes the large language models (LLMs) by just forward passes in a zeroth-order SGD optimizer (ZO-SGD), demonstrating…

The Muon optimizer is consistently faster than Adam in training Large Language Models (LLMs), yet the mechanism underlying its success remains unclear. This paper demystifies this mechanism through the lens of associative memory. By…

We present Amos, a stochastic gradient-based optimizer designed for training deep neural networks. It can be viewed as an Adam optimizer with theoretically supported, adaptive learning-rate decay and weight decay. A key insight behind Amos…

An algorithm is presented for momentum gradient descent optimization based on the first-order differential equation of the Newtonian dynamics. The fictitious mass is introduced to the dynamics of momentum for regularizing the adaptive…

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 vast majority of modern deep learning models are trained with momentum-based first-order optimizers. The momentum term governs the optimizer's memory by determining how much each past gradient contributes to the current convergence…

This paper investigates the distributed stochastic nonconvex and nonsmooth composite optimization problem. Existing stochastic typically rely on uniform step size strictly bounded by global network parameters, such as the maximum node…

We introduce two complementary techniques for efficient optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm step size, generalizes AdaGrad-Norm and…

Gradient orthogonalization is a simple strategy that shows great utility in speeding up gradient descent. The Muon optimizer (Jordan, Jin, et al., 2024) combines gradient orthogonalization with first-order momentum and achieves significant…

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…

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…

Fine-tuning Large Language Models (LLMs) is essential for adapting pre-trained models to downstream tasks. Yet traditional first-order optimizers such as Stochastic Gradient Descent (SGD) and Adam incur prohibitive memory and computational…

Training a modern machine learning architecture on a new task requires extensive learning-rate tuning, which comes at a high computational cost. Here we develop new Polyak-type adaptive learning rates that can be used on top of any momentum…

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 identify and formalize an underexplored phenomenon in deep learning optimization: directional alignment and loss convergence can be decoupled. An optimizer can exhibit near-perfect directional consistency (cc_t -> 1, measured via…