A Physical Response-and-Memory Model for Muon Optimization
作者: Yinze Hu, Hongjun Xiang, Xingao Gong, Hongyu Yu
分类: cs.LG, cond-mat.dis-nn, cond-mat.stat-mech, cs.AI, physics.comp-ph
发布日期: 2026-08-24
备注: 44 pages, 8 figures
💡 一句话要点
提出物理响应与记忆模型以优化Muon算法
🎯 匹配领域: 支柱九：具身大模型 (Embodied Foundation Models)
关键词: 优化器 深度学习 物理模型 记忆机制 动量更新 大型语言模型 训练效率
📋 核心要点
- 现有优化器如Muon在训练大型语言模型时，动量的有效性和历史平均长度仍主要依赖经验，缺乏理论支持。
- 本文提出将权重矩阵视为具有记忆的响应介质，构建物理模型以解释优化器的有效性，并提出Bi-Maxwell优化器。
- 实验结果表明，替换记忆核从单一时间尺度到双时间尺度，显著减少了达到目标损失所需的训练步骤。
📝 摘要（中文）
训练大型语言模型的成本高昂，优化器在梯度转化为权重更新的过程中起着关键作用。现有的优化器如SGD、AdamW和Muon等主要依赖工程直觉和基准测试进行选择。Muon通过半正交化动量矩阵来进行更新，尽管在公共训练基准上屡创佳绩，但其有效性及动量平均的历史长度仍主要依赖经验。本文将权重矩阵视为具有记忆的响应介质，构建了相应的物理模型，解释了半正交化方向的有效性及动量的放松时间尺度。基于此，提出了Bi-Maxwell优化器，并通过实验验证了其在训练阶段的记忆长度应随之增长。
🔬 方法详解
问题定义：本文旨在解决现有优化器在动量更新方向和历史平均长度选择上的理论不足，现有方法主要依赖经验，缺乏系统性分析。
核心思路：通过将权重矩阵视为具有记忆的响应介质，构建物理模型，解释半正交化动量更新方向的有效性，并提出Bi-Maxwell优化器以优化动量的历史平均长度。
技术框架：该框架包括两个主要模块：一是半正交化动量矩阵的更新机制，二是基于介质放松的记忆长度调整机制，整体流程通过动态调整动量更新方向和历史长度来提升训练效率。
关键创新：最重要的创新在于将优化器的设计与物理模型相结合，提出了动量的内应力概念及其放松时间尺度的多重性，显著区别于传统优化器的单一时间尺度设计。
关键设计：在Bi-Maxwell优化器中，动量更新的参数设置和损失函数设计基于物理模型的推导，采用双时间尺度的记忆核设计，以适应不同训练阶段的需求。
🖼️ 关键图片
📊 实验亮点
实验结果显示，Bi-Maxwell优化器在公共大型语言模型优化基准上，替换记忆核后，训练达到目标损失所需的步骤显著减少，具体提升幅度为XX%，验证了理论模型的有效性。
🎯 应用场景
该研究的潜在应用领域包括大型语言模型的训练、深度学习优化器的设计以及其他需要高效梯度更新的机器学习任务。通过优化器的改进，可以显著降低训练成本，提高模型性能，具有广泛的实际价值和未来影响。
📄 摘要（原文）
Training large language models is costly. How low a loss the same compute can ultimately reach depends on how each step's gradient is converted into a weight update; the rule that performs this conversion is the optimizer. From SGD and AdamW to the recent Muon, effective update rules have mostly been shaped by engineering intuition and then selected on benchmarks. Muon semi-orthogonalizes the momentum matrix before applying the update and has kept breaking records on public training benchmarks; yet why the semi-orthogonalized direction works, and over how long a history the momentum should average, are two questions at present answered mainly by experience. Here we treat the weight matrix during training as a responsive medium with memory and build a physical model for it, in which both questions find answers: the semi-orthogonalized direction is the maximally dissipative response under an output-side safety budget, which explains why it works; momentum is the internal stress accumulated by the medium; how long it should average is set by the relaxation of this stress, and a real medium relaxes on more than one timescale, the simplest form being one fast and one slow. On this basis we propose the Bi-Maxwell optimizer. The framework further yields a testable consequence: gradient directions change fast early in training and more slowly later, so the optimal memory length should grow with training stage; step-by-step measurements of a proxy for it by a read-only probe across 8 independent training trajectories are consistent with this consequence. Replacing the memory kernel alone, from a single timescale to two, brings training to the target loss in noticeably fewer steps on a public large-language-model optimizer benchmark.