发表机构
NVIDIA Research（英伟达研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出GO-MUON算法，将谱几何与曲率结合，通过复用匹配的依赖数据的几何实现精确更新，量化了softmax交叉熵中相关因子的接近情况，揭示了惰性几何是计算-统计的权衡。
AI 中文摘要
Muon的极性更新对于未加权谱几何是精确的。我们引入GO-MUON，其使用匹配的、依赖数据的几何并在多个优化步骤中复用该几何。在任意正定左右映射的条件下，其原始更新可精确求解对应的加权谱神谕；该结论与映射的估计方式或最近刷新的时间无关。对于softmax交叉熵，我们量化了观测标签反向因子接近模型Fisher和广义高斯-牛顿因子的情况。我们还表明，四步刷新几乎保留了缓慢变化几何的跟踪延迟，同时增加了平稳因子噪声，这使得惰性几何成为计算-统计的权衡，而非去噪机制。
英文摘要
Muon's polar update is exact for an unweighted spectral geometry. We introduce GO-MUON, which uses a matched data-dependent geometry and reuses it across several optimization steps. Conditioned on any positive-definite left and right maps, its raw update exactly solves the corresponding weighted spectral oracle; this statement is independent of how the maps are estimated or how recently they were refreshed. For softmax cross-entropy, we quantify when the observed-label backward factor approaches the model Fisher and generalized Gauss--Newton factor. We also show that four-step refresh nearly preserves the tracking delay of slowly changing geometry while increasing stationary factor noise, making lazy geometry a compute--statistics tradeoff rather than a denoising mechanism.
