The Loss Does Not See the Basis, but Adam Does: Gauge Equivariance Determines Optimizers' Low-Rank Bias
Score
77首次从规范等变性角度系统区分Adam与梯度下降的低秩偏好，理论统一性强，信息增量大。
包含结构定理、迁移定理和九种更新规则的实验，分析严谨且覆盖全面。
为选择或设计具有低秩偏置的优化器（如shared-scalar Adam、Muon）提供了可操作判据。
预印本紧跟优化器隐式偏置与低秩恢复的研究热点，对当前大模型训练有直接参考价值。
来自arXiv AI/CL/LG类别，方法描述完整，虽未经同行评审但可信度较高。
Base 85 × T2 Source weight ×0.9 = 77
Curation line 70Selected
Why selected
Combining theoretical depth, experimental breadth, and timeliness, this is a high-value paper, though its applicability to general non-factorized tasks still needs further validation.
AI commentary
「I think the most elegant part of this paper is turning optimizer coordinate choice into a provable mathematical condition rather than an empirical black box. For engineering teams who want to replace Adam with Muon or Shampoo, this is especially important: if the goal is low-rank compression or more compact interpolants, equivariant optimizers are the right direction.」
Quick overview
This paper shows that in factored models W = UV^T, gradient descent is implicitly biased toward low-rank solutions, while Adam is not. The difference is traced to gauge symmetry: the loss is invariant under (U,V)->(UQ,VQ), but only gauge-equivariant optimizers can inherit gradient flow's low-rank mechanism. Gradient descent, momentum, shared-scalar Adam, Muon, and Shampoo satisfy this condition; Adam, RMSProp, and other coordinate-wise methods do not. The authors prove a structure theorem identifying memoryless equivariant rules with Gram-determined left preconditioners, and a transfer theorem for common-scalar flows. Experiments on matrix sensing, Transformers, and hyperspectral datasets confirm that basis choice determines which interpolant the optimizer selects, with gradient descent cutting held-out error 43-44% at the lowest sampling density.
- Gauge equivariance determines whether an optimizer inherits gradient flow's low-rank bias: GD, momentum, shared-scalar Adam, Muon, and Shampoo satisfy it, while Adam, RMSProp, and coordinate-wise methods do not.
- A structure theorem shows memoryless equivariant update rules are exactly Gram-determined left preconditioners, and a transfer theorem extends gradient-flow path properties to common-scalar flows.
- In Transformers, Adam separates gauge-equivalent initializations at the first step; on hyperspectral data, gradient descent reduces held-out error by 43-44% at the lowest sampling density.
Full text
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