变换器学习动态中的低维执行流形：来自模运算任务的证据
Low-Dimensional Execution Manifolds in Transformer Learning Dynamics: Evidence from Modular Arithmetic Tasks
AI总结：
研究发现变换器学习动态在高维参数空间中坍缩到低维流形，揭示了核心计算发生在大幅减少的子空间中，对模型解释性和训练设计有重要影响。
摘要：
我们通过精心控制的模运算任务，研究了过度参数化变换器模型的学习动态的几何结构。我们的主要发现是，尽管在高维参数空间（d=128）中运行，变换器训练轨迹迅速坍缩到低维执行流形上，维度为3-4。这种维度坍缩在随机种子和中等任务难度下具有鲁棒性，尽管在参数空间中流形的方向在不同运行中有所不同。我们证明这种几何结构下了一些经验观察到的现象：（1）在执行流形内路由坐标饱和时，出现尖锐的注意力集中；（2）SGD交换子在训练早期优先对齐于执行子空间（相对于随机基线高出10倍），超过92%的非交换性被限制在正交的 staging 方向，且随着训练收敛，这种对齐性逐渐减弱；（3）稀疏自编码器捕捉到辅助路由结构，但无法隔离执行本身，执行仍分布于低维流形上。我们的结果表明，变换器学习有一个统一的几何框架，其中大部分参数用于吸收优化干扰，而核心计算发生在大幅减少的子空间中。这些发现对可解释性、训练课程设计以及理解神经网络学习中过度参数化的作用具有影响。
英文摘要：
We investigate the geometric structure of learning dynamics in overparameterized transformer models through carefully controlled modular arithmetic tasks. Our primary finding is that despite operating in high-dimensional parameter spaces ($d=128$), transformer training trajectories rapidly collapse onto low-dimensional execution manifolds of dimension $3$--$4$. This dimensional collapse is robust across random seeds and moderate task difficulties, though the orientation of the manifold in parameter space varies between runs. We demonstrate that this geometric structure underlies several empirically observed phenomena: (1) sharp attention concentration emerges as saturation along routing coordinates within the execution manifold, (2) SGD commutators are preferentially aligned with the execution subspace (up to $10\times$ random baseline) early in training, with $>92\%$ of non-commutativity confined to orthogonal staging directions and this alignment decreasing as training converges, and (3) sparse autoencoders capture auxiliary routing structure but fail to isolate execution itself, which remains distributed across the low-dimensional manifold. Our results suggest a unifying geometric framework for understanding transformer learning, where the vast majority of parameters serve to absorb optimization interference while core computation occurs in a dramatically reduced subspace. These findings have implications for interpretability, training curriculum design, and understanding the role of overparameterization in neural network learning.
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