Decomposing SGD Dynamics in Neural Networks: Teacher-Induced Spikes and Variance Inflation

We study two-layer neural networks trained by stochastic gradient descent (SGD) on a multi-index teacher model, focusing on how first-layer parameters orthogonal to the teacher subspace affect generalization. We decompose the first-layer weights into a signal component aligned with the teacher and an orthogonal complement, and analyze their joint SGD dynamics together with the second layer in the proportional limit. We show that although SGD updates to the complement are initially isotropic, the training dynamics induce teacher-dependent spectral spikes in the complement Gram matrix. This emergent anisotropy provides a concrete mechanism by which representation variance is amplified, and generalization degrades. To isolate this effect, we introduce Decomposed Dynamics, an analytically tractable surrogate whose asymptotic test-error dynamics coincide with those of SGD. Using this surrogate, we characterize regimes in which freezing the orthogonal complement strictly improves generalization relative to training it, despite identical training error. Finally, we show that the signal component rapidly aligns with the second-layer weights, revealing strong cross-layer coupling even in this shallow setting. Together, our results provide a precise dynamical explanation for when and how feature learning outside the teacher subspace harms generalization.