The success of deep neural networks largely depends on the statistical structure of the training data. While learning dynamics and generalization on isotropic data are well-established, the impact of pronounced anisotropy on these crucial aspects is not yet fully understood. We examine the impact of data anisotropy, represented by a spiked covariance structure, a canonical yet tractable model, on the learning dynamics and generalization error of a two-layer linear network in a linear regression setting. Our analysis reveals that the learning dynamics proceed in two distinct phases, governed initially by the input-output correlation and subsequently by other principal directions of the data structure. Furthermore, we derive an analytical expression for the generalization error, quantifying how the alignment of the spike structure of the data with the learning task improves performance. Our findings offer deep theoretical insights into how data anisotropy shapes the learning trajectory and final performance, providing a foundation for understanding complex interactions in more advanced network architectures.
Neural scaling laws describe the predictable power-law dependence of model performance on training time, dataset size, and model scale, with the exponents quantifying how performance changes with resources. Recent theory has shown how scaling exponents are shaped by the data distribution and the structure of the learning task, but whether network depth can improve training-time scaling has remained unsettled. Through a refined analysis of deep linear networks, we clarify that the task spectrum determines when increasing network depth improves training-time scaling. For spectrally difficult tasks, increasing depth improves the asymptotic training-time scaling exponent, whereas for easy tasks, the exponent remains independent of depth, with depth affecting only a logarithmic correction at the boundary between these regimes. Remarkably, we find that a reparameterization of time governed by the predictor norm provides a unified description of the dynamics across task difficulties. Under this reparameterization, the loss follows a common scaling law whose exponent is independent of depth. Numerical experiments on a task derived from benchmark image data and with nonlinear networks exhibit qualitatively consistent task-dependent effects of depth on training-time scaling. Our findings advance the theoretical understanding of how task structure shapes the effects of network architecture on learning and offer a basis for studying scaling behavior across broader classes of neural networks.