Super Consistency of Neural Network Landscapes and Learning Rate Transfer

Recently, there has been growing evidence that if the width and depth of a neural network are scaled toward the so-called rich feature learning limit ($\mu$P and its depth extension), then some hyperparameters --- such as the learning rate --- exhibit transfer from small to very large models. From an optimization perspective, this phenomenon is puzzling, as it implies that the loss landscape is consistently similar across very different model sizes. In this work, we study the landscape through the lens of the Hessian, with a focus on its largest eigenvalue (i.e. the sharpness), and find that certain spectral properties under $\mu$P are largely independent of the width and depth of the network along the training trajectory. We name this property *super consistency* of the landscape. On the other hand, we show that in the Neural Tangent Kernel (NTK) and other scaling regimes, the sharpness exhibits very different dynamics at different scales.