Spectral Evolution and Invariance in Linear-width Neural Networks

We investigate the spectral properties of linear-width feed-forward neural networks, where the sample size is asymptotically proportional to network width. Empirically, we show that the spectra of weight in this high dimensional regime are invariant when trained by gradient descent for small constant learning rates; we provide a theoretical justification for this observation and prove the invariance of the bulk spectra for both conjugate and neural tangent kernels. We demonstrate similar characteristics when training with stochastic gradient descent with small learning rates. When the learning rate is large, we exhibit the emergence of an outlier whose corresponding eigenvector is aligned with the training data structure. We also show that after adaptive gradient training, where a lower test error and feature learning emerge, both weight and kernel matrices exhibit heavy tail behavior.