Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks
This provides a theoretical framework for understanding learning dynamics in neural networks, which is incremental but clarifies spectral learning stages.
The authors derived analytical expressions for generalization error in kernel regression and wide neural networks, showing that as training data increases, these models learn target function modes in stages, with verification on synthetic data and MNIST.
We derive analytical expressions for the generalization performance of kernel regression as a function of the number of training samples using theoretical methods from Gaussian processes and statistical physics. Our expressions apply to wide neural networks due to an equivalence between training them and kernel regression with the Neural Tangent Kernel (NTK). By computing the decomposition of the total generalization error due to different spectral components of the kernel, we identify a new spectral principle: as the size of the training set grows, kernel machines and neural networks fit successively higher spectral modes of the target function. When data are sampled from a uniform distribution on a high-dimensional hypersphere, dot product kernels, including NTK, exhibit learning stages where different frequency modes of the target function are learned. We verify our theory with simulations on synthetic data and MNIST dataset.