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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.
Joint work with Blake Bordelon and Abdulkadir Canatar
Cengiz (pronounced “Jen·ghiz”) Pehlevan is an assistant professor of applied mathematics at the Harvard John A. Paulson School of Engineering and Applied Sciences. He received his undergraduate degrees in physics and electrical engineering from Bogazici University of Istanbul in 2004 and his doctorate in theoretical physics from Brown University in 2011. He was a Swartz Fellow at Harvard University, a postdoctoral associate at Janelia Research Campus, and a research scientist in the neuroscience group at the Flatiron Institute. His research interests are in theoretical neuroscience, theory of deep learning, biologically-inspired machine learning and neuromorphic computing.