Asymptotic learning curves of kernel methods: empirical data versus teacher–student paradigm. (21st December 2020) - Spigler, Stefano; Geiger, Mario; Wyart, Matthieu
Abstract: How many training data are needed to learn a supervised task? It is often observed that the generalization error decreases as n − β where n is the number of training examples and β is an exponent that depends on both data and algorithm. In this work we measure β when applying kernel methods to real datasets. For MNIST we find β ≈ 0.4 and for CIFAR10 β ≈ 0.1, for both regression and classification tasks, and for Gaussian or Laplace kernels. To rationalize the existence of non-trivial exponents that can be independent of the specific kernel used, we study the teacher–student framework for kernels. In this scheme, a teacher generates data according to a Gaussian random field, and a student learns them via kernel regression. With a simplifying assumption—namely that the data are sampled from a regular lattice—we derive analytically β for translation invariant kernels, using previous results from the kriging literature. Provided that the student is not too sensitive to high frequencies, β depends only on the smoothness and dimension of the training data. We confirm numerically that these predictions hold when the training points are sampled at random on a hypersphere. Overall, the test error is found to be controlled by the magnitude of the projection of the true function on the kernel eigenvectors whose rank is larger than n . Using this idea we predict the exponent β from real data by performing kernel PCA, leading to β ≈ 0.36 for MNIST and β ≈ 0.07 for CIFAR10, in good agreement with observations. We argue that these rather large exponents are possible due to the small effective dimension of the data.