Asymptotic learning curves of kernel methods: empirical data versus teacher-student paradigm

Abstract

How many training data are needed to learn a supervised task? It is often observed that the generalization error decreases as n(-beta) where n is the number of training examples and beta is an exponent that depends on both data and algorithm. In this work we measure beta when applying kernel methods to real datasets. For MNIST we find beta approximate to 0.4 and for CIFAR10 beta approximate to 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 beta for translation invariant kernels, using previous results from the kriging literature. Provided that the student is not too sensitive to high frequencies, beta 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 beta from real data by performing kernel PCA, leading to beta approximate to 0.36 for MNIST and beta approximate to 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