Locality defeats the curse of dimensionality in convolutional teacher–student scenarios

By Alessandro Favero, Francesco Cagnetta, Matthieu Wyart. 2021

33 citations recorded by the source corpus.

Convolutional neural networks perform a local and translationally-invariant treatment of the data: quantifying which of these two aspects is central to their success remains a challenge. We study this problem within a teacher–student framework for kernel regression, using ‘convolutional’ kernels inspired by the neural tangent kernel of simple convolutional architectures of given filter size. Using heuristic methods from physics, we find in the ridgeless case that locality is key in determining the learning curve exponent β (that relates the test error ϵ t ∼ P −β to the size of the training set P), whereas translational invariance is not. In particular, if the filter size of the teacher t is smaller than that of the student s, β is a function of s only and does not depend on the input dimension. We confirm our predictions on β empirically. We conclude by proving, under a natural universality assumption, that performing kernel regression with a ridge that decreases with the size of the training set leads to similar learning curve exponents to those we obtain in the ridgeless case.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistical and Nonlinear Physics

Citations in the corpus

Learning curves for continual learning in neural networks: Self-knowledge transfer and forgetting (2021, background): “Studies on NTK eigenvalues have made remarkable progress, covering shallow and deep ReLU neural networks (Geifman et al., 2020; Chen & Xu, 2020), skip connections (Belfer et al., 2021), and CNNs (Favero et al., 2021).”