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

TL;DR

This paper demonstrates that local receptive fields are crucial for the learning efficiency of convolutional neural networks, showing that locality, rather than translational invariance, primarily determines the learning curve exponent in teacher-student kernel regression models.

Contribution

It introduces a theoretical framework analyzing the roles of locality and invariance in CNNs, revealing locality's dominant influence on learning curves and providing empirical validation.

Findings

Locality determines the learning curve exponent in convolutional models.

Translational invariance does not significantly affect the learning rate.

Kernel regression with adaptive ridge yields similar learning behavior as ridgeless case.

Abstract

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 $\beta$ (that relates the test error $\epsilon_t\sim P^{-\beta}$ 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 $s$, $\beta$ is a function of $s$ only and does not depend on the input dimension. We confirm our predictions on $\beta$ empirically. We conclude by proving, using 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.