Locality defeats the curse of dimensionality in convolutional teacher-student scenarios
This work addresses a fundamental problem in understanding convolutional neural networks for researchers in machine learning theory, though it is incremental as it builds on existing teacher-student frameworks.
The study investigates the roles of locality and translational invariance in convolutional neural networks using a teacher-student framework for kernel regression, finding that locality is crucial for determining the learning curve exponent, which does not depend on input dimension when the teacher's filter size is smaller than the student's, and empirical results confirm these predictions.
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 $ε_t\sim P^{-β}$ 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, 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.