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 $epsilon_tsim P^{-beta}$ 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. Theoretically, in some cases (including when teacher and student are equal) it can be shown that this prediction is an upper bound on performance. 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.
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