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

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.

32 pages, 7 figures

References in corpus (5)

• Deep Learning Scaling is Predictable, Empirically
• Learning curves of generic features maps for realistic datasets with a teacher-student model
• Learning with invariances in random features and kernel models
• Kernel Alignment Risk Estimator: Risk Prediction from Training Data
• Approximation and Learning with Deep Convolutional Models: a Kernel Perspective

Cited by in corpus (3)

• Data-driven emergence of convolutional structure in neural networks
• CNNs Avoid Curse of Dimensionality by Learning on Patches
• Learning with convolution and pooling operations in kernel methods