Empirical evidence shows that fully-connected neural networks in the infinite-width limit (lazy training) eventually outperform their finite-width counterparts in most computer vision tasks; on the other hand, modern architectures with convolutional layers often achieve optimal performances in the finite-width regime. In this work, we present a theoretical framework that provides a rationale for these differences in one-hidden-layer networks; we derive an effective action in the so-called proportional limit for an architecture with one convolutional hidden layer and compare it with the result available for fully-connected networks. Remarkably, we identify a completely different form of kernel renormalization: whereas the kernel of the fully-connected architecture is just globally renormalized by a single scalar parameter, the convolutional kernel undergoes a local renormalization, meaning that the network can select the local components that will contribute to the final prediction in a data-dependent way. This finding highlights a simple mechanism for feature learning that can take place in overparametrized shallow convolutional neural networks, but not in shallow fully-connected architectures or in locally connected neural networks without weight sharing.
