Newest papers

The role of locality and weight sharing in convolutional neural networks

Francesco Mori, Melanie Weber · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.30118 · cited by 0 works (OpenAlex) · open-access PDF

French

Le taux de réussite des réseaux neuronaux convolutifs est souvent attribué à la localisation et au partage du poids. Cependant, comment ces deux propriétés forment séparément le taux de localisation de l'apprentissage, et comment leurs effets dépendent des corrélations spatiales tardives des mêmes images naturelles, restent mal compris. Nous traitons ces questions dans un modèle enseignant-étudiant avec un enseignant convolutif agissant sur les positions d'entrée de $P$ avec la corrélation $C$ entre les positions. Nous comparons les élèves convolutifs et locaux formés par le partage de gradients stochastiques en ligne et dérivons des équations différentielles ordinaires qui décrivent leur dynamique exactement dans la limite de la grande

Machine translation (Meta M2M100); the English original follows.

Abstract

The success of convolutional neural networks is often attributed to locality and weight sharing. However, how these two properties separately shape learning, and how their effects depend on the spatial correlations of natural images, remain poorly understood. We address these questions in a teacher-student model with a convolutional teacher acting on $P$ input positions with correlation $c$ between positions. We compare convolutional and locally connected students trained by online stochastic gradient descent, and derive ordinary differential equations that describe their dynamics exactly in the limit of large feature dimension $D$. For uncorrelated positions ($c = 0$), the two students follow the same learning trajectory, with the convolutional student learning exactly $P$ times faster. We also find that locality affects the sample complexity: a student that matches the kernel size of the teacher recovers it from $\mathcal{O}_{D}(D)$ samples, whereas $\mathcal{O}_{D}(D\log(D))$ samples are needed with a larger kernel. For $c > 0$, spatial correlations can reduce the advantage of weight sharing. The asymptotic error of the convolutional student grows with $c$, whereas that of the locally connected student is independent of $c$ and overall lower when evaluated at the same learning rate. When the learning rate is instead tuned to the training horizon, however, the locally connected student wins at early times and the convolutional student at late times, with a transition that we predict analytically.

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Topics: Convolutional neural network, Locality, Limit (mathematics), Computer science, Uncorrelated, Dimension (graph theory)

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