The Role of Locality and Weight Sharing in Convolutional Neural Networks

Forum topic · 小凯 · 2026-10-23

Summary

A new arXiv paper (2610.30118) by Francesco Mori and Melanie Weber disentangles the effects of locality and weight sharing in convolutional neural networks using a teacher-student framework. A convolutional teacher operates on P input positions with spatial correlation c, while convolutional and locally connected students are trained via online stochastic gradient descent. The authors derive ODEs that exactly describe the learning dynamics in the large feature-dimension D limit. For uncorrelated inputs (c=0), both students follow identical learning trajectories, but the convolutional student learns exactly P times faster. Sample complexity is also affected by locality: a student whose kernel size matches the teacher recovers it with O_D(D) samples, whereas larger kernels require O_D(D log(D)) samples. When spatial correlations are present (c>0), the advantage of weight sharing diminishes: the convolutional student's asymptotic error grows with c, while the locally connected student's error is independent of c and lower overall at equal learning rates. With learning rates tuned to training duration, locally connected students perform better early and convolutional students later, with an analytically predictable transition point.

Paper Overview

Field: Machine Learning Authors: Francesco Mori, Melanie Weber Published: 2026-10-21 arXiv: 2610.30118

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.

The authors address these questions in a teacher-student model with a convolutional teacher acting on P input positions with correlation c between positions. They 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.

Key Findings

For uncorrelated positions (c = 0), the two students follow the same learning trajectory, but the convolutional student learns exactly P times faster.
Locality affects sample complexity: a student whose kernel size matches the teacher's can recover the teacher with O_D(D) samples, while using a larger kernel requires O_D(D log(D)) samples.
When c > 0, spatial correlations reduce the benefit of weight sharing: the convolutional student's asymptotic error grows with c, whereas the locally connected student's asymptotic error is independent of c and overall lower at equal learning rates.
When the learning rate is adjusted according to training duration, locally connected students perform better early in training and convolutional students perform better later; the transition point can be predicted analytically.

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