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

arXiv: https://arxiv.org/abs/2106.08619 · HTML: https://mlsanity.com/paper/2106.08619/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Placed in General knowledge distillation methods, one of 12 directions in Knowledge distillation.
- Builds on 27 of its 47 references that are in the map.
- First posted on arXiv 16 Jun 2021.

## Placed in

- Knowledge distillation / [General knowledge distillation methods](https://mlsanity.com/distill/general-knowledge-distillation-methods/)

## Builds on (most cited first, 27 of 27)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Learning with invariances in random features and kernel models](https://mlsanity.com/paper/2102.13219/) (2021)
- [On the Similarity between the Laplace and Neural Tangent Kernels](https://mlsanity.com/paper/2007.01580/) (2020)
- [Theoretical Issues in Deep Networks: Approximation, Optimization and Generalization](https://mlsanity.com/paper/1908.09375/) (2019)
- [A jamming transition from under- to over-parametrization affects loss landscape and generalization](https://mlsanity.com/paper/1810.09665/) (2018)
- [Implicit Regularization of Random Feature Models](https://mlsanity.com/paper/2002.08404/) (2020)
- [Kernel Alignment Risk Estimator: Risk Prediction from Training Data](https://mlsanity.com/paper/2006.09796/) (2020)
- [Towards Learning Convolutions from Scratch](https://mlsanity.com/paper/2007.13657/) (2020)
- [End-to-End Kernel Learning with Supervised Convolutional Kernel Networks](https://mlsanity.com/paper/1605.06265/) (2016)
- [Asymptotic learning curves of kernel methods: empirical data v.s. Teacher-Student paradigm](https://mlsanity.com/paper/1905.10843/) (2019)
- [Computational Separation Between Convolutional and Fully-Connected Networks](https://mlsanity.com/paper/2010.01369/) (2020)
- [Building Bayesian Neural Networks with Blocks: On Structure, Interpretability and Uncertainty](https://mlsanity.com/paper/1806.03563/) (2018)

## Built on by (most cited first, 26 of 26)

- [Task Arithmetic in the Tangent Space: Improved Editing of Pre-Trained Models](https://mlsanity.com/paper/2305.12827/) (2023)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model](https://mlsanity.com/paper/2307.02129/) (2023)
- [The SSL Interplay: Augmentations, Inductive Bias, and Generalization](https://mlsanity.com/paper/2302.02774/) (2023)
- [Data-driven emergence of convolutional structure in neural networks](https://mlsanity.com/paper/2202.00565/) (2022)
- [Learning with convolution and pooling operations in kernel methods](https://mlsanity.com/paper/2111.08308/) (2021)
- [Mapping of attention mechanisms to a generalized Potts model](https://mlsanity.com/paper/2304.07235/) (2023)
- [Local Kernel Renormalization as a mechanism for feature learning in overparametrized Convolutional Neural Networks](https://mlsanity.com/paper/2307.11807/) (2023)
- [Eigenspace Restructuring: a Principle of Space and Frequency in Neural Networks](https://mlsanity.com/paper/2112.05611/) (2021)
- [U-Nets as Belief Propagation: Efficient Classification, Denoising, and Diffusion in Generative Hierarchical Models](https://mlsanity.com/paper/2404.18444/) (2024)
- [Learning curves theory for hierarchically compositional data with power-law distributed features](https://mlsanity.com/paper/2505.07067/) (2025)
- [On the Spectral Bias of Convolutional Neural Tangent and Gaussian Process Kernels](https://mlsanity.com/paper/2203.09255/) (2022)
- [What Can Be Learnt With Wide Convolutional Neural Networks?](https://mlsanity.com/paper/2208.01003/) (2022)
- [Learning Curves for Continual Learning in Neural Networks: Self-Knowledge Transfer and Forgetting](https://mlsanity.com/paper/2112.01653/) (2021)
- [A Kernel Perspective of Skip Connections in Convolutional Networks](https://mlsanity.com/paper/2211.14810/) (2022)
- [Strong inductive biases provably prevent harmless interpolation](https://mlsanity.com/paper/2301.07605/) (2023)
- [On the Universal Approximation Property of Deep Fully Convolutional Neural Networks](https://mlsanity.com/paper/2211.14047/) (2022)
- [How Deep Networks Learn Sparse and Hierarchical Data: the Sparse Random Hierarchy Model](https://mlsanity.com/paper/2404.10727/) (2024)
- [Synergy and Symmetry in Deep Learning: Interactions between the Data, Model, and Inference Algorithm](https://mlsanity.com/paper/2207.04612/) (2022)
- [Scaling Laws and Representation Learning in Simple Hierarchical Languages: Transformers vs. Convolutional Architectures](https://mlsanity.com/paper/2505.07070/) (2025)
- [CNNs Avoid Curse of Dimensionality by Learning on Patches](https://mlsanity.com/paper/2205.10760/) (2022)
- [Short-Term Memory Convolutions](https://mlsanity.com/paper/2302.04331/) (2023)
- [The impact of memory on learning sequence-to-sequence tasks](https://mlsanity.com/paper/2205.14683/) (2022)
- [Inference of a Rumor's Source in the Independent Cascade Model](https://mlsanity.com/paper/2205.12125/) (2022)
- [Kernels, Data & Physics](https://mlsanity.com/paper/2307.02693/) (2023)
- [Cognitive Convergence: Deep Similarities Between Large Language Models and Human Cognition](https://mlsanity.com/paper/2607.26179/) (2026)

## Abstract (arXiv)

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 β (that relates the test error εt~ P-β 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, β is a function of s only and does not depend on the input dimension. We confirm our predictions on β 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.