mlsanityData to 14 Oct 2026

Optimal learning of high-dimensional classification problems using deep neural networks

arXiv 2112.12555 ↗First posted 23 Dec 2021math.FA cs.LG stat.MLcited by 9

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Abstract (arXiv)

We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish universal lower bounds for this estimation problem, for general classes of continuous decision boundaries. For the class of locally Barron-regular decision boundaries, we find that the optimal estimation rates are essentially independent of the underlying dimension and can be realized by empirical risk minimization methods over a suitable class of deep neural networks. These results are based on novel estimates of the L1 and L^∞ entropies of the class of Barron-regular functions.

Key facts. Cited by 9 papers in the map (data through 14 Oct 2026). Builds on 17 of its 49 references that are in the map. First posted on arXiv 23 Dec 2021.

Builds on

17 of 49 references.

References · 17

  1. 2014Deep Learning in Neural Networks: An Overviewcited by 770
  2. 2017Nonparametric regression using deep neural networks with ReLU activation functioncited by 433
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  4. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 179
  5. 2020Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don'tcited by 95
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  7. 2016Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controlscited by 84
  8. 2018A Priori Estimates of the Population Risk for Two-layer Neural Networkscited by 76
  9. 2021The Modern Mathematics of Deep Learningcited by 63
  10. 2020Representation formulas and pointwise properties for Barron functionscited by 62
  11. 2018Deep Neural Networks Learn Non-Smooth Functions Effectivelycited by 57
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5 more
  1. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 45
  2. 2020On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamicscited by 39
  3. 2021Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spacescited by 30
  4. 2020Neural network approximation and estimation of classifiers with classification boundary in a Barron classcited by 28
  5. 2016Risk Bounds for High-dimensional Ridge Function Combinations Including Neural Networkscited by 26

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 3; 2023: 1; 2024: 2; 2025: 2; 2026: 1.

9 papers.

  1. 2022A general approximation lower bound in Lp norm, with applications to feed-forward neural networkscited by 12
  2. 2022Lp sampling numbers for the Fourier-analytic Barron spacecited by 6
  3. 2023Deep ReLU networks and high-order finite element methods II: Chebyshev emulationcited by 3
  4. 2024High-dimensional classification problems with Barron regular boundaries under margin conditionscited by 2
  5. 2022Optimal Learningcited by 0
  6. 2024Dimension-independent learning rates for high-dimensional classification problemscited by 0
  7. 2025Super-fast Rates of Convergence for Neural Network Classifiers under the Hard Margin Conditioncited by 0
  8. 2025Minimax learning rates for estimating binary classifiers under margin conditionscited by 0
  9. 2026Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networkscited by 0