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Dimension-independent learning rates for high-dimensional classification problems

arXiv 2409.17991 ↗First posted 26 Sep 2024cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

We study the problem of approximating and estimating classification functions that have their decision boundary in the RBV2 space. Functions of RBV2 type arise naturally as solutions of regularized neural network learning problems and neural networks can approximate these functions without the curse of dimensionality. We modify existing results to show that every RBV2 function can be approximated by a neural network with bounded weights. Thereafter, we prove the existence of a neural network with bounded weights approximating a classification function. And we leverage these bounds to quantify the estimation rates. Finally, we present a numerical study that analyzes the effect of different regularity conditions on the decision boundaries.

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References · 17

  1. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
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  4. 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. 2021What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theorycited by 60
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  10. 2018Fast convergence rates of deep neural networks for classificationcited by 46
  11. 2021Near-Minimax Optimal Estimation With Shallow ReLU Neural Networkscited by 41
  12. 2018A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientscited by 31
5 more
  1. 2020Neural network approximation and estimation of classifiers with classification boundary in a Barron classcited by 28
  2. 2020Advantage of Deep Neural Networks for Estimating Functions with Singularity on Hypersurfacescited by 17
  3. 2021Optimal learning of high-dimensional classification problems using deep neural networkscited by 9
  4. 2020A priori estimates for classification problems using neural networkscited by 5
  5. 2022Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problemscited by 3

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