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Minimax rates for learning spectral Barron functions by deep ReLU neural networks

  • Songqiu Ma
  • Yunfei Yang
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Abstract

We study how well deep neural networks approximate and learn spectral Barron functions. Recent studies have shown that these function classes can be efficiently approximated by shallow neural networks without suffering from the curse of dimensionality. We complement these results by providing new approximation bounds for deep networks with ReLU activation and establishing the minimax rates for learning these function classes. Specifically, we show that d-dimensional spectral Barron functions with smoothness index s>0 can be approximated by deep ReLU neural networks with approximation rate π’ͺΜƒ (Sβˆ’1/2βˆ’s/d), where S denotes the number of nonzero parameters in the network. Using this approximation result, we further show that deep ReLU neural networks can learn spectral Barron functions in a fast rate nβˆ’(d+2s)/(2d+2s) with n training samples. Finally, we prove that this convergence rate is minimax optimal up to logarithmic factors.

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Cite

BibTeXRIS
BibTeX
@misc{ma2026minimax,
  title = {{Minimax rates for learning spectral Barron functions by deep ReLU neural networks}},
  author = {Songqiu Ma and Yunfei Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39020},
  eprint = {2609.39020},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39020},
}
RIS
TY  - UNPB
TI  - Minimax rates for learning spectral Barron functions by deep ReLU neural networks
AU  - Songqiu Ma
AU  - Yunfei Yang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39020
UR  - https://arxiv.org/abs/2609.39020
AB  - We study how well deep neural networks approximate and learn spectral Barron functions. Recent studies have shown that these function classes can be efficiently approximated by shallow neural networks without suffering from the curse of dimensionality. We complement these results by providing new approximation bounds for deep networks with ReLU activation and establishing the minimax rates for learning these function classes. Specifically, we show that d-dimensional spectral Barron functions with smoothness index s>0 can be approximated by deep ReLU neural networks with approximation rate π’ͺΜƒ (S^βˆ’1/2βˆ’s/d), where S denotes the number of nonzero parameters in the network. Using this approximation result, we further show that deep ReLU neural networks can learn spectral Barron functions in a fast rate n^βˆ’(d+2s)/(2d+2s) with n training samples. Finally, we prove that this convergence rate is minimax optimal up to logarithmic factors.
ER  -