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This paper establishes minimax optimal convergence rates for learning spectral Barron functions using deep ReLU neural networks, advancing the theoretical understanding of approximation capabilities in high-dimensional settings. By proving that these networks avoid the curse of dimensionality with specific smoothness indices, the work clarifies the fundamental limits and efficiency of modern architectures for complex function classes.
Minimax rates for learning spectral Barron functions by deep ReLU neural networks
arXiv:2609.39020v1 Announce Type: cross 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 $\widetilde{\mathcal{O}} (S^{-\frac{1}{2}-\frac{s}{d}})$, 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 $n^{-\frac{d+2s}{2d+2s}}$ with $n$ training samples. Finally, we prove that this convergence rate is minimax optimal up to logarithmic factors.
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 $\widetilde{\mathcal{O}} (S^{-\frac{1}{2}-\frac{s}{d}})$, 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 $n^{-\frac{d+2s}{2d+2s}}$ with $n$ training samples. Finally, we prove that this convergence rate is minimax optimal up to logarithmic factors.
Machine Learning (stat.ML); Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2609.39020 [stat.ML]
(or arXiv:2609.39020v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2609.39020 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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