Sharp uniform approximation for spectral Barron functions by deep neural networks

This work explores the neural network approximation capabilities for functions within the spectral Barron space $\mathscr{B}^s$, where $s$ is the smoothness index. We demonstrate that for functions in $\mathscr{B}^{1/2}$, a shallow neural network (a single hidden layer) with $N$ units can achieve an $L^p$-approximation rate of $\mathcal{O}(N^{-1/2})$. This rate also applies to uniform approximation, differing by at most a logarithmic factor. Our results significantly reduce the smoothness requirement compared to existing theory, which necessitate functions to belong to $\mathscr{B}^1$ in order to attain the same rate. Furthermore, we show that increasing the network's depth can notably improve the approximation order for functions with small smoothness. Specifically, for networks with $L$ hidden layers, functions in $\mathscr{B}^s$ with $0 < sL \le 1/2$ can achieve an approximation rate of $\mathcal{O}(N^{-sL})$. The rates and prefactors in our estimates are dimension-free. We also confirm the sharpness of our findings, with the lower bound closely aligning with the upper, with a discrepancy of at most one logarithmic factor.