arXiv:2605.18468v4 Announce Type: replace-cross
Abstract: This paper studies approximation by shallow ReLU$^s$ networks, $sigma_s(t)=max{0,t}^s$, together with their generalization behavior under $ell_1$ path-norm control. For the $L^p$-type integral spaces $widetilde{mathcal{F}}_{p,tau_d,s}$, $1le ple2$, spherical harmonic analysis yields approximation bounds for shallow networks. In particular, when $tau_d$ is the uniform measure and $1le p<2$, the approximation rate is $O!left(m^{-frac{p(2s+2d+1)-2d}{2dp}}right)$ for $1le ple p^*$ and $O!left(m^{-frac{p(4s+3d-1)-2d+2}{4dp}}right)$ for $p^*<p<2$, where $p^*=frac{2d+2}{d+3}$. Approximation bounds for Sobolev spaces $W^{alpha,p}$, $1le p<2$, are obtained through embeddings into spectral Barron spaces. For nonparametric regression with sub-Gaussian noise, path-norm-regularized shallow ReLU$^s$ networks achieve minimax-optimal rates $O!left(n^{-frac{d+2s+1}{2d+2s+1}}log nright)$ over $mathscr{B}_s$ and $O!left(n^{-frac{2alpha}{2alpha+d}}log nright)$ over $W^{alpha,infty}$, with matching lower bounds up to logarithmic factors.