Shallow ReLU^s Networks in L^p-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization

arXiv:2605.18468v1 Announce Type: cross Abstract: We study approximation by shallow ReLU^s networks, sigma_s(t)=max{0,t}^s, and the generalization behavior of such networks under ell_1 path-norm control. For the L^p-type integral spaces widetilde{F}_{p,au_d,s}, 1le ple2, we establish approximation bounds for shallow networks using spherical harmonic analysis. In particular, when the parameter measure is the uniform measure au_d and p<p^*=(2d+2)/(d+3), we obtain the rate O(m^{-1/2-d(2-p)/(2d(2-p)+2p(2s+d+1))}log^{3/2}m), which improves the corresponding random-feature rate. We also derive approximation rates for Sobolev spaces W^{alpha,p} in the range 1le p<2 by embedding them into spectral Barron spaces. Finally, for nonparametric regression with sub-Gaussian noise, we prove minimax-optimal generalization bounds for path-norm-regularized shallow ReLU^s networks over Barron and Sobolev spaces, with matching lower bounds up to logarithmic factors.

Source: arXiv cs.LG | 2026-05-19