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

• URL: http://arxiv.org/abs/2605.18468v4
• Date: Mon, 25 May 2026 18:12:10 GMT
• Title: Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization
• Abstract summary: In particular, when $_d$ is the uniform measure and $1le p2$, the approximation rate is $O!left(m-fracp(2s+2d+1)-2d2dpright)$ for $1le p*$ and $O!left(m-fracp(4s+3d-1)-2d+24dpright)$ for $p*p2$, where $p*=frac2d+2d+3$.<n>For non
• Score: 12.871748162999062
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper studies approximation by shallow ReLU$^s$ networks, $σ_s(t)=\max\{0,t\}^s$, together with their generalization behavior under $\ell_1$ path-norm control. For the $L^p$-type integral spaces $\widetilde{\mathcal{F}}_{p,τ_d,s}$, $1\le p\le2$, spherical harmonic analysis yields approximation bounds for shallow networks. In particular, when $τ_d$ is the uniform measure and $1\le p<2$, the approximation rate is $O\!\left(m^{-\frac{p(2s+2d+1)-2d}{2dp}}\right)$ for $1\le p\le p^*$ 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 $W^{α,p}$, $1\le p<2$, 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 $O\!\left(n^{-\frac{d+2s+1}{2d+2s+1}}\log n\right)$ over $\mathscr{B}_s$ and $O\!\left(n^{-\frac{2α}{2α+d}}\log n\right)$ over $W^{α,\infty}$, with matching lower bounds up to logarithmic factors.

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