Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization
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| Format: | Preprint |
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2026
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| _version_ | 1866914601694658560 |
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| author | Li, Weizhao Liu, Fanghui Shi, Lei |
| author_facet | Li, Weizhao Liu, Fanghui Shi, Lei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_18468 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization Li, Weizhao Liu, Fanghui Shi, Lei Machine Learning 41A25, 41A46, 68T07, 62G08 I.2.6; G.1.2; G.3 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. |
| title | Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization |
| topic | Machine Learning 41A25, 41A46, 68T07, 62G08 I.2.6; G.1.2; G.3 |
| url | https://arxiv.org/abs/2605.18468 |