Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization
Authors: Weizhao Li, Fanghui Liu, Lei Shi
Summary
arXiv:2605. 18468v1 Announce Type: new 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.
Relevance
Read next because Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization overlaps with clean result "Language-mismatch LoRA SFT on Qwen2.5-7B leaks the trained completion language into bystander directives the model was never trained on, absent under same-language SFT (LOW confidence)", clean result "Only continuous soft prefixes hit both EM axes at once on Qwen-2.5-7B-Instruct: discrete prompt searches split between the alignment objective and the distributional objective, and both discretizations of the soft prefix collapse (MODERATE confidence)", clean result "The marker is a representational handle, not a behavioural one — sharing it between a villain persona and the assistant transfers no misalignment (HIGH confidence)". Matching terms: under, alpha, rate, control, factor. Source: arxiv stat.ML (Machine Learning).
Abstract
arXiv:2605.18468v1 Announce Type: new 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 $\widetilde{\mathcal{F}}_{p,\tau_d,s}$, $1\le p\le2$, we establish approximation bounds for shallow networks using spherical harmonic analysis. In particular, when the parameter measure is the uniform measure $\tau_d$ 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 $1\le p<2$ 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.