研究浅层ReLU^s网络在L^p和Sobolev空间中的逼近与泛化，给出最优误差率。
Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization
- 用球谐分析和谱Barron空间推导逼近界
- 在子高斯噪声下达到最小最大泛化率，对数项可忽略
- 适用于高维函数逼近与理论分析的科研人员
本文研究浅层ReLU^s网络σ_s(t)=max{0,t}^s的逼近能力及其在ℓ_1路径范数控制下的泛化行为。对于L^p型积分空间̃𝐹_{p,τ_d,s}（1≤p≤2），通过球谐分析获得逼近界：当τ_d为均匀测度且1≤p<2时，逼近率为𝑂(𝑚^{−𝑝(2𝑠+2𝑑+1)−2𝑑⁄2𝑑𝑝})（1≤p≤p^*）和𝑂(𝑚^{−𝑝(4𝑠+3𝑑−1)−2𝑑+2⁄4𝑑𝑝})（p^*<p<2），其中p^*= (2d+2)/(d+3)。通过嵌入到谱Barron空间，得到Sobolev空间𝑊^{α,𝑝}（1≤p<2）的逼近界。在子高斯噪声下的非参数回归中，路径范数正则化的浅层ReLU^s网络在̑𝐵_𝑠上达到最小最大率𝑂(𝑛^{−(𝑑+2𝑠+1)⁄2𝑑+2𝑠+1} log 𝑛)，在𝑊^{𝛼,∞}上为𝑂(𝑛^{−2𝛼⁄2𝛼+𝑑} log 𝑛)，下界匹配至对数因子。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。