将对称结构融入神经网络，可显著提升对称函数的泛化性能。
Statistical Learning Guarantees for Group-Invariant Barron Functions
- 基于Barron框架分析对称神经网络的逼近误差
- 对称性使逼近率提升，关键因子δ_{G,Γ,σ}可低至|G|^{-1}
- 适合学习具有群对称性的目标函数的研究者
我们研究了在Barron框架下，群不变神经网络的泛化误差。分析表明，引入群不变结构会带来一个依赖于群的因子δ_{G,Γ,σ} ≤ 1，该因子越小，逼近精度提升越明显。在估计方面，我们证明群不变类的Rademacher复杂度不超过非不变类，说明对称性不会增加估计误差。因此，当目标函数本身具有群对称性时，整体泛化误差可显著改善。我们进一步提供了典型例子：在有利情况下，δ_{G,Γ,σ} ≈ |G|^{-1}；在不利情况下，δ_{G,Γ,σ} ≈ 1。结果为在神经网络中编码群不变结构提供了严格的理论依据，表明其对对称目标函数具有明确的统计优势。
原文摘要 · Abstract (English)
We investigate the generalization error of group-invariant neural networks within the Barron framework. Our analysis shows that incorporating group-invariant structures introduces a group-dependent factor $δ_{G,Γ,σ} \le 1$ into the approximation rate. When this factor is small, group invariance yields substantial improvements in approximation accuracy. On the estimation side, we establish that the Rademacher complexity of the group-invariant class is no larger than that of the non-invariant counterpart, implying that the estimation error remains unaffected by the incorporation of symmetry. Consequently, the generalization error can improve significantly when learning functions with inherent group symmetries. We further provide illustrative examples demonstrating both favorable cases, where $δ_{G,Γ,σ}\approx |G|^{-1}$, and unfavorable ones, where $δ_{G,Γ,σ}\approx 1$. Overall, our results offer a rigorous theoretical foundation showing that encoding group-invariant structures in neural networks leads to clear statistical advantages for symmetric target functions.
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。