Statistical Learning Guarantees for Group-Invariant Barron Functions
This provides a rigorous theoretical foundation for encoding group-invariant structures in neural networks, benefiting researchers in machine learning theory and applications involving symmetric data.
The paper tackles the generalization error of group-invariant neural networks in the Barron framework, showing that group invariance can improve approximation accuracy with a factor δ_{G,Γ,σ} ≤ 1, while estimation error remains unaffected, leading to significant statistical advantages for symmetric functions.
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.