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An infinitely wide model is a weighted integration $\int \varphi(x,v) d\mu(v)$ of feature maps. This model excels at handling an infinite number offeatures, and thus it has been adopted to the theoretical study of deeplearning. Kernel quadrature is a kernel-based numerical integration schemedeveloped for fast approximation of expectations $\int f(x) d p(x)$. In thisstudy, regarding the weight $\mu$ as a signed (or complex/vector-valued)distribution of parameters, we develop the general kernel quadrature (GKQ) forparameter distributions. The proposed method can achieve a fast approximationrate $O(e^{-p})$ with parameter number $p$, which is faster than thetraditional Barron's rate, and a fast estimation rate $\widetilde{O}(1/n)$ withsample size $n$. As a result, we have obtained a new norm-based complexitymeasure for infinitely wide models. Since the GKQ implicitly conducts theempirical risk minimization, we can understand that the complexity measure alsoreflects the generalization performance in the gradient learning setup.

Fast Approximation and Estimation Bounds of Kernel Quadrature for Infinitely Wide Models