Convergence guarantees for kernel-based quadrature rules in misspecified settings

By Motonobu Kanagawa, Bharath K. Sriperumbudur, Kenji Fukumizu. 2016

57 citations recorded by the source corpus.

Kernel-based quadrature rules are becoming important in machine learning and statistics, as they achieve super-$\sqrt{n}$ convergence rates in numerical integration, and thus provide alternatives to Monte Carlo integration in challenging settings where integrands are expensive to evaluate or where integrands are high dimensional. These rules are based on the assumption that the integrand has a certain degree of smoothness, which is expressed as that the integrand belongs to a certain reproducing kernel Hilbert space (RKHS). However, this assumption can be violated in practice (e.g., when the integrand is a black box function), and no general theory has been established for the convergence of kernel quadratures in such misspecified settings. Our contribution is in proving that kernel quadratures can be consistent even when the integrand does not belong to the assumed RKHS, i.e., when the integrand is less smooth than assumed. Specifically, we derive convergence rates that depend on the (unknown) lesser smoothness of the integrand, where the degree of smoothness is expressed via powers of RKHSs or via Sobolev spaces.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistical Learning Theory, Mathematical Optimization

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Kanagawa et al. (2016), for example, proposes a filtering method with KBR in the situation where the observation model must be estimated nonlinearly and nonparametrically with data.”
Consistent Kernel Mean Estimation for Functions of Random Variables (2016, methodology): “We now state Theorem 1 from Kanagawa et al. (2016), which serves as basis to our proof.”
Convergence rates for a class of estimators based on Stein’s method (2016, background): “This might proceed in a similar vein to the related work of [21,32].”
Bayesian Probabilistic Numerical Methods (2017, unknown): “See also Kanagawa et al. (2016); Karvonen and Särkkä (2017).”