Geometric Rates of Convergence for Kernel-based Sampling Algorithms
This work addresses convergence analysis for kernel-based sampling algorithms, providing theoretical insights with practical validation, but it appears incremental as it builds on existing methods.
The paper investigates the convergence rates of weighted kernel herding and sequential Bayesian quadrature for integral estimation, establishing near-geometric convergence under specific conditions and showing comparable performance to optimal algorithms, supported by empirical tests on simulated and real-world data.
The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geometric rate of convergence for target measures that are nearly atomic. Furthermore, we show these algorithms perform comparably to the theoretical best possible sampling algorithm under the maximum mean discrepancy. An analysis is also conducted in a distributed setting. Our theoretical developments are supported by empirical observations on simulated data as well as a real world application.