On the Sampling Problem for Kernel Quadrature
This addresses a practical bottleneck in numerical integration for researchers and practitioners using Kernel Quadrature, though it is incremental as it builds on existing methods.
The paper tackles the problem of selecting optimal sampling distributions for Kernel Quadrature (Bayesian Monte Carlo), which significantly affects convergence rates but lacks closed-form solutions. It proposes an adaptive tempering and sequential Monte Carlo method, achieving up to 4 orders of magnitude reduction in integration error.
The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio $s/d$, where $s$ and $d$ encode the smoothness and dimension of the integrand. However, an empirical investigation reveals that the rate constant $C$ is highly sensitive to the distribution of the random points. In contrast to standard Monte Carlo integration, for which optimal importance sampling is well-understood, the sampling distribution that minimises $C$ for Kernel Quadrature does not admit a closed form. This paper argues that the practical choice of sampling distribution is an important open problem. One solution is considered; a novel automatic approach based on adaptive tempering and sequential Monte Carlo. Empirical results demonstrate a dramatic reduction in integration error of up to 4 orders of magnitude can be achieved with the proposed method.