Optimally-Weighted Herding is Bayesian Quadrature
This work provides theoretical insights and improved performance for sample selection in probabilistic integration, which is incremental but relevant for machine learning and statistical applications.
The paper tackled the problem of selecting samples for summarizing probability distributions and estimating integrals, showing that kernel herding minimizes the same criterion as Bayesian quadrature's posterior variance and that sequential Bayesian quadrature outperforms other weighted herding methods with a convergence rate faster than O(1/N).
Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves performance superior to any other weighted herding method. We demonstrate empirically a rate of convergence faster than O(1/N). Our results also imply an upper bound on the empirical error of the Bayesian quadrature estimate.