Kernel herding belongs to a family of deterministic quadratures that seek to
minimize the worst-case integration error over a reproducing kernel Hilbert
space (RKHS). In spite of strong experimental support, it has revealed
difficult to prove that this worst-case error decreases at a faster rate than
the standard square root of the number of quadrature nodes, at least in the
usual case where the RKHS is infinite-dimensional. In this theoretical paper,
we study a joint probability distribution over quadrature nodes, whose support
tends to minimize the same worst-case error as kernel herding. We prove that it
does outperform i.i.d. Monte Carlo, in the sense of coming with a tighter
concentration inequality on the worst-case integration error. While not
improving the rate yet, this demonstrates that the mathematical tools of the
study of Gibbs measures can help understand to what extent kernel herding and
its variants improve on computationally cheaper methods. Moreover, we provide
early experimental evidence that a faster rate of convergence, though not
worst-case, is likely.

核心聚类属于一族确定性定积分方法之一，旨在通过最小化再生核希尔伯特空间（RKHS）上的最坏情况积分误差来实现。本文研究了一种关于定积分节点的联合概率分布，其支撑集趋向于最小化与核心聚类相同的最坏情况误差。我们证明其在最坏情况积分误差的集中不等式方面优于独立同分布的蒙特卡洛方法。虽然尚未改进速率，但这表明吉布斯测度研究的数学工具有助于理解核心聚类及其变体在计算便宜方法上的改进程度。此外，我们提供了早期的实验证据，表明收敛速度更快（虽然不是最坏情况），这很可能实现。

基于核方法的 Monte Carlo 与 Gibbs 测度：概率采样的保证

Monte Carlo with kernel-based Gibbs measures: Guarantees for  probabilistic herding

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.

本文研究了采样方法和贝叶斯积分估计在选取样本时的损失函数关系，进而证明了带权的 kernel herding 是所有带权采样方法中最优的方法，同时也得出了贝叶斯积分估计的经验误差上界。