Bayesian inference problems require sampling or approximating high-dimensional probability distributions. The focus of this paper is on the recently introduced Stein variational gradient descent methodology, a class of algorithms that rely on iterated steepest descent steps with respect to a reproducing kernel Hilbert space norm. This construction leads to interacting particle systems, the mean-field limit of which is a gradient flow on the space of probability distributions equipped with a certain geometrical structure. We leverage this viewpoint to shed some light on the convergence properties of the algorithm, in particular addressing the problem of choosing a suitable positive definite kernel function. Our analysis leads us to considering certain nondifferentiable kernels with adjusted tails. We demonstrate significant performance gains of these in various numerical experiments.

翻译：贝叶斯推断问题需要采样或近似高维概率分布。本文聚焦于近期提出的Stein变分梯度下降方法，这是一类依赖迭代最速下降步（基于再生核希尔伯特空间范数）的算法。该构造形成了相互作用粒子系统，其平均场极限是在具有特定几何结构的概率分布空间上的梯度流。我们利用此视角阐明算法的收敛性质，特别针对选择合适正定核函数的问题展开研究。通过分析，我们提出需考虑具有调整尾部的非可微核函数。数值实验表明，所提出的核函数在性能上具有显著提升。