The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information, also called the squared Stein discrepancy, as a duality pairing between $H^{-1}(\mathbb{R}^d)$ and $H^{1}(\mathbb{R}^d)$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.

翻译：Stein变分梯度下降方法是统计学中一种变分推断方法，近年来受到广泛关注。该方法通过引入核函数的非局部相互作用，为目标分布提供确定性近似。尽管备受关注，连续方法的指数收敛速率一直是一个悬而未决的问题，这源于建立相关Stein-对数Sobolev不等式的困难。本文证明，该不等式在每个空间维度下均成立，且适用于所有傅里叶变换在无穷远处具有二次衰减、在局部远离零和无穷大的核函数。此外，我们构造了相关偏微分方程的弱解，并证明其以指数速率收敛至平衡态。我们方法的主要创新点在于将Stein-Fisher信息（亦称平方Stein差异）解释为$H^{-1}(\mathbb{R}^d)$与$H^{1}(\mathbb{R}^d)$空间之间的对偶配对，从而得以运用傅里叶变换。我们还提供了若干不满足Stein-对数Sobolev不等式的核函数示例，部分证明了假设条件的必要性。