Stein-Log-Sobolev inequalities for the continuous Stein variational gradient descent method
数学专题报告
报告题目(Title)：Stein-Log-Sobolev inequalities for the continuous Stein variational gradient descent method
报告人(Speaker)：José A. Carrillo (University of Oxford)
地点(Place)：ZOOM ID: 954 7274 3525 PWD：174291
时间(Time)：2025年10月23日（周四）16:00-17:00
邀请人(Inviter)：熊金钢
报告摘要
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 \(H^{-1}(\mathbb{R}^n)\) and \(H^{1}(\mathbb{R}^n)\), 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.
* This PDE seminar is co-organized with Tianling Jin at HKUST and Juncheng Wei at CUHK. See the seminar webpage: https://www.math.hkust.edu.hk/~tianlingjin/PDEseminar.html
