Stein变分梯度下降的标度极限：平均场区
Scaling limit of the Stein variational gradient descent: the mean field regime
摘要：
我们研究了$\mathbf{R}^d$中的相互作用粒子系统。Liu和D. Wang，NIPS 2016]，一种确定性算法，用于从具有未知归一化的给定概率密度进行采样。我们证明了在大粒子极限下，粒子系统的经验测度收敛于非局部非线性偏微分方程的解。证明了极限偏微分方程解的整体存在性、唯一性和正则性。最后证明了方程的解在长时间内收敛于唯一不变解。
英文摘要：
We study an interacting particle system in $\mathbf{R}^d$ motivated by Stein variational gradient descent [Q. Liu and D. Wang, NIPS 2016], a deterministic algorithm for sampling from a given probability density with unknown normalization. We prove that in the large particle limit the empirical measure of the particle system converges to a solution of a non-local and nonlinear PDE. We also prove global existence, uniqueness and regularity of the solution to the limiting PDE. Finally, we prove that the solution to the PDE converges to the unique invariant solution in long time limit.
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