In this paper we propose and analyze a novel multilevel version of Stein variational gradient descent (SVGD). SVGD is a recent particle based variational inference method. For Bayesian inverse problems with computationally expensive likelihood evaluations, the method can become prohibitive as it requires to evolve a discrete dynamical system over many time steps, each of which requires likelihood evaluations at all particle locations. To address this, we introduce a multilevel variant that involves running several interacting particle dynamics in parallel corresponding to different approximation levels of the likelihood. By carefully tuning the number of particles at each level, we prove that a significant reduction in computational complexity can be achieved. As an application we provide a numerical experiment for a PDE driven inverse problem, which confirms the speed up suggested by our theoretical results.

翻译：本文提出并分析了一种新颖的多级斯坦因变分梯度下降（SVGD）方法。SVGD是一种基于粒子的近期变分推断方法。对于似然函数计算代价昂贵的贝叶斯逆问题，该方法可能因需要演化多个时间步的离散动力系统而变得不可行——每个时间步均需在所有粒子位置处计算似然函数。为解决此问题，我们引入了一种多级变体，该变体并行运行多个相互作用的粒子动力学过程，每个过程对应似然函数的不同近似层级。通过精心调整每个层级的粒子数量，我们证明可显著降低计算复杂度。作为应用，我们针对一个偏微分方程驱动的逆问题进行了数值实验，其结果证实了理论分析所揭示的加速效果。