Improved Stein Variational Gradient Descent with Importance Weights

Stein Variational Gradient Descent~(\algname{SVGD}) is a popular sampling algorithm used in various machine learning tasks. It is well known that \algname{SVGD} arises from a discretization of the kernelized gradient flow of the Kullback-Leibler divergence $D_{KL}\left(\cdot\mid\pi\right)$, where $\pi$ is the target distribution. In this work, we propose to enhance \algname{SVGD} via the introduction of {\em importance weights}, which leads to a new method for which we coin the name \algname{$\beta$-SVGD}. In the continuous time and infinite particles regime, the time for this flow to converge to the equilibrium distribution $\pi$, quantified by the Stein Fisher information, depends on $\rho_0$ and $\pi$ very weakly. This is very different from the kernelized gradient flow of Kullback-Leibler divergence, whose time complexity depends on $D_{KL}\left(\rho_0\mid\pi\right)$. Under certain assumptions, we provide a descent lemma for the population limit \algname{$\beta$-SVGD}, which covers the descent lemma for the population limit \algname{SVGD} when $\beta\to 0$. We also illustrate the advantages of \algname{$\beta$-SVGD} over \algname{SVGD} by simple experiments.