Improved Stein Variational Gradient Descent with Importance Weights

Stein Variational Gradient Descent (SVGD) is a popular sampling algorithm used in various machine learning tasks. It is well known that 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 SVGD via the introduction of importance weights, which leads to a new method for which we coin the name $\beta$-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 $\beta$-SVGD, which covers the descent lemma for the population limit SVGD when $\beta\to 0$. We also illustrate the advantages of $\beta$-SVGD over SVGD by simple experiments.