Local KL Convergence Rate for Stein Variational Gradient Descent with Reweighted Kernel

Xunpeng Huang, Hanze Dong, Cong Fang

Keywords: SVGD, asymptotic analysis

Abstract: We study convergence properties of Stein Variational Gradient Descent (SVGD) algorithm for sampling from a non-normalized probabilistic distribution $p_*({x})\propto\exp(-f_*({x}))$. Compared with Kernelized Stein Discrepancy (KSD) convergence analyzed in previous literature, KL convergence as a more convincing criterion can better explain the effectiveness of SVGD in real-world applications. In the population limit, SVGD performs smoothed gradient descent with kernel integral operator. Notably, SVGD with smoothing kernels suffers from gradient vanishing in low-density areas, which makes the error term between smoothed gradient and Wasserstein gradient not controllable. In this context, we introduce a reweighted kernel to amplify the smoothed gradient in low-density areas, which leads to a bounded error term to Wasserstein gradient. When the $p_*({x})$ satisfies log-Sobolev inequality, we develop the convergence rate for SVGD in KL divergence with the reweighted kernel. Our analysis points out the defects of conventional smoothing kernels in SVGD and provides the convergence rate for SVGD in KL divergence.

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