Convergence of SVGD in KL divergence via approximate gradient flow

Masahiro Fujisawa, Futoshi Futami

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

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Keywords: Stein variational gradient descent, SVGD, gradient flow, approximate inference

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TL;DR: We show the sub-linear convergence of SVGD in Kullback–Leibler divergence based on a novel concept called $(\epsilon,\delta)$-approximate gradient flow.

Abstract: This study investigates the convergence of Stein variational gradient descent (SVGD), which is used to approximate a target distribution based on a gradient flow on the space of probability distributions. The existing studies mainly focus on the convergence in the kernel Stein discrepancy, which doesn't imply the weak convergence in many practical settings. To address this issue, we propose to introduce a novel analytical approach called $(\epsilon,\delta)$-approximate gradient flow, extending conventional concepts of approximation error for Wasserstein gradient. With this approach, we show the sub-linear convergence of SVGD in Kullback-Leibler divergence under the discrete-time and infinite particle settings. Finally, we validate our theoretical findings through several numerical experiments.

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Submission Number: 5426