Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space

• URL: http://arxiv.org/abs/2304.05398v1
• Date: Mon, 10 Apr 2023 19:49:50 GMT
• Title: Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space
• Authors: Michael Diao, Krishnakumar Balasubramanian, Sinho Chewi, Adil Salim
• Abstract summary: Variational inference (VI) seeks to approximate a target distribution $pi$ by an element of a tractable family of distributions. We develop the Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when $pi$ is log-smooth and log-concave.
• Score: 19.19325201882727
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Variational inference (VI) seeks to approximate a target distribution $\pi$ by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates $\pi$ by minimizing the Kullback-Leibler (KL) divergence to $\pi$ over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when $\pi$ is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when $\pi$ is only log-smooth.

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