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

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.