Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space
This work addresses a key challenge in statistics and machine learning for efficiently approximating complex distributions, though it appears incremental as it builds on existing variational inference methods with improved convergence results.
The paper tackles the problem of approximating a target distribution using Gaussian variational inference by developing the Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm, which achieves state-of-the-art convergence guarantees for log-smooth and log-concave distributions and provides the first convergence guarantees for first-order stationary solutions when the distribution is only log-smooth.
Variational inference (VI) seeks to approximate a target distribution $π$ by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates $π$ by minimizing the Kullback-Leibler (KL) divergence to $π$ 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 $π$ is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when $π$ is only log-smooth.