Proximal Gradient Algorithms for Gaussian Variational Inference: Optimization in the Bures–Wasserstein Space
thesis
My MEng thesis. Translates the machinery of Euclidean optimization to the Bures-Wasserstein space to develop state-of-the-art algorithms for Gaussian variational inference.
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. Additionally, in the setting where the potential admits a representation as the average of many smooth component functionals, we develop and analyze a variance-reduced extension to (Stochastic) FB–GVI with improved complexity guarantees.
Links
See my thesis on MIT DSpace (2023).
The work done in this thesis was published at ICML 2023 (2023).
References
Diao, Michael Ziyang. 2023. “Proximal Gradient Algorithms for Gaussian Variational Inference: Optimization in the Bures–Wasserstein Space.” Master’s thesis, Massachusetts Institute of Technology.
Diao, Michael Ziyang, Krishna Balasubramanian, Sinho Chewi, and Adil Salim. 2023. “Forward-Backward Gaussian Variational Inference via JKO in the Bures-Wasserstein Space.” In Proceedings of the 40th International Conference on Machine Learning, 202:7960–91. Proceedings of Machine Learning Research. PMLR. https://proceedings.mlr.press/v202/diao23a.html.