Stochastic variance-reduced Gaussian variational inference on the Bures-Wasserstein manifold

• URL: http://arxiv.org/abs/2410.02490v1
• Date: Thu, 3 Oct 2024 13:55:49 GMT
• Title: Stochastic variance-reduced Gaussian variational inference on the Bures-Wasserstein manifold
• Authors: Hoang Phuc Hau Luu, Hanlin Yu, Bernardo Williams, Marcelo Hartmann, Arto Klami,
• Abstract summary: We propose a novel variance-reduced estimator for optimization in the Bures-Wasserstein space. We show that this estimator has a smaller variance than the Monte-Carlo estimator in scenarios of interest. We demonstrate that the proposed estimator gains order-of-magnitude improvements over the previous Bures-Wasserstein methods.
• Score: 4.229248343585333
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Optimization in the Bures-Wasserstein space has been gaining popularity in the machine learning community since it draws connections between variational inference and Wasserstein gradient flows. The variational inference objective function of Kullback-Leibler divergence can be written as the sum of the negative entropy and the potential energy, making forward-backward Euler the method of choice. Notably, the backward step admits a closed-form solution in this case, facilitating the practicality of the scheme. However, the forward step is no longer exact since the Bures-Wasserstein gradient of the potential energy involves "intractable" expectations. Recent approaches propose using the Monte Carlo method -- in practice a single-sample estimator -- to approximate these terms, resulting in high variance and poor performance. We propose a novel variance-reduced estimator based on the principle of control variates. We theoretically show that this estimator has a smaller variance than the Monte-Carlo estimator in scenarios of interest. We also prove that variance reduction helps improve the optimization bounds of the current analysis. We demonstrate that the proposed estimator gains order-of-magnitude improvements over the previous Bures-Wasserstein methods.

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