Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space

• URL: http://arxiv.org/abs/2312.02849v2
• Date: Sun, 9 Jun 2024 01:19:10 GMT
• Title: Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space
• Authors: Yiheng Jiang, Sinho Chewi, Aram-Alexandre Pooladian,
• Abstract summary: We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which seeks to approximate a distribution $pi$ over $mathbbRd$ by a product measure $pistar$.
• Score: 10.292118864147097
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which seeks to approximate a distribution $\pi$ over $\mathbb{R}^d$ by a product measure $\pi^\star$. When $\pi$ is strongly log-concave and log-smooth, we provide (1) approximation rates certifying that $\pi^\star$ is close to the minimizer $\pi^\star_\diamond$ of the KL divergence over a \emph{polyhedral} set $\mathcal{P}_\diamond$, and (2) an algorithm for minimizing $\text{KL}(\cdot\|\pi)$ over $\mathcal{P}_\diamond$ with accelerated complexity $O(\sqrt \kappa \log(\kappa d/\varepsilon^2))$, where $\kappa$ is the condition number of $\pi$.

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