Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space
Autor: | Jiang, Yiheng, Chewi, Sinho, Pooladian, Aram-Alexandre |
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Rok vydání: | 2023 |
Předmět: | |
Druh dokumentu: | Working Paper |
Popis: | 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 $\pi$. Comment: 47 pages |
Databáze: | arXiv |
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