Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912402716491776 |
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| author | Jiang, Yiheng Chewi, Sinho Pooladian, Aram-Alexandre |
| author_facet | Jiang, Yiheng Chewi, Sinho Pooladian, Aram-Alexandre |
| contents | 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 $π$ over $\mathbb{R}^d$ by a product measure $π^\star$. When $π$ is strongly log-concave and log-smooth, we provide (1) approximation rates certifying that $π^\star$ is close to the minimizer $π^\star_\diamond$ of the KL divergence over a \emph{polyhedral} set $\mathcal{P}_\diamond$, and (2) an algorithm for minimizing $\text{KL}(\cdot\|π)$ over $\mathcal{P}_\diamond$ based on accelerated gradient descent over $\R^d$. As a byproduct of our analysis, we obtain the first end-to-end analysis for gradient-based algorithms for MFVI. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02849 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space Jiang, Yiheng Chewi, Sinho Pooladian, Aram-Alexandre Statistics Theory Machine Learning Optimization and Control 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 $π$ over $\mathbb{R}^d$ by a product measure $π^\star$. When $π$ is strongly log-concave and log-smooth, we provide (1) approximation rates certifying that $π^\star$ is close to the minimizer $π^\star_\diamond$ of the KL divergence over a \emph{polyhedral} set $\mathcal{P}_\diamond$, and (2) an algorithm for minimizing $\text{KL}(\cdot\|π)$ over $\mathcal{P}_\diamond$ based on accelerated gradient descent over $\R^d$. As a byproduct of our analysis, we obtain the first end-to-end analysis for gradient-based algorithms for MFVI. |
| title | Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space |
| topic | Statistics Theory Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2312.02849 |