Understanding Stochastic Natural Gradient Variational Inference
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909215904235520 |
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| author | Wu, Kaiwen Gardner, Jacob R. |
| author_facet | Wu, Kaiwen Gardner, Jacob R. |
| contents | Stochastic natural gradient variational inference (NGVI) is a popular posterior inference method with applications in various probabilistic models. Despite its wide usage, little is known about the non-asymptotic convergence rate in the \emph{stochastic} setting. We aim to lessen this gap and provide a better understanding. For conjugate likelihoods, we prove the first $\mathcal{O}(\frac{1}{T})$ non-asymptotic convergence rate of stochastic NGVI. The complexity is no worse than stochastic gradient descent (\aka black-box variational inference) and the rate likely has better constant dependency that leads to faster convergence in practice. For non-conjugate likelihoods, we show that stochastic NGVI with the canonical parameterization implicitly optimizes a non-convex objective. Thus, a global convergence rate of $\mathcal{O}(\frac{1}{T})$ is unlikely without some significant new understanding of optimizing the ELBO using natural gradients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_01870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Understanding Stochastic Natural Gradient Variational Inference Wu, Kaiwen Gardner, Jacob R. Machine Learning Stochastic natural gradient variational inference (NGVI) is a popular posterior inference method with applications in various probabilistic models. Despite its wide usage, little is known about the non-asymptotic convergence rate in the \emph{stochastic} setting. We aim to lessen this gap and provide a better understanding. For conjugate likelihoods, we prove the first $\mathcal{O}(\frac{1}{T})$ non-asymptotic convergence rate of stochastic NGVI. The complexity is no worse than stochastic gradient descent (\aka black-box variational inference) and the rate likely has better constant dependency that leads to faster convergence in practice. For non-conjugate likelihoods, we show that stochastic NGVI with the canonical parameterization implicitly optimizes a non-convex objective. Thus, a global convergence rate of $\mathcal{O}(\frac{1}{T})$ is unlikely without some significant new understanding of optimizing the ELBO using natural gradients. |
| title | Understanding Stochastic Natural Gradient Variational Inference |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2406.01870 |