Understanding Stochastic Natural Gradient Variational Inference

Kaiwen Wu, Jacob R. Gardner
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:53398-53421, 2024.

Abstract

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 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 (a.k.a. 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-wu24f, title = {Understanding Stochastic Natural Gradient Variational Inference}, author = {Wu, Kaiwen and Gardner, Jacob R.}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {53398--53421}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/wu24f/wu24f.pdf}, url = {https://proceedings.mlr.press/v235/wu24f.html}, abstract = {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 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 (a.k.a. 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.} }
Endnote
%0 Conference Paper %T Understanding Stochastic Natural Gradient Variational Inference %A Kaiwen Wu %A Jacob R. Gardner %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-wu24f %I PMLR %P 53398--53421 %U https://proceedings.mlr.press/v235/wu24f.html %V 235 %X 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 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 (a.k.a. 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.
APA
Wu, K. & Gardner, J.R.. (2024). Understanding Stochastic Natural Gradient Variational Inference. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:53398-53421 Available from https://proceedings.mlr.press/v235/wu24f.html.

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