Black-Box Variational Inference as a Parametric Approximation to Langevin Dynamics

Matthew Hoffman, Yian Ma
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:4324-4341, 2020.

Abstract

Variational inference (VI) and Markov chain Monte Carlo (MCMC) are approximate posterior inference algorithms that are often said to have complementary strengths, with VI being fast but biased and MCMC being slower but asymptotically unbiased. In this paper, we analyze gradient-based MCMC and VI procedures and find theoretical and empirical evidence that these procedures are not as different as one might think. In particular, a close examination of the Fokker-Planck equation that governs the Langevin dynamics (LD) MCMC procedure reveals that LD implicitly follows a gradient flow that corresponds to a variational inference procedure based on optimizing a nonparametric normalizing flow. This result suggests that the transient bias of LD (due to the Markov chain not having burned in) may track that of VI (due to the optimizer not having converged), up to differences due to VI’s asymptotic bias and parameterization. Empirically, we find that the transient biases of these algorithms (and their momentum-accelerated counterparts) do evolve similarly. This suggests that practitioners with a limited time budget may get more accurate results by running an MCMC procedure (even if it’s far from burned in) than a VI procedure, as long as the variance of the MCMC estimator can be dealt with (e.g., by running many parallel chains).

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-hoffman20a, title = {Black-Box Variational Inference as a Parametric Approximation to {L}angevin Dynamics}, author = {Hoffman, Matthew and Ma, Yian}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {4324--4341}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/hoffman20a/hoffman20a.pdf}, url = {https://proceedings.mlr.press/v119/hoffman20a.html}, abstract = {Variational inference (VI) and Markov chain Monte Carlo (MCMC) are approximate posterior inference algorithms that are often said to have complementary strengths, with VI being fast but biased and MCMC being slower but asymptotically unbiased. In this paper, we analyze gradient-based MCMC and VI procedures and find theoretical and empirical evidence that these procedures are not as different as one might think. In particular, a close examination of the Fokker-Planck equation that governs the Langevin dynamics (LD) MCMC procedure reveals that LD implicitly follows a gradient flow that corresponds to a variational inference procedure based on optimizing a nonparametric normalizing flow. This result suggests that the transient bias of LD (due to the Markov chain not having burned in) may track that of VI (due to the optimizer not having converged), up to differences due to VI’s asymptotic bias and parameterization. Empirically, we find that the transient biases of these algorithms (and their momentum-accelerated counterparts) do evolve similarly. This suggests that practitioners with a limited time budget may get more accurate results by running an MCMC procedure (even if it’s far from burned in) than a VI procedure, as long as the variance of the MCMC estimator can be dealt with (e.g., by running many parallel chains).} }
Endnote
%0 Conference Paper %T Black-Box Variational Inference as a Parametric Approximation to Langevin Dynamics %A Matthew Hoffman %A Yian Ma %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-hoffman20a %I PMLR %P 4324--4341 %U https://proceedings.mlr.press/v119/hoffman20a.html %V 119 %X Variational inference (VI) and Markov chain Monte Carlo (MCMC) are approximate posterior inference algorithms that are often said to have complementary strengths, with VI being fast but biased and MCMC being slower but asymptotically unbiased. In this paper, we analyze gradient-based MCMC and VI procedures and find theoretical and empirical evidence that these procedures are not as different as one might think. In particular, a close examination of the Fokker-Planck equation that governs the Langevin dynamics (LD) MCMC procedure reveals that LD implicitly follows a gradient flow that corresponds to a variational inference procedure based on optimizing a nonparametric normalizing flow. This result suggests that the transient bias of LD (due to the Markov chain not having burned in) may track that of VI (due to the optimizer not having converged), up to differences due to VI’s asymptotic bias and parameterization. Empirically, we find that the transient biases of these algorithms (and their momentum-accelerated counterparts) do evolve similarly. This suggests that practitioners with a limited time budget may get more accurate results by running an MCMC procedure (even if it’s far from burned in) than a VI procedure, as long as the variance of the MCMC estimator can be dealt with (e.g., by running many parallel chains).
APA
Hoffman, M. & Ma, Y.. (2020). Black-Box Variational Inference as a Parametric Approximation to Langevin Dynamics. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:4324-4341 Available from https://proceedings.mlr.press/v119/hoffman20a.html.

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