Quasi-Monte Carlo Variational Inference

Alexander Buchholz, Florian Wenzel, Stephan Mandt
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:668-677, 2018.

Abstract

Many machine learning problems involve Monte Carlo gradient estimators. As a prominent example, we focus on Monte Carlo variational inference (MCVI) in this paper. The performance of MCVI crucially depends on the variance of its stochastic gradients. We propose variance reduction by means of Quasi-Monte Carlo (QMC) sampling. QMC replaces N i.i.d. samples from a uniform probability distribution by a deterministic sequence of samples of length N. This sequence covers the underlying random variable space more evenly than i.i.d. draws, reducing the variance of the gradient estimator. With our novel approach, both the score function and the reparameterization gradient estimators lead to much faster convergence. We also propose a new algorithm for Monte Carlo objectives, where we operate with a constant learning rate and increase the number of QMC samples per iteration. We prove that this way, our algorithm can converge asymptotically at a faster rate than SGD . We furthermore provide theoretical guarantees on qmc for Monte Carlo objectives that go beyond MCVI , and support our findings by several experiments on large-scale data sets from various domains.

Cite this Paper

BibTeX
@InProceedings{pmlr-v80-buchholz18a, title = {Quasi-{M}onte {C}arlo Variational Inference}, author = {Buchholz, Alexander and Wenzel, Florian and Mandt, Stephan}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {668--677}, year = {2018}, editor = {Dy, Jennifer and Krause, Andreas}, volume = {80}, series = {Proceedings of Machine Learning Research}, month = {10--15 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v80/buchholz18a/buchholz18a.pdf}, url = {https://proceedings.mlr.press/v80/buchholz18a.html}, abstract = {Many machine learning problems involve Monte Carlo gradient estimators. As a prominent example, we focus on Monte Carlo variational inference (MCVI) in this paper. The performance of MCVI crucially depends on the variance of its stochastic gradients. We propose variance reduction by means of Quasi-Monte Carlo (QMC) sampling. QMC replaces N i.i.d. samples from a uniform probability distribution by a deterministic sequence of samples of length N. This sequence covers the underlying random variable space more evenly than i.i.d. draws, reducing the variance of the gradient estimator. With our novel approach, both the score function and the reparameterization gradient estimators lead to much faster convergence. We also propose a new algorithm for Monte Carlo objectives, where we operate with a constant learning rate and increase the number of QMC samples per iteration. We prove that this way, our algorithm can converge asymptotically at a faster rate than SGD . We furthermore provide theoretical guarantees on qmc for Monte Carlo objectives that go beyond MCVI , and support our findings by several experiments on large-scale data sets from various domains.} }
Endnote
%0 Conference Paper %T Quasi-Monte Carlo Variational Inference %A Alexander Buchholz %A Florian Wenzel %A Stephan Mandt %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-buchholz18a %I PMLR %P 668--677 %U https://proceedings.mlr.press/v80/buchholz18a.html %V 80 %X Many machine learning problems involve Monte Carlo gradient estimators. As a prominent example, we focus on Monte Carlo variational inference (MCVI) in this paper. The performance of MCVI crucially depends on the variance of its stochastic gradients. We propose variance reduction by means of Quasi-Monte Carlo (QMC) sampling. QMC replaces N i.i.d. samples from a uniform probability distribution by a deterministic sequence of samples of length N. This sequence covers the underlying random variable space more evenly than i.i.d. draws, reducing the variance of the gradient estimator. With our novel approach, both the score function and the reparameterization gradient estimators lead to much faster convergence. We also propose a new algorithm for Monte Carlo objectives, where we operate with a constant learning rate and increase the number of QMC samples per iteration. We prove that this way, our algorithm can converge asymptotically at a faster rate than SGD . We furthermore provide theoretical guarantees on qmc for Monte Carlo objectives that go beyond MCVI , and support our findings by several experiments on large-scale data sets from various domains.
APA
Buchholz, A., Wenzel, F. & Mandt, S.. (2018). Quasi-Monte Carlo Variational Inference. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:668-677 Available from https://proceedings.mlr.press/v80/buchholz18a.html.

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