Quasi-Monte Carlo Variational Inference

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Alexander Buchholz, Florian Wenzel, Stephan Mandt ;
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:667-676, 2018.

Abstract

Many machine learning problems involve MonteCarlo gradient estimators. As a prominent example, we focus on Monte Carlo variational inference (MCVI) in this paper. The performanceof MCVI crucially depends on the variance of itsstochastic 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 deterministicsequence of samples of length N. This sequencecovers the underlying random variable space moreevenly than i.i.d. draws, reducing the variance ofthe 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 forMonte Carlo objectives, where we operate witha constant learning rate and increase the numberof QMC samples per iteration. We prove that thisway, our algorithm can converge asymptoticallyat a faster rate than SGD . We furthermore providetheoretical guarantees on qmc for Monte Carloobjectives that go beyond MCVI , and support ourfindings by several experiments on large-scaledata sets from various domains.

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