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
Copyright Notice
The documents contained in these directories are included by the contributing authors as a means to ensure timely dissemination of scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author’s copyright. These works may not be reposted without the explicit permission of the copyright holder.