Variance Reduction Properties of the Reparameterization Trick

Ming Xu, Matias Quiroz, Scott A Sisson, Robert Kohn

Abstract: The reparameterization trick is widely used in variational inference as it yields more ac- curate estimates of the gradient of the varia- tional objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relatively little research exploring why the reparameterization trick is so effective. We explore this under the idealized assumptions that the variational approximation is a mean- field Gaussian density and that the log of the joint density of the model parameters and the data is a quadratic function that depends on the variational mean. From this, we show that the marginal variances of the reparame- terization gradient estimator are smaller than those of the score function gradient estima- tor. We apply the result of our idealized anal- ysis to real-world examples