Abstract

We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the for malism of optimal transport. We use this perspective to compute (approximate) pathwise gradients for probability distributions not directly amenable to the reparameterization trick: Gamma, Beta, and Dirichlet. We further observe that when the reparameterization trick is applied to the Choleskyfactorized multivariate Normal distribution, the resulting gradients are suboptimal in the sense of optimal transport. We derive the optimal gradients and show that they have reduced variance in a Gaussian Process regression task. We demonstrate with a variety of synthetic experiments and stochastic variational inference tasks that our pat wise gradients are competitive with other methods.