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We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the formalism 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 Cholesky-factorized 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 pathwise gradients are competitive with other methods.
Author Information
Martin Jankowiak (Uber AI Labs)
Fritz Obermeyer (Uber AI Labs)
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2018 Oral: Pathwise Derivatives Beyond the Reparameterization Trick »
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2019 Oral: Tensor Variable Elimination for Plated Factor Graphs »
Fritz Obermeyer · Elias Bingham · Martin Jankowiak · Neeraj Pradhan · Justin Chiu · Alexander Rush · Noah Goodman