Pathwise Derivatives Beyond the Reparameterization Trick
This work addresses a bottleneck in variational inference for researchers and practitioners by enabling more efficient gradient estimation for specific distributions, though it is incremental in extending existing methods.
The paper tackled the problem of computing pathwise gradients for probability distributions not amenable to the reparameterization trick, such as Gamma, Beta, and Dirichlet, by leveraging optimal transport theory, and showed that their derived gradients reduce variance in Gaussian Process regression tasks.
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.