  Efficient low-variance gradient estimation enabled by the reparameterization
trick (RT) has been essential to the success of variational autoencoders.
Doubly-reparameterized gradients (DReGs) improve on the RT for multi-sample
variational bounds by applying reparameterization a second time for an
additional reduction in variance. Here, we develop two generalizations of the
DReGs estimator and show that they can be used to train conditional and
hierarchical VAEs on image modelling tasks more effectively. We first extend
the estimator to hierarchical models with several stochastic layers by showing
how to treat additional score function terms due to the hierarchical
variational posterior. We then generalize DReGs to score functions of arbitrary
distributions instead of just those of the sampling distribution, which makes
the estimator applicable to the parameters of the prior in addition to those of
the posterior.
