  The Black Box Variational Inference (Ranganath et al. (2014)) algorithm
provides a universal method for Variational Inference, but taking advantage of
special properties of the approximation family or of the target can improve the
convergence speed significantly. For example, if the approximation family is a
transformation family, such as a Gaussian, then switching to the
reparameterization gradient (Kingma and Welling (2014)) often yields a major
reduction in gradient variance. Ultimately, reducing the variance can reduce
the computational cost and yield better approximations.
  We present a new method to extend the reparameterization trick to more
general exponential families including the Wishart, Gamma, and Student
distributions. Variational Inference with Numerical Derivatives (VIND)
approximates the gradient with numerical derivatives and reduces its variance
using a tight coupling of the approximation family. The resulting algorithm is
simple to implement and can profit from widely known couplings. Our experiments
confirm that VIND effectively decreases the gradient variance and therefore
improves the posterior approximation in relevant cases. It thus provides an
efficient yet simple Variational Inference method for computing non-Gaussian
approximations.
