respect to the parameters of the approximation consists of two parts. One part is the gradient of the model itself, evaluated at parameter values drawn from the approximation, whereas the other part is the gradient of the transformation used in the reparameterization estimate. We show that the gradient required for optimization can be computed for the newly updated approximation without re-computing the first part, which is computationally heavier. Instead, we can re-use existing computation by appropriately modifying and re-weighting the available terms.

We show how to formulate this idea in a justified manner, by constructing an importance sampling estimate for the gradient. Importance sampling is typically used for cases where one cannot sample from the distribution of interest but instead has to resort to sampling from a related proposal distribution. In our case we could sample from the distribution of interest – the current approximation – but choose not to, since by using an earlier approximation as a proposal we can avoid costly computation. The idea is conceptually similar to the way @ep reuses samples from previous iterations in expectation propagation.

Since the advances in our case are related to the computation of the gradient itself, the idea can readily 