importance-sampling estimate for the gradient that often only needs to evaluate the computationally cheaper part to provide the estimate. Skipping the computationally costly evaluation of the gradient of the model itself as often as possible lead to a practical speedup that is independent of other improvements provided by more advanced optimization algorithms [@svrg; @saga]. Our method relies on the inverse transformation being unique and efficient to compute. This might not be the case for complex structured approximations or approximations parameterized by neural networks; we leave more efficient extensions for such cases as future work.

We demonstrated the core idea in creating a more efficient stochastic gradient descent algorithm for both reparameterization [@dsvi; @advi] and score function [@bbvi] estimates used for variational inference. In addition, we formulated a theoretically justified variant of stochastic average gradients [@sage] applicable for variational inference. The idea, however, extends well beyond these special cases. For example, the rejection sampling variational inference [@naesseth] can be readily combined with our importance sampling strategy and is expected to result in a similar speedup.

Our main focus was in practically applicable algorithms, with much of the theoretical analysis left for future work. Two particular directions are immediately apparent: (a) The 