  Variational inference approximates the posterior distribution of a
probabilistic model with a parameterized density by maximizing a lower bound
for the model evidence. Modern solutions fit a flexible approximation with
stochastic gradient descent, using Monte Carlo approximation for the gradients.
This enables variational inference for arbitrary differentiable probabilistic
models, and consequently makes variational inference feasible for probabilistic
programming languages. In this work we develop more efficient inference
algorithms for the task by considering importance sampling estimates for the
gradients. We show how the gradient with respect to the approximation
parameters can often be evaluated efficiently without needing to re-compute
gradients of the model itself, and then proceed to derive practical algorithms
that use importance sampled estimates to speed up computation.We present
importance sampled stochastic gradient descent that outperforms standard
stochastic gradient descent by a clear margin for a range of models, and
provide a justifiable variant of stochastic average gradients for variational
inference.
