  For training an encoder network to perform amortized variational inference,
the Kullback-Leibler (KL) divergence from the exact posterior to its
approximation, known as the inclusive or forward KL, is an increasingly popular
choice of variational objective due to the mass-covering property of its
minimizer. However, minimizing this objective is challenging. A popular
existing approach, Reweighted Wake-Sleep (RWS), suffers from heavily biased
gradients and a circular pathology that results in highly concentrated
variational distributions. As an alternative, we propose SMC-Wake, a procedure
for fitting an amortized variational approximation that uses
likelihood-tempered sequential Monte Carlo samplers to estimate the gradient of
the inclusive KL divergence. We propose three gradient estimators, all of which
are asymptotically unbiased in the number of iterations and two of which are
strongly consistent. Our method interleaves stochastic gradient updates, SMC
samplers, and iterative improvement to an estimate of the normalizing constant
to reduce bias from self-normalization. In experiments with both simulated and
real datasets, SMC-Wake fits variational distributions that approximate the
posterior more accurately than existing methods.
