  Black box variational inference (BBVI) with reparameterization gradients
triggered the exploration of divergence measures other than the
Kullback-Leibler (KL) divergence, such as alpha divergences. In this paper, we
view BBVI with generalized divergences as a form of estimating the marginal
likelihood via biased importance sampling. The choice of divergence determines
a bias-variance trade-off between the tightness of a bound on the marginal
likelihood (low bias) and the variance of its gradient estimators. Drawing on
variational perturbation theory of statistical physics, we use these insights
to construct a family of new variational bounds. Enumerated by an odd integer
order $K$, this family captures the standard KL bound for $K=1$, and converges
to the exact marginal likelihood as $K\to\infty$. Compared to
alpha-divergences, our reparameterization gradients have a lower variance. We
show in experiments on Gaussian Processes and Variational Autoencoders that the
new bounds are more mass covering, and that the resulting posterior covariances
are closer to the true posterior and lead to higher likelihoods on held-out
data.
