Perturbative Black Box Variational Inference
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 → ∞. 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.
Summary:  
The authors present a new variational objective for approximate Bayesian inference.  The variational objective is nicely framed as an interpolation between classic importance sampling and the traditional ELBO-based variational inference.  Properties of the variance of importance sampling estimator and ELBO estimators are studied and leveraged to create a better marginal likelihood bound with tractable variance properties.  The new bound is based on a low-degree polynomial of the log-importance weight (termed the interaction energy).  The traditional ELBO estimator is expressed as a first order polynomial in their more general framework. 
The authors then test out the idea on a Gaussian process regression problem and a Variational autoencoder. 
Quality:  I enjoyed this paper --- I thought the idea was original, the presentation is clear, and the experiments were convincing.  The paper appears to be technically correct, and the method itself appears to be effective. 
Clarity:  This paper was a pleasure to read.  Not only was this paper extremely clear and well written, the authors very nicely frame their work in the context of other current and previous research.
Originality:  While establishing new lower bounds on the marginal likelihood is a common subject at this point, the authors manage to approach this with originality. 
Impact:  I think this paper has the potential to be highly impactful --- the spectrum drawn from importance sampling to KLVI is an effective way of framing these ideas for future research.
Questions/Comments: 
  - Figure 1: should the legend text "KLVI: f(x) = 1 + log(x)" read "f(x) = log(x)" ?  I believe that would cohere with the bullet point on line 124. 
  - How does the variance of the marginal likelihood bound estimators relate to the variance of the gradients of those estimators wrt variational params?  KLVI reparameterization gradients can have some unintuitive irreducibility (Roeder et al, https://arxiv.org/abs/1703.09194); is this the case for PVI?