Paper:
Variational inference (VI) plays an essential role in approximate Bayesian inference due to its computational efficiency and general applicability. Crucial to the performance of VI is the selection of the divergence measure in the optimization objective, as it affects the properties of the approximate posterior significantly. In this paper, we propose a meta-learning algorithm to learn (i) the divergence measure suited for the task of interest to automate the design of the VI method; and (ii) initialization of the variational parameters, which reduces the number of VI optimization steps drastically. We demonstrate the learned divergence outperforms the hand-designed divergence on Gaussian mixture distribution approximation, Bayesian neural network regression, and partial variational autoencoder based recommender systems.

Review:
Summary of the paper:
This paper proposes to meta-learn a parametric divergence measure for variational inference (VI). Specifically, the paper focuses on alpha-divergence and f-divergence, aims at choosing a good alpha or f for a particular task leveraging the experience learned from previous tasks. The meta-learned divergence is applied to approximate a mixture of two Gaussians, regress sinusoid function with Bayesian neural networks, and learn recommender systems with partial VAE. 
Summary of my opinion:
I am leaning towards reject this paper because 
1) I'm not convinced by their motivation -- "this line of work has also shown that the optimal divergence can vary depending on tasks". There are many key factors to achieve good performance in VI, such as a good likelihood model, a good variational posterior, a good optimizer and so on. The divergence itself, in my opinion, is less significant. If you really care about the flexibility, the Wasserstein distance may be a better choice.
2) Choosing the alpha or f is a hyper-parameter search problem to me. What you need to do is to prepare a validation set and select the best hyper-parameters based on the performance there. It is impractical to collect M tasks and search for hyper-parameters according to some meta-losses. 
3) The experimental part is weak -- two toy problems plus an unusual recommendation system problem is not the typical way that people choose to evaluate VI methods. I suggest to evaluate on standard VAE/BNN benchmarks. 
Major comments: 
1. From eq.(4) and eq.(8), it seems one has to compute the value of the density function p(theta, D) in order to compute the gradients, which is obviously not possible for BNNs and VAEs. More details on how to perform VI with alpha-divergence and f-divergence should be covered. 
2. Can you let the meta-loss to be equal to D_eta? Meta-learning algorithms often optimize the same loss function in the inner and outer level. I couldn't see the point why you want to learn a f-divergence while setting the meta-loss to be D_0.5. 
3. Do you have a validation set for the Bayesian optimization (BO) baseline or do you use cross-validation? The details of this baseline should be elaborated. In fact, it is possible to choose the best alpha for each task using BO. Should this be compared?
4. What does VB mean in Section 4.2? If VB uses KL rather than alpha-divergence or f-divergence, the outperformance may only suggest that KL is insufficient there. 
5. Have you considered input convex neural networks (Amos et al. 2016) for implementing f? 
Minor comments:
1. The proofs of Prop 1 & 2 are missing.   
2. "However, using this KL divergence for VI has been criticized for under-estimating the uncertainty..." Any reference to this?
3. Figure 1: alpha is used before it is defined. 
4. "Other existing definitions of alpha-divergences have dis-continuous gradients at the alpha values corresponding to KL divergences" -- missing reference.
5. "When D_0.5 is in use as the meta-loss, the corresponding log f*'' is analytical" -- could you elaborate on this? 
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After rebuttal:
I would like to thank the authors for their extremely detailed responses! The paper is now greatly improved. 
I would like to increase my score from 3 to 5, however, there is no such option this year. 
The reason why I am still hesitated to accept this paper is mainly because I am not sure how useful learning the divergence to VI is. For unsupervised learning, people use GAN or VQ-VAE to obtain realistic samples, but which is not attributed to better divergence measures. Plus, what we refer to "better" depends on the supplied meta-loss, which may not be better at all. Another unsatisfactory point of this paper in my opinion is that it only demonstrated results on toy datasets and MNIST. Although the proposed method is valid for alpha-divergence and f-divergence, I suggest the authors to find a real application which demonstrates the merit of the idea in practice.