Paper:
Variational inference (VI) is a popular approach for approximate Bayesian inference that is particularly promising for highly parameterized models such as deep neural networks. A key challenge of variational inference is to approximate the posterior over model parameters with a distribution that is simpler and tractable yet sufficiently expressive. In this work, we propose a method for training highly flexible variational distributions by starting with a coarse approximation and iteratively refining it. Each refinement step makes cheap, local adjustments and only requires optimization of simple variational families. We demonstrate theoretically that our method always improves a bound on the approximation (the Evidence Lower BOund) and observe this empirically across a variety of benchmark tasks. In experiments, our method consistently outperforms recent variational inference methods for deep learning in terms of log-likelihood and the ELBO. We see that the gains are further amplified on larger scale models, significantly outperforming standard VI and deep ensembles on residual networks on CIFAR10.

Review:
The paper proposes to improve standard variational inference by increasing the flexibility of the variational posterior by introducing a finite set of auxiliary variables. Motivated by the limited expressivity of mean field variational inference the author suggests to iteratively refine a ‘proposal’ posterior by conditioning on a sequence of auxiliary variables with decreasing variance. The key requirement to set the variance of the auxiliary variables such that the integrating over them leaves the original model unchanged. As noted by the authors this is a variant of auxiliary variables introduced by Barber & Agakov. The motivation and theoretical sections seems sound and the experimental results are encouraging, however maybe not completely supporting the claim of new ‘state of the art’ on uncertainty prediction. 
Overall i find the motivation and theoretical contribution interesting. However I do not find the experimental section completely comprehensive why I currently think the paper is borderline acceptance. 
Comments
1) The computational demand using the method seems quite large by adding O(NumSamples * NumAuxiliary) additional computational cost on top of the standard VI. Here each factor M is quite large e.g. 200 epochs for CIFAR10 (if i understand the algorithm correctly?)
2) For the UCI experiments the comparison is only made against DeepEnsembels or other VI methods, however to the best of my knowledge MCMC methods are superior in this setting given the small dataset size? 
3) The results on CIFAR10 do seem to demonstrate that the proposed method is superior to DeepEnsembles and standard VI in one particular setting where VI is only performed over a small subset of layers in a ResNet (why doesn’t it work for when doing VI on all the parameters?). However generally looking at the best obtained results of ~86% acc this is quite far from current best probabilistic models (see e.g. Heek2019 that gets 94% acc). Some of this can probably be attributed to differences in data-augmentation and model architecture however in general it makes it very hard to compare with other methods when the baselines are not competitive. 
Minor Comments:
 In relation to comment 3) above I think you should reword the sentence “It sets a new state-of-the-art in uncertainty estimation at ResNet scale on CIFAR10” in the conclusion.
“In  order  to  get  independent  samples  from  the  variational  posterior,we have to repeat the iterative refinement for each ensemble member”: Does this imply that if we want M samples we first have to optimize using the standard VI and then to M optimizations to get q_k(w)?
How sensitive is the method to sequence of variances for a?
[Heek2019]: Bayesian Inference for Large Scale Image Classification