Paper:
Bayesian inference offers a theoretically grounded and general way to train neural networks and can potentially give calibrated uncertainty. It is, however, challenging to specify a meaningful and tractable prior over the network parameters. More crucially, many existing inference methods assume mean-field approximate posteriors, ignoring interactions between parameters in high-dimensional weight space. To this end, this paper introduces two innovations: (i) a Gaussian process-based hierarchical model for the network parameters based on recently introduced unit embeddings that can flexibly encode weight structures, and (ii) input-dependent contextual variables for the weight prior that can provide convenient ways to regularize the function space being modeled by the network through the use of kernels. Furthermore, we develop an efficient structured variational inference scheme that alleviates the need to perform inference in the weight space whilst retaining and learning non-trivial correlations between network parameters. We show these models provide desirable test-time uncertainty estimates, demonstrate cases of modeling inductive biases for neural networks with kernels and demonstrate competitive predictive performance of the proposed model and algorithm over alternative approaches on a range of classification and active learning tasks.

Review:
**Summary**: This paper proposes a hierarchical Bayesian approach to hyper-networks by placing a Gaussian process prior over the latent representation for each weight. A stochastic variational inference scheme is then proposed to infer the posterior over both the Gaussian process and the weights themselves. Experiments are performed on toy regression, classification, (edit: post rebuttal) and transfer learning tasks, as well as an uncertainty quantification experiment on MNIST.
post rebuttal (noticed recently): Many apologies for updating the review the day before the deadline; however, I recently remembered that Kronecker inference is often used in variational methods - particularly within the vein of literature of deep kernel learning. Indeed, structure exploiting SVI was proposed in Stochastic Variational Deep Kernel Learning, https://arxiv.org/pdf/1611.00336.pdf, and this method is currently the default in Gpytorch: https://github.com/cornellius-gp/gpytorch/tree/master/examples/08_Deep_Kernel_Learning .
Furthermore, Kronecker inference for non-Gaussian likelihoods for Laplace approximations was proposed back in 2015: http://proceedings.mlr.press/v37/flaxman15.pdf. 
I am not updating my score because it would be unfair; however, the record should be set somewhat straight here.
post rebuttal: Thank you for the many clarifications and detailed responses. I'm now satisfied with their many changes and and tend to accept this paper despite the experimental results being somewhat limited. I would really encourage the authors to fix the color schemes (please less black and more brighter colors) on their decision boundaries plots however. 
**tldr**: While I appreciate the concept of this paper, I tend to reject this paper because I find the experimental results to be on too small scale of datasets. Specifically, I would like to see either a larger scale problem being solved with this kind of approach or a tough to model applied problem that is solved with this approach.
**Originality**: As far as I can tell, this seems to be a novel approach to hyper-networks. Neural processes (Garnelo et al; 2018) propose a somewhat similar approach to training – with a latent process over some stored weight space. However, even that is quite distinct from the method proposed in this paper, and I tend to prefer this approach. 
**Quality**: I really appreciate the merging of neural network and Gaussian process methods; however, tragically, I do wonder if the proposed approach combines the worst of both worlds – the necessity of architecture search for neural networks with the choice of kernel function (as illustrated in Figure 5). 
If the method is truly kernel dependent, is it also architecture dependent? That is, is it robust to different settings of nonlinearities and depths?
Active learning experiment: While I appreciate the comparison here, it seems like here standard HMC should be trainable over well-designed priors on these architectures. So why not include a comparison instead of just MFVI?
**Significance**: Unfortunately, I think that the experiments section is just a bit too limited to warrant acceptance right now. This is despite the fact that I really do appreciate the thoroughness and thoughtfulness of the experiments as they are. 
Specifically, in Section 6.2 why is the metaGP prior only applied to the last layer of the network? If as I suspect, it is due to the complexity and difficulty of inference, that makes the method doubly tough to use in practice. With that being said, to only have experiments on the last layer implies that one should compare to Bayesian logistic regression and linear regression on the last layer of neural networks (e.g Perrone et al, 2018 and Riquelme et al, 2018). Experiments with other methods that combine Gaussian processes with representations on the final layer (e.g. Wilson et al, 2015) are also probably worth running. 
Figure 4 is a very well-done experiment, if a bit tough to read. I’d suggest that the out of distribution examples get their own figure, with the in distribution examples going into the appendix. I’d also suggest computing the expected calibration error (Naeini et al, 2015) for in and out of distribution examples on the test sets for both MNIST and K-MNIST in order to have quantitative results on the entire test set. 
To recommend acceptance, I’d really have to see experiments on either a CIFAR sized dataset for classification or a larger scale regression experiment. A larger dataset on either transfer learning (after all you do have a meta-representation over functions that the NN can learn), a larger active learning experiment, or semi-supervised learning. 
**Clarity**: Overall, the paper is well-written and mostly easy to follow. The meat of the paper is found in Section 4, which I found a bit difficult to follow. 
(edit: post rebuttal. This concern is somewhat resolved due to the field not being well developed in this area, although it is a useful place to possibly extend the method in the future.) My primary concern here is that the prior ends up becoming Kronecker structured (after Eq. 7), so it isn’t clear to my why dense matrices and dense variational bounds have to be derived in this setting. Can one not follow the lead of the Gaussian process literature (e.g. Saatci 2012, Wilson & Nickisch, 2015) to exploit the Kronecker structure here to make computation of the log likelihoods fast?
(edit: post rebuttal. This concern is somewhat resolved.) As a result, it’s not immediately clear to me why a diagonal approximation (Eq. 10) is even necessary? 
Furthermore, this may be a setting where iterative methods (e.g conjugate gradients and Lanczos decompositions as in Pleiss et al, 2018) for the predictive means and variances may shine and be fast.
I do agree that the approximation in Figure 2 does seem to be relatively accurate, although I would ask the authors to compute a relative error for that plot if possible. Additionally, what is the strange high off diagonal correlations in the marginal covariances?
(edit: post rebuttal. Thank you for the clarifications here.) Finally, I was a bit confused by the effect of adding the input dependent kernel in Section 3; this seems to make the weights much more complicated to model – now each data point has its own set of weights and therefore, we might have to store considerably more weight matrices over time. Could the authors perform a set of experiments showing the necessity of this kernel matrix in the rebuttal?
**Minor Comments**: 
-	Above Eq. 9, “splitted” should be split.
-	Figure 3: could the data points be plotted in a brighter fashion? On a dark background, they are quick tough to see. Additionally, what is the difference between the two levels of classification plots?
References:
Naeini, et al. Obtaining Well Calibrated Probabilities by Bayesian Binning, AAAI, 2015. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4410090/
Perrone, V, et al. Scalable Hyperparameter Transfer Learning, NeurIPS, 2018. http://papers.nips.cc/paper/7917-scalable-hyperparameter-transfer-learning
Pleiss, G, et al. Constant Time Predictive Distributions for Gaussian Processes, ICML, 2018. https://arxiv.org/abs/1803.06058
Riquelme, C, Tucker, G, Snoek, J. Deep Bayesian Bandits Showdown, ICLR, 2018. https://arxiv.org/abs/1802.09127
Saatci, Y. Scalable Inference for Structured Gaussian process models, PhD Thesis, U. of Cambridge, 2011. http://mlg.eng.cam.ac.uk/pub/pdf/Saa11.pdf
Wilson, AG and Nickisch, H. Kernel Interpolation for Scalable Structured Gaussian Processes, ICML, 2015. http://proceedings.mlr.press/v37/wilson15.pdf
Wilson, AG, et al. Deep Kernel Learning, AISTATS, 2015. https://arxiv.org/abs/1511.02222