__ Summary and Contributions__: The authors propose using doubly stochastic variational inference to compute the relationship between the input and output signals on the vertices of a graph under a Gaussian process prior.

__ Strengths__: The method appears to be novel & potentially quite powerful.
It allows 'side information' provided by the graph structure to be introduced into the model.
Good selection of empirical experiments.
Interesting proof for the variance scale.
Finally it may improve on the rather fragile optimisation that complex models in this field can suffer from.

__ Weaknesses__: One thing that worries me a lot is that all the evaluations are about the posterior mean: How well does it handle the posterior (co)variance? Does it properly reach across the relevant part of the domain or is the posterior distribution very concentrated? VI suffers from this anyway, I imagine that a graph structure like this might compound the problem.

__ Correctness__: To my knowledge this seems valid.
The empirical assessment seems correct from what we are told.
I'm not sure about the proof for the variance.
Do the numbers in Table 2 & 3 have units?
I'm not sure what cross-validation split was conducted for the experiments?

__ Clarity__: I think it would benefit from a simple example and/or some clarity about what the task is (in words, before getting to the problem statement's maths? What are the input/output for example wasn't immediately clear.

__ Relation to Prior Work__: I found the background to be thorough (to the best of my knowledge) & puts the paper in context.

__ Reproducibility__: Yes

__ Additional Feedback__: It's unclear if the inducing points were optimised?
Also how long (how many iterations) did these models need? (fig 2 shows this for one example).
ln 218 - "In total, we generate" might be better than "We totally..."
==After discussions and reading the author feedback==
My assessment was pulled in two directions:
At first, having read the other reviews & feedback, I was going to downgrade my assessment as I've read a bit more background and think the application of DGP over graphs is a relatively straightforward step in terms of the surrounding literature. But not enough to make me 'reject' but not feel strongly about the accept.
But then I read the author's point that they did look at uncertainty, reread that section and feel that actually my criticism was largely unwarranted on that count.
So my overall score will remain unchanged (although I'm not going to increase my confidence score, as I don't feel I'm an expert in this area).

__ Summary and Contributions__: This paper introduces a deep GP model that makes predictions on nodes of a graph. It takes advantage of the graph structure to create a purposefully constrained model to help make predictions.
The paper provides an empirical evaluation on several noisy datasets with graph structure, with impressive

__ Strengths__: The paper develops the model based on the well-developed theoretical principles of variational inference in GP models. By applying these techniques to a sensible model, the authors achieved good performance. The model is generally applicable to many regression on tasks to grids, so it is likely to be useful for many researchers and practitioners in the community.

__ Weaknesses__: - Limitations of theorem 1
The theorem on the variance of the predictions (theorem 1) is a bit odd, and it is unclear how much value it really adds. The theorem effectively tries to prove that the predictive variance of the approximate posterior is smaller for the graph-aware model than for a plain DGP. This depends only on the covariance of the inducing point distribution q(u). This covariance is learned by optimising the ELBO, and so it is hard to prove anything about it short of simply running the training procedure. Assumption 2 doesn't provide much additional clarity, as it effectively places a constraint on \tilde{S} as a function of S. This assumption also cannot be verified without running the algorithm.
In short, it is not clear that Theorem 1 makes any statements about the algorithm that cannot be made simply by empirically keeping track of the predictive variance while running the algorithm.
In addition, the empirical evaluation isn't making a fair comparison between the two models. The empirical evaluation claims that the graph-aware DGP converges faster than the normal DGP. This is not at all obvious from figure 2. If anything, it seems (as expected) that the normal DGP simply converges to a much lower value, which is explained by it not being a suitable model.
- Test on simpler models
I wonder to what extent the non-linearity of the GPs are really needed, given the small number of inducing points that are used. For the small datasets M = 5. For such small numbers of inducing points, GPs often behave like linear models. It would be good to include an experiment using a linear kernel to see how much the non-linearity of the GP really helps.
In addition, it would be good to include a short discussion on how the number of inducing points was chosen. Also, how were the inducing points initialized?

__ Correctness__: The methodology and empirical evaluation are broadly correct.

__ Clarity__: The paper is broadly clearly written, with a few spots of lack of clarity.
- §2.2: The description of a DGP as a probabilistic model is very short, and could do with more details.
- §3: The connectivity structure of what nodes in the graph a GP depends on is the single key contribution of this paper. It would be good to have an illustration showing a graph, and what nodes are used as input to the next-layer GP.
- §3: The recursive sampling scheme is presented in an unclear way, to the point of being incorrect. A better way to present this would be simply to explicitly describe the recursive procedure for obtaining a sample from the function values at the last layer. Equation 5 does not correctly represent the correlations of function evaluations between layers. A confusing aspect of Salimbeni's DGP formulation (which is confusingly presented in the orignal paper), is that there is a distinction between the approximate GPs for each layer, which are independent, and the function evaluations between layers, which are dependent. See e.g. Thang Bui's thesis (http://mlg.eng.cam.ac.uk/thang/docs/papers/thesis-thang.pdf) page 94.

__ Relation to Prior Work__: Prior work is well discussed. 2 comments:
- It would be helpful for a reader to have a discussion on why this graph-based structure was chosen. Were there any approaches in the literature that directly inspired the choices that were made? The structure of the proposed method does look a lot like those from the Graph Convolutional Network literature. Although these papers are cited, if these were an inspiration, it would be good to have a more explicit discussion.
- Around line 72 there is a discussion of sparse methods helping with the complexity burden of GPs. [1] provides proofs for regression that these approximations are very accurate with only small numbers of inducing points.
[1] http://proceedings.mlr.press/v97/burt19a.html

__ Reproducibility__: Yes

__ Additional Feedback__: Overall, the paper presents a sensible approach to an important problem, with impressive empirical results. While I recommend accepting, I hope that my issues will be addressed in the rebuttal so I can raise my score.

__ Summary and Contributions__: The paper modifies a Deep Gaussian Process to operate on a graph (input and outputs on a Graph). Inference proceeds by a modification of the doubly stochastic VI framework by Salimbeni & Deisenroth, which is modified to consider the neighbourhood of a vertex/node when computing the marginal at layer l leading to a recursive sampling scheme. There are a few synthetic, small and large real world datasets being used based on prior work [20,45] from Graph NN and shallow GP approaches on Graphs. The paper offers a main theorem (similar to [20]) that the DGPG has a reduced sampling variance (compared to a vanilla DGP) because it exploits the Markov property/graph structure.

__ Strengths__: The paper does examine multiple datasets and compares to a shallow GP [20] on a small dataset and with Graph ANNs on the large dataset. The approach seems technically correct and it is an improvement over ignoring the graph structure and naively running a DGP. The standard ARD kernel in this graph setting naturally allows (given the parents from the graph structure) to learn relative importance between these parents and also relative importance between features in each node.

__ Weaknesses__: I think the novelty is somewhat limited and summarised in the recursive sampling scheme; It is just a DGP model with the doubly stochastic inference framework slightly modified to only consider parents of a vertex in the previous layer during updates/sampling. The main Theorem is somewhat trivial as it is against a standard DGP that does not use any of the graph information hence it has full dependency (and hence larger resulting sampling variance) whereas by slightly restricting the dependency structure (via a 1st order Markov assumption on parents as in DGPG) you can reduce that variance. It is somewhat obvious that this is the case.
All comparisons on real data are against Graph ANN approaches and [20] on the small dataset; It would help to include probabilistic deep learning models such as a Deep Gaussian Markov Random Fields (ICML 2020, arxived since February) or a Bayesian ANN approach or a Convolutional Gaussian Process or [18] or or [19] Graph kernels, that would better exploit the graph structure.

__ Correctness__: As far as I examined its a correct adaptation of the DGP model and doubly stochastic inference to work with Graphs but it is somewhat trivial. Empirical results don't compare against other DGP/Deep GMRF/Bayesian graph ANN/Convolutional GP methods.

__ Clarity__: Yes

__ Relation to Prior Work__: Somewhat yes, although Deep GRMFs [ref above], standard GMRFs [Rue & Held] and Convolutional GPs, (and [18, 19]) are very relevant to such graph structures, as is the general area of Graph Kernels [Journal of Machine Learning Research 11 (2010) 1201-1242].

__ Reproducibility__: Yes

__ Additional Feedback__: Summarising from above, I think the paper is technically correct but lacks in novelty and offers a minor adjustment of a standard DGP and corresponding doubly stochastic inference method to encode the graph structure. The novelty and contribution offered is contained in upper half of page 5 and its a straightforward modification to exploit parts of the graph structure within a DGP and resulting ELBO. Unfortunately I don't have any good ideas on how you would improve the novelty further, but perhaps looking at related work and the connections to GMRFs and Graph kernels would help to at least explore connections/analogies and to improve the experimental comparisson. Also, I don't think the ICASSP paper [20] (which again and similarly to this paper only compares with a standard GP) is the state of the art for this problem in GP land, but looking at [18] and the aforementioned literature on Graph Kernels, and (Deep) GMRFs is.
In the synthetic experiment I have specific questions for the authors:
You are training a DGPG with 1 layer, is the neighbourhood used in the DGPG defined to be the same 7 parents that you have used to construct the output data of each node? If so, isn't it obvious that a DGP which is closer to the data generating process by construction (encodes that neighbourhood dependency), would perform better and have less variance from another DGP which tries to model a larger dependency structure (that does not exist by construction)? Isn't the conclusion hence trivial?
Finally, the main focus of comparison on the probabilistic side is between a general vanilla DGP that is completely ignorant of the graph structure, and the proposed Graph DGP version. If I have a fully connected graph, is the GDGP equivalent to a DGP and hence just a special case of it?
In rebuttal I invite the authors to convince me to raise my score by focusing on two key points mentioned above in various sections and questions:
1) The novelty of the approach and the strength of the contribution in relation to the underlying DGP model and inference scheme
2) The experimental comparison is against state of the art GP/probabilistic approaches for Graphs as I am not convinced that running [20] on the small dataset is convincing.
Minor:
- various spelling/grammar mistakes like L35 natural->nature, L147 edged-edges, etc; check again
- ARD - I thought initials are Automatic Relevance Determination, not "Discovery"
- Line 89: "In [11]...(DGP)" Thats wrong, the correct reference is [7] where the DGP was first introduced and in the accompanying PhD thesis by Damianou.
- Line 66-67: "..assuming both the prior and the likelihood .. are Gaussian" Thats also not true, a GP is a prior over functions and IF the likelihood is Gaussian you have GPR which is closed form. GPs don't assume the likelihood is Gaussian necessarily.
- Why you need the word "Stochastic" in the method/title? a GP is a stochastic process eitherway so it does not add any new info? Are you trying to highlight that inference in the model follows the doubly stochastic framework? Its not needed.
- All the metrics in experiments are focusing on accuracy (MAE/MAPE/RMSE) and none capture uncertainty quantification which would be the main benefit of a Bayesian approach like a (D)GP.
## Post-rebuttal
I have read the author's response and appreciate the responses and additional study on corrupting graph connectivity. I do appreciate that modifying a GMRF to compare against in their examples might not be as trivial as I thought. However, I still think this work is a special case of a DGP - as the authors agree that if graph is fully connected you get back the DGP model and associated doubly stochastic inference - with limited novelty on model or inference beyond just restricting connectivity/dependency in the model. More importantly the paper would benefit with a comparison against a proper probabilistic kernel-based graph approach. You are proposing a probabilistic kernel-method for graphs without comparing against the state of the art in this specific area e.g. DGP & GP with proper graph kernels (not just linear/rbf/matern). Given the limited technical/model/inference novelty, the empirical comparisons should be at least complete and compelling for NeurIPS. Graph kernels where introduced a decade or more ago. I will not change my score of 5 and if the paper gets accepted I would strongly encourage inclusion of comparison against GPs & DGPs with graph kernels at least which should be trivial to do.

__ Summary and Contributions__: Stochastic Deep Gaussian Processes over Graphs
The manuscript proposes a variant of the stochastic deep Gaussian processes (DGP) which uses the graph structure (encoded in input and output signals) to define the conditional independence structure of the DGP priors (in eq.3, F^{l,k} is independent from the unconnected nodes in layer l-1 given F^{l-1,pa(k)}). The proposed model can be learned by the recursive sampling method which is modified from that of the original doubly stochastic DGP paper [11].

__ Strengths__: Significance and relevance: Using the graph structure in DGP is a quite interesting idea, and the proposed method outperforms the previous GP-based work (which also is based on the graph structure) and shows comparative performances to a state-of-the-art-method (DCRNN) for the nontrivial prediction problem. I think that the manuscript shows a good application of DGP to real-world problems.

__ Weaknesses__: Novelty & contributions
1. My main concern is that the novelty and contributions of the proposed method do not seem to be significant. It seems to me that the proposed method is a direct application of the stochastic DGP [11] to a certain problem domain where input and output signals have the graph structure. All the inference steps appear to be straightforward from the previous work [11].

__ Correctness__: I think the method is technically correct.

__ Clarity__: I think that certain points are unclear and need to be modified.
- In line 148, I think that the term ‘parents’ can be a little bit misleading. This term refers to just connected nodes (neighbors). The term ‘parents’ is more commonly used for directed graphs (e.g., trees or directly acyclic graphs) than for undirected graphs. I think that “neighbors” of a given node would be enough (and more appropriate).
I think that the proposed method mainly utilizes the locality information from the graph to make prediction (in eq. 3, functions values at the node k in the layer l are dependent on those at the parental nodes of the node k in the l-1 layer). Thus, it does not make a sense to me that the parent node set pa(v_k) does not include v_k itself (based on Figure 1, parents nodes pa(v2), v_1 and v_3, do not include v_2). For example, a single layer DGP (L=1), the first term of the GP prior in eq. 3 becomes p(F^{1,k}| U^{1,k}; X^{pa(k)}, Z^{pa(k)}), and it is more reasonable that the GP prior defined on the node k should involve the features associated with the node k.
- In the experimental result section, it is not clear about what the datasets look like. A table containing the basic statistics (number of samples, dimension, the average number of neighbors of a node in the graph, and so on) of the datasets should have been included. In addition, it is not clear about the sentence in the lines 232-233. It seems that all the small (and large) datasests are not graph datasets because the authors mentioned in the line 234 (and 253) that they built an adjacent matrix for each dataset. Is it correct? I think that the authors should include further explanations about the datasets and how to build graphs from them (e.g., how to define the distance between two nodes?).

__ Relation to Prior Work__: I think that the authors should more clearly explain the previous work based GP [20] and how the proposed method differs from it (because this work is one of main baseline methods)

__ Reproducibility__: Yes

__ Additional Feedback__: Typo: in the line 208: DGPG should be DGP?