Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: The paper discusses stable deep Markov models (DMM), which can be used to model stochastic dynamical systems, restricting to the case where state transition probabilities are given by normal distributions with mean and covariance matrices parameterized by two deep neural nets.
One of the main contributions of this paper is to study sufficient conditions under which such DMMs are stable (in the mean-square sense), the second is to provide a list of practical (re)parameterization of the DNNs that satisfy the sufficient conditions. 
The paper concludes with a series of small scale numerical experiments aimed at showing the possible behaviours of the class of proposed models.

main_review: Clarity:
I found the paper very well written (found only one typo in Eq (7), missing t=0) and generally easy to follow.  
The work is well motivated; I appreciated that the proofs of the main results are in the paper, and  I found section 3.4., particularly informative and well organized, linking theoretical findings to practical design principles.
A few comments regarding clarity:
- In the proof of lemma 1, in eq (4), it is not clear whether the bias terms are included or not, especially since in line 95 there is written $z_{l+1} = A_l h_l$ whereas I would have expected to see also the bias in $z$.
- In theorems 4 and 5, since the statements are about mean-square stability, perhaps it would be better to state in the hypotheses that K<1 and $c^A c^{\Lambda}<1$ and then in the proof (or in a remark after those) relax these conditions referring to eq (11).
- (Question) It is not clear to me how the parametric stability constraints (also used in sec 4.3) relate to the hypotheses of the theorems. 
- The experimental section is the only section that I found rather difficult to follow, this is because: (i) there is no description, however short, of the experimental settings, and (ii) I think the discussion could be a bit more focused, better accompanying the reader on what the should look at. For (i), I would recommend making a bit of space for short descriptions of the setting in the main paper.

Originality: 
I find the contributions of this work overall novel, although not surprising, as they are in a way adaptations of known facts that have been studied in other circumstances (as per related section and also see added references below, which could be added to the related work). I think however the results are not obvious, and hence they deserve publication. In section 3.4, the paragraph which I found most original (and interesting) is the last one, which probably deserves some more discussion. 

Significance:
I believe the theoretical results and especially the design principles will be useful in the ML community and beyond; this paper may become a handy reference for DMM designers.

Quality:
The mathematical treatment is clear and precise, and the results look correct to me. 
The experimental section could be improved, by describing the experimental settings (as in the clarity section) and by including some quantitative comparative results about optimizing with various parameterizations (ideally this should be addressed in the rebuttal).
Related literature is discussed well enough, but perhaps the authors could consider including also works with deterministic models e.g. [1] and [2] (in particular, in [2] there is a discussion about equilibrium models and stability thorough projections which could complement the discussion of section 4.3.)


References:
[1] Miller, J. and Hardt, M.  Stable recurrent models. ICLR, 2019.
[2] Grazzi, Riccardo, et al. "On the iteration complexity of hypergradient computation." International Conference on Machine Learning. PMLR, 2020.

____
Post-rebuttal

Thanks for your reply. I am keeping my score and recommend acceptance.
limitations_and_societal_impact: Addressed
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper analyses stochastic stability of Deep Markov Models (DMMs) based on the contraction of probabilistic maps modeled by Deep Neural Networks (DNNs). To this end, DMMs are modeled using latent space representations of a dynamical system, and Gaussian distribution is used in modeling the transition and emission dynamics through DNN representation. With this modeling assumption, the authors perform stochastic stability analysis of DMM focused on bounding the norm of the model parameters of DNN. They state sufficient conditions for achieving stability and outline techniques for doing so. Finally, their claim is substantiated through small-scale experimentation of DMMs.
main_review: The paper analyses the stability of Deep Markov Models (DMMs) and provides sufficient conditions for stochastic stability of DMM. The authors perform mean-squared based stability analysis by using Deep Neural Networks (DNNs) to parameterize the probabilistic state transitions. In doing so, deterministic stability analysis is used in a probabilistic setting.

Their proof is based on modeling DMMs as a dynamical system with latent space variables and using the property that a dynamical system is said to be mean-square stable if its first and second moment converge with time. This leads to equivalent stability conditions outlined in Theorem 4 and Corollary 5 relying on contractive mapping of the DNN (modeling the DMM). These conditions are sufficient for the convergence of a DMM to a stable equilibrium. The proof is simple and naturally flows from existing literature dealing with probabilistic transitions in a dynamical system. 

In my opinion, the main crux of the paper lies in using the sufficient conditions (in Theorem 4 and Corollary 5) to outline the methods for designing stable DMMs by enforcing the contractivity of the weight matrices through different matrix factorizations. More importantly, the discussions on how some design considerations can have a negative impact on the expressivity of DMMs and can be ameliorated using parametric stability constraints have value in real-life applications. Going along with this, the numerical case studies in Section 4 shows a nice intuitive illustration of the ideas that have been theoretically proven earlier.

While the paper has potential in motivating design of (mean)-stable DMMs for real-life applications, there is little discussion connecting how much validity mean-stability analysis has in real-life applications. Also, there could have been more literature review and basics of spectral analysis of DNNs. This is not very clear at the moment. 

In spite of the strong points, my main concern about the paper remains about the applicability of the design choices (Section 3.4) with respect to large-scale dynamical systems. It would have been better to discuss the limitations with more reference to the literature and state the possible use cases where this kind of design might not be feasible.

Minor comments: The paper requires proofreading as there are typos, repeated abbreviations, and grammatical errors.

limitations_and_societal_impact: While the authors have done a good job in outlining a good theoretical framework for their work, the practical implications are somewhat lacking. From a practical point of view, it would have been good if they would have shown how their findings could be applied to a DMM in real-life. My main concern is that in large-scale systems where it might be computationally challenging to apply the design principles that have been stated in the Section “Design of Stable Deep Markov Models”. It would have been better if there was a section titled “Threats to Validity” for discussing the above issue.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper analyzes deep Markov models from a theoretical perspective. More specifically, it derives conditions on the construction/parameters of models which allow for a specific notion of stochastic stability. This is accomplished by first proving that a multilayer perceptron can be equivalently represented as a pointwise affine map (PWA). It is then proven that achieving specific properties on the norms of these PWAs leads to global stability, and that bounding the norms of the layers in neural networks leads to achieving these desired properties within the corresponding PWAs. There is a discussion on how one can achieve these properties when training a neural network, followed by empirical analysis demonstrating the connections between the properties of DMMs and their stability.

main_review: ### Originality: 
I am unaware of prior work which analyzes DMMs in this manner. Connections to prior work are made clear.

### Quality: 
As far as I could tell, the analysis done in this paper is sound. The conclusions regarding the analysis, and suggestions are made/analyzed regarding how to construct/train networks that achieve the required properties. The required assumptions are made very clear, and the authors are open and honest about the limitations of the current analysis.

### Clarity: 
The paper is clearly written and straightforward to follow. All important properties are defined, and the proofs are organized in a logical order. The empirical analysis is well-visualized, and the points being made are conveyed well by the plots.

### Significance: 
Assessing the properties of deep neural networks and their use in a variety of modeling frameworks is an open field, and understanding the theoretical properties of networks and how to achieve desired configurations can help lead to better performing models. The analysis presented here is another step in this direction, and thus I imagine many will find it interesting.

### Miscellaneous Comments
Equation 7 needs to be rewritten slightly - for example, $P(\mathbf{y}_0|\mathbf{x}_0)$ appears twice on the right side the way it is currently written.

# POST-REBUTTAL UPDATE
Thanks for the responses. After reading other reviews and the responses to each of them, I am maintaining the original score I gave. I think this paper makes a valuable contribution and should be accepted.

limitations_and_societal_impact: The limitations are presented clearly and thoroughly. I agree with the authors that this work is basic science and does not have any direct connections to societal impact.
review_title: null

Meta-review: This paper analyzes the stability properties of _deep Markov models_ (DMMs), which are nonlinear dynamical systems that are parameterized by deep neural networks (basically, replacing the transition/emission functions of an HMM with neural nets). The form of stability used in this paper is _stochastic stability_, which essentially means that the first and second moments of a dynamical process asymptotically converge. (Note that this is different from the notion of _algorithmic stability_ used in learning theory.) The paper provides sufficient conditions under which a DMM is stochastically stable, and then analyzes how various architectural considerations (activation, weights, depth, etc.) affect stability. A numerical study rounds out the picture.

The reviews were unanimously positive, with 7s (Accept) across the board. The theory is sound; the work feels complete; and the paper is well written and easy to follow, with minor typos. None of the reviewers were all that confident in their assessments (perhaps this is a niche subject for the NeurIPS audience), but all agreed that it's definitely a solid paper.

In my opinion, the main weakness (echoed by Reviewer fyK8) is that not much discussion is devoted to motivating stability. Why is stochastic stability important in applications of dynamical systems? The introduction ignores this question and starts from the assumption that stability is important and, therefore, worth studying. As far as I can tell, the only discussion of why stability is important comes up at the end, in the Broader Impact statement, which only says, "Stability is the major concern of many safety-critical systems. ... [T]he methods presented in this paper have the potential for practical impact in many real-world applications such as unmanned autonomous vehicles, robotics, or process control applications." OK, but why? What would happen if an _unstable_ system were deployed in an autonomous vehicle? How specifically does stochastic stability ensure safety in an autonomous vehicle? I can kind of fill in the blanks myself, but I'd like the paper to be more explicit about how this highly theoretical study has practical impact.

One other thing I'll note is that the term "stability," in the context of ML Theory, typically implies something different from stochastic stability (which comes from Control Theory). Readers familiar with _algorithmic stability_ (see Bousquet & Elisseeff, JMLR 2002) may be confused or misled by the title/abstract. I recommend adding something in the abstract and intro that (a) briefly describes stochastic stability and (b) makes it clear that it is not the same as algorithmic stability.
Decision: Accept (Poster)