__ Summary and Contributions__: This paper aims to develop a non-cooperative game setting whose aim is to simultaneously design signal priors and corresponding optimization methods, all within the context of a neural type architecture trainable by automatic differentiation.
The authors consider both gradient l1 (TV) an l2 (Laplacian) type regularization methods, as well as "standard" l1 and l2 methods, for image restoration tasks, and demonstrate the efficacy of their approaches via PSNR values; actual images are shown in the supplement.
UPDATE: The authors commented their framework is actually a bit broader.

__ Strengths__: The framework proposed is fairly flexible, and the results arising from it here are compelling.
The overall learning formulation being posed as a cooperative game seems to be novel.

__ Weaknesses__: The exposition was, in my opinion, a bit too "high level" throughout. For example, despite this being fundamentally an algorithmic paper that uses a standard prototype algorithm, it was correspondingly difficult to parse specifically what that algorithm was. While I appreciate the authors including source code, I personally would prefer that pseudocode for the algorithm be clearly spelled out in the paper too.
Along related lines, the exposition went back and forth between a more general, high level, abstract formulation of the overall problem, and a more specific image patch-based example. Maybe the "Tricks of the Trade" could be moved somewhere closer to the beginning of the paper where the more general framework was discussed.
UPDATE: The former comments weren't addressed by the authors.
In my opinion, PSNR values by themselves don't really provide an overly compelling justification for image-based methods. It would be much better to actually see the differences between the competing methods visually, in the main body of the paper.
UPDATE: The authors pledged to include some additional justification in a revision.
UPDATE: I still hold that this is a unique perspective worth exploring, but could/should be more clearly exposed.

__ Correctness__: There are no significant analytical claims here. The methodology seems correct at a high level, but more specific details are missing outside of the included simulation code.

__ Clarity__: Some restructuring, along the lines of the comments above, would improve the readability in my opinion.

__ Relation to Prior Work__: The paper was well positioned with respect to existing prior works.

__ Reproducibility__: Yes

__ Additional Feedback__: N/A

__ Summary and Contributions__: This contribution proposes a framework to learn specific priors for convex optimization problems. This framework is a generalization of the bi-level formulation for the unrolled optimization of [36]. Instead of averaging the effect of the regularization over a sample, a non cooperative convex game is used.
The authors demonstrate their framework on several classic priors: TV, l1, non-local TV,...
The paper also provides several tricks to train these special networks.
The advantages of this technique are twofolds. First the obtained models are more interpretable as the forward pass is equivalent to solve well-known convex optimization problem. Second, the amount of computation and parameters is much smaller than classical neural net. However, they compromise a bit in term of performance.
The experiment section is convincing.

__ Strengths__: * Relatively novel
* Technically accurate
* Meaningful contribution for Green AI and interpretability

__ Weaknesses__: * In several practical cases, the non-cooperative game can be solved by minimizing the sum of all the convex terms, i.e. the setting called a potential game. In this case we recover a classical convex optimization problem.
If I understand correctly, we then recover the classical setting or eq.1? It also seems to me that all priors and problems within the context of this paper are convex. (I am not sure about the dictionary learning case for this particular framework.)
If this is the case, the generalization to non-cooperative game seems a bit artificial to me. I might also be wrong. Could you please elaborate?
# Post review comment:
I am still not convinced by the author response and I am not sure that I fully understand this central point in the paper. Hence my score as well as my confidence score.

__ Correctness__: I believe it is. However, I have not checked the Appendix or the code.

__ Clarity__: I definitely feel the Neurips page limit. Overall, I believe the authors did a relatively good job. I would emphasize a bit more the difference between the non-cooperative game setup and eq. 1 as this part was not very clear to me.

__ Relation to Prior Work__: Overall yes. I found a bit confusing that sometimes the original (convex optimization) paper is cited although I would expect the unrolled neural network version of it. For example after eq 1.

__ Reproducibility__: Yes

__ Additional Feedback__: Do you need fewer samples? Maybe you should mention that this work goes in the direction of green AI?
# Post review edit:
Thank you for the additional experiment. I believe it is very insideful...
Line 69: Eq 1. you use the symbol \in and not =, is p>m?, is there multiple solutions?
Line 75: add a reference?
Line 76-79: this is a long sentence...
Line 111: at some point the patches overlap. Maybe that should be mentioned before.
Line 113: I think P_j should *not* be transposed here
Line 119: I think it should not be x_j but x in the norm.
Line 132-133 Total variation: this is fairly different from the TV from Chambolle. Maybe mention that the gradient is done by the difference of the z. What is \mathcal{N}? Is it the same as \mathcal{N}_j, you probably share the weights, maybe mention it.
Line 132: No caption in the table?
Line 144: convolutional sparse coding-> Convolutional Sparse Coding
Line 163: I think you should specify that h is convex for z given the other parameters fixed.
Line 177: and then to use?
In Table 1: GD is not defined, I guess you mean Gradient

__ Summary and Contributions__: This paper considers learning for a representation of the image signal with domain-specific priors (through regularizers, either smooth or non-smooth) as part of the end-to-end training pipeline for final prediction tasks. By incorporating informative priors, extensive empirical results demonstrate its effectiveness compared to existing methods with much larger number of parameters. The optimization for the encoding step is modeled as a non-cooperative convex game and first-order algorithm for solving it is unrolled for computing updates to the parameter using auto-differentiation. Heuristic tricks are also discussed for improving practical performance.

__ Strengths__: Empirical evaluation of the paper is quite thorough and strong. The setup under study could see a lot of applications in a diverse range of image processing applications.

__ Weaknesses__: The proposed framework draws upon existing literature on unrolling optimization for end-to-end training; some of the extensions/generalizations seem relatively straightforward.
[Discussion phase addition: I would like to thank the authors for their response and the clarification. Overall I think the presentation of the paper could be improved but otherwise think it's a good submission and would like to leave my view unchanged. ]

__ Correctness__: The method and claims look correct.

__ Clarity__: Paper is mostly clearly written, well-motivated and easy to follow.

__ Relation to Prior Work__: The related work section adequately surveys the prior works.

__ Reproducibility__: Yes

__ Additional Feedback__: \item Notation: In equation (4), or tracing back to equation (3), no $\theta$ dependence is explicitly written, which makes the decision variable $\theta$ in the optimization problem a bit implicit.
\item From the discussion in Section 3.2, it's elaborated how one would learn $z_\theta^*$ so the optimization w.r.t $\theta$ part is clear, but the update rule for $W$ is never mentioned - is it updated through backpropagation along with $\theta$? I ask because it looks like $a_{j,k}$ depends on $W$ through $\hat{y}$, which makes $z_j^*(x)$ a function of both $\kappa$ and $W$. But looking at the last column of the table on page 4, for the Non-local total variation for example, only $\kappa$ is considered as the model parameter $\theta$ in the (convex?) regularization term $\psi_\theta$, which is slightly confusing.

__ Summary and Contributions__: In this paper the authors present a framework for image denoising, basically by designing a neural network based on optimization algorithms, but with more general prior functions than the predecessors, and including in the cost function a more general approach.
The paper is well written in general, it includes examples of each type of functions like models, loss functions, regularization terms, etc.
I'm quite far from being an expert in this filed, so it took me a while to understand the big picture, what the whole method was doing in the end.
The introduction is correct, and so are the next sections, and I got the feeling that I was following step by step the procedure, but yet in the end it was hard for me to understand what the method was doing.
The main contribution is the the general framework, with a solid ground on models/optimization techniques, that allows to train an interpretable model with few parameters, obtaining very good results on image processing tasks.

__ Strengths__: Although there are no theoretical results in the paper, the theoretical ground is a strength, in the sense of the deductions, models, and methods used to get to the proposed framework.
This is not a paper trying to explain theoretically why neural networks work, but it is neither just a model that works, without proper justification.
The experimental results are promising as well, with fewer parameters than other methods, and the interpretability that comes from the optimization algorithms used as inspiration.
To the best of my knowledge, the generality of the model within this framework is new, and it is definitively relevant to the NeurIPS community.

__ Weaknesses__: The weakness is the lack of theoretical guarantees, although its not a weakness of the paper itself, but of the family of methods in general.
This being said, the general theoretical grounding, in terms of optimization for instance, is sound.

__ Correctness__: The deductions made through the paper are standard, and they're correct. The empirical methodology is correct as well. The comparisons seem to be fair.

__ Clarity__: The paper is well written in general, with the observations made in point 1.

__ Relation to Prior Work__: The bibliography in general is abundant.
The pior work is clearly presented, and the puntual differences are stated. I would use the opportunity in that part to explain the differences more globally.

__ Reproducibility__: Yes

__ Additional Feedback__: Minor comments:
- Line 89. It's weird that just after saying "a more general point of view", it assumes a patch structure. I know what you meant, it's just that the placing is weird.
- Line 177 "and then us to use auto"
- Line 197 "typically be adapted handle" -> "typically be adapted to handle"
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I I've read the authors' response, but my review doesn't change.