__ Summary and Contributions__: This paper investigates a novel regularization technique based on introducing a proximal mapping transformation into a deep neural network. The authors first show an example of a proximal mapping being used in the optimization of a shallow model. Here a conventional regularized objective is replaced with an objective that uses proximal mapping instead (explicit rather than implicit regularization; for a positive semi-definite quadratic regularizer this internal optimization for the proximal mapping can be done in a closed form). The proximal mapping approach is then applied to robust learning in LSTMs and for multiview learning.
I thank the authors for their rebuttal. Having read it and the other reviews, I keep my recommendation as "accept".

__ Strengths__: After briefly reviewing theoretical derivations, I think that they are sound and well grounded. The method itself is interesting and well-justified and I believe it to be both novel and significant. In my opinion, the proposed regularization technique may potentially find many interesting applications in the field and I am looking forward to seeing them. Even though I am not particularly well familiar with multiview learning, presented results (both ProxLSTM and multiview learning) appear to be sound and promising. Since this article both presents a novel idea and shows several promising deep learning applications, I believe it to be of relevance for the NeurIPS community.

__ Weaknesses__: While the approach itself and specific applications to LSTMs and ProxNet look promising, the paper does not explore more straightforward applications to per-layer regularization in deep neural networks (thus laying a foundation for more complex applications), potentially expanding a toy example shown in Figure 1. Mentioning this seemingly fundamental application, the authors do not study it just saying that it is straightforward to implement. This makes the reader wonder whether this method can actually show promising results in this setting (akin those hinted at in Figure 1).

__ Correctness__: I did not check most of the derivations in the supplementary materials, but I read them quickly and I did not see any immediate issues, which makes me believe that to the best of my knowledge, most claims appear to be correct. Similarly, I did not read the source code and do not know all of the details of the implementations, but the overall empirical methodology outlined in Section 6 is sound.

__ Clarity__: The paper is well written. I assume Eq. (4) should read argmin instead of argmax, but I did not see any other obvious issues. The case of a general R in Section 3 was a little difficult to follow without referring to supplementary materials (same also applies to Section 5.1), but otherwise the text is understandable even if a little overloaded.

__ Relation to Prior Work__: While I am not thoroughly familiar with related work, after a short literature overview, I believe this work to be novel. While there are numerous examples of proximal mapping applications in the field of deep learning (including those in optimization methods), specific ideas and designs developed in this work are novel to the best of my knowledge. The paper itself provides a very brief overview of related literature with a much more detailed discussion presented in Appendix A.

__ Reproducibility__: No

__ Additional Feedback__: The optimization of objective (22) with the proximal map (21) is not discussed in detail in Appendix D. Despite the claim that the Matlab code is available on GitHub, I could not find it in the GitHub repository. This led to me reply "No" to the reproducibility question.

__ Summary and Contributions__: Thanks for the clarifications in the rebuttal. This is a good paper and I've increased my score from a marginal accept to an accept.
---
This paper studies model regularization via proximal operators.
They show how these relate to regularized risk minimization in general
and present how to practice use these in theory.
As examples they instantiate an LSTM layer that uses a proximal mapping
to encourage invariance to perturbations to the inputs, and
study proximal operations for multiview learning.

__ Strengths__: Better-understanding model regularization is an important topic to address
and this paper presents a nice collection of general ideas paired with
instantiations in non-trivial domains.
Their regularized models almost always surpass the baseline methods
on Sketchy, XRMB, HAR, and other sequential datasets.
The interpretation of existing operations as proximal mappings as
discussed in S2 is insightful.

__ Weaknesses__: Proximal operators provide an extremely large function class to to
optimize and it seems like this flexibility could hurt the model's
performance if they are not instantiated correctly for the problem
and model that are being considered.
I do not have much intuition on how the ProxNet improvements on the
datasets they consider compare to other regularization approaches
considered on the tasks beyond the RRM baselines presented
in this paper.

__ Correctness__: I see no correctness errors in this paper

__ Clarity__: The paper is clear and well-written in most parts. The one area
I did not find clear is in the experiments, it's difficult to understand
exactly how the proximal operator is instantiated and paramaterized
for every experiment.

__ Relation to Prior Work__: The paper already discusses a significant amount of related literature
and methods and discusses how existing regularization approaches can
be interpreted as proximal operators. One area that may be worth adding
is the existing use cases of proximal operators for regularizing
other optimization problems, such as:
Meinhardt, T., Moller, M., Hazirbas, C., & Cremers, D.
Learning proximal operators: Using denoising networks for
regularizing inverse imaging problems. CVPR 2017.

__ Reproducibility__: Yes

__ Additional Feedback__: The finite-difference derivative approximation in L178 seems like it can cause
instability during learning. Does care need to be taken to properly
setting up the proximal operator and setting the \epsilon term to ensure
the derivative approximation is well-behaved?

__ Summary and Contributions__: The authors present ProxNet, an approach to regularized optimization that replaces regularization terms in the loss with regularization steps in the model. They show that many (data-independent) regularizers, as well as "data-dependent regularizers" like the regularization of multiple embeddings towards each other in multi-view learning, can be reformulated with ProxNet. They demonstrate substantially improved performance over regularization-in-the-loss (regularized risk minimization) and alternative multi-view learning approaches on four practical benchmarks.

__ Strengths__: I'm broadly convinced by the results of the paper: in cases where the proximal mapping doesn't have a closed form, ProxNet does more "optimization work" on, and gets more "optimization value" out of, each example or minibatch. In cases where there is a closed form, or the inner optimization is cheap, ProxNet performs better without adding additional training cost. It's widely applicable, and the kind of idea that seems obvious in retrospect.

__ Weaknesses__: I would have liked to see more discussion and analysis of the role of annealing lambda over training, and how the 1/t annealing schedule was chosen. Would it (for instance) be helpful to anneal the RRM term in over time as the ProxNet term is annealed out?
The discussion of ProxNet equivalents to common regularizers and neural network primitives seems a little hollow: _not_ being able to construct an optimization problem whose solution is a particular given function would be the surprising case, as far as I can tell.
I would have appreciated seeing multi-view learning results on larger and more recent benchmarks (e.g. semi-supervised ImageNet comparing with SimCLR), as contrastive learning seems to be picking up pace in various application areas.

__ Correctness__: The empirical results cover a wide array of problems and include solid baseline methods and good experimentation practices (e.g. reporting mean/std).
The results could benefit from more clarity about optimizers used and hyperparameter tuning (e.g., how aggressively was the lambda schedule tuned, given that it accounts for two new degrees of freedom).

__ Clarity__: The paper is a little opaque to readers (like me) without a learning theory background, but it rewards deliberate reading as all the conceptual building blocks are included. It's also forced to be relatively concise in reporting experiment details and results, as there are four different benchmarks and three pages to fit them in.
But overall the paper does a good job summing up what's essentially an entire small-scale research program with many connections to different areas of ML theory and practice.

__ Relation to Prior Work__: The comparison to OptNet in the main text is quite limited, but there's a lot more detail in the appendix. Both OptNet and ProxNet are highly general abstractions, and ProxNet can be seen as a special case of OptNet, but the authors are clearly introducing a novel category of optimizing layer with a novel set of applications, and they discuss this fairly.
Similarly, the appendix also includes theoretical treatment of the relationship with RRM (which I did not review in detail), as well as a reframing of ProxNet in meta-learning terms (again, it's a distinctly-novel special case).

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This work proposes the use of proximal mapping to introduce certain data-dependent regularizers on neural network activations. The authors introduce two different regularization methods based on this idea.
The first is a regularization on outputs of a recurrent network (LSTM) to encourage robustness to perturbations in the input. This regularizer has a closed form solution, though second order derivatives are required.
The second regularization method introduced controls correlation between activations of hidden layers on two different data sets, similar to deep CCA (DCCA). The proposed method improves over DCCA by allowing the CCA objective to be jointly optimized with a final classification layers on top of the correlated representation, and is shown to be more effective than including a correlation objective as part of the overall loss. This multiview regularizer does not have a closed form solution and requires an inner optimization involving L-BFGS and SVD.

__ Strengths__: Good results across several different tasks/datasets for each model, compared to reasonable baselines.
I think these regularizers could apply to a fairly large number of problems (though not completely general like dropout etc.)

__ Weaknesses__: Implementation appears to be quite involved, which could limit more widespread use.

__ Correctness__: To my knowledge, everything seems reasonable.

__ Clarity__: Reasonably clear.

__ Relation to Prior Work__: Relation to prior work like virtual adversarial learning and deep CCA is clearly discussed. I'm not aware of any missing citations.

__ Reproducibility__: Yes

__ Additional Feedback__: I don't see any aspect of the robustness method for RNNs that actually relies on the recurrent structure. Am I missing something, or could it be placed on any type of neural network hidden layer?