Summary and Contributions: DEQ is a very interesting work, which models infinite depth with a constant memory. DEQ does not save the intermediate hidden layers and updates the weights with only the fixed point. However, it did not solve the problem that does the fixed point stable and unique? The paper used the theory of monotone operators and presented a solution to this problem.
Strengths: The paper used the theory of monotone operators and presented a practical solution to obtain unique fixed points.
Weaknesses: Compared with DEQ, the fixed points of MOE are much more stable and unique, which needs experiments to justify.
Correctness: Yes. Yes.
Relation to Prior Work: Yes
Additional Feedback: After reading the reviews and the rebuttal, I will keep my score (here is 7). DEQ is an important work and the presented extension is promising. The authors claimed in the rebuttal that they will add the comparisons with DEQ. However, it is not enough for me. For example, they need to investigate whether the more stable of the fixed point the better the test performance. Since it is an extension of DEQ, the paper should test the advantages over DEQ thoroughly. This prevents me from raising my score further. ------------------------------------------------------------------ Overall, the paper is well-written and is easy to follow. I have the following comments. 1. The paper argued that DEQ requires careful initialization and regularization and MOE does not require much tuning. However, the paper failed to verify this claim. Since MOE is the improvement of DEQ, the extensive comparison between the two methods is need. For example, compared with DEQ, the fixed points of MOE are much more stable and unique, which needs experiments to justify. 2. In Theorem 1, the function f should be convex closed proper (CCP). The paper needs to highlight the requirement. All non-decreasing activation functions can be represented as proximal operators. However, the function f in the proximal operators may be not CCP. The paper needs more discussions on this issue. 3. In Theorem 1, A and B are operators, while they are matrices in Proposition 1. It is better to use other letters in Proposition 1. 4. In line 399 of the Appendix, A should be G.
Summary and Contributions: The paper develops a principled approach for training fixed-point networks. Given input $x$, fixed point networks define the output of a network as a fixed point $z$ of a computational block $f(x, z)$. The authors consider computational blocks consisting of a linear operator and a component-wise non-linearity; for such blocks, the authors propose a parametrization for the block to ensure a sufficient condition for the existence of a fixed point. Then the authors adopt two different operator splitting algorithms for the forward and backward passes of the fixed-point network. For the evaluation, the authors develop several architectures and compare them against Neural ODE. With the same parameter count, the proposed approach outperforms Neural ODE in terms of accuracy and computational efficiency. Additionally, the authors show the role of the fixed point solver hyperparameters and the scalability to more parameters.
Strengths: The original work on deep equilibrium models (DEQ) did not give any convergence guarantees for the underlying iterative process. The paper fills the gap with an alternative parametrization, an elegant and easy to implement solution. Similarly to DEQ, Monotone networks use implicit differentiation and do not store the intermediate steps of the underlying solver. As a result, the model training loop is more memory-efficient compared to conventional deep architectures. Another appealing feature is that the considered class of fixed-point networks include a variety of non-linear activations and, as the authors show, is extendable beyond fully-connected matrices. Monotone networks outperform NeuralODE (another implicit depth architecture that is, unlike DEQ, well-defined) with a similar number of parameters and demonstrate a room for further performance improvement.
Weaknesses: Compared to DEQ, Monotone Networks consider a less versatile class of functions (a price to pay for the convergence?). In particular, NeuralODE and DEQ extend to series data, but it is not clear how to extend MON to such data.
Correctness: The method and claims are correct. In the experimental section, it would be nice to see the results of NODE and ANODE for bigger model size along with the results of MON.
Clarity: The paper is very well written.
Relation to Prior Work: The authors clearly discuss the relation to DEQ and NeuralODE. Could the authors comment on how their approach compares to the "Implicit Deep Learning" paper?
Additional Feedback: [Post rebuttal edit: Thank you for the thorough feedback. After the rebuttal, I am keeping my original score.] Is there any intuition on whether such architectures are suitable for regression tasks?
Summary and Contributions: Authors have used the theory of monotone operators to develop a novel implicit-depth model. Unlike previous approaches such as NODE and ANODE, authors show that their proposed approach has stable convergence.
Strengths: The major impressive strength of the paper is the fact that the proposed approach significantly outperforms state-of-the-art results from NODE and ANODE papers.
Weaknesses: It is great that authors have compared against NODE and ANODE. However, there are still some comparisons that are missing compared to the reported results in NODE and ANODE. One more comment about simulation results is that it would be great if authors could report the standard error for their results similar to NODE and ANODE papers.
Correctness: Both claims and methodologies are correct.
Clarity: The paper is very well written except few places that have grammatical errors and typos. One example is line 127 in which authors have written: "be formalized in a the following theorem".
Relation to Prior Work: It is clearly discussed how this work is different than the previous ones. Also, authors have compared how their work is different than the previous state-of-the-art results from NODE and ANODE papers.
Additional Feedback: Post Rebuttal: Thanks for additional feedback and comments and congrats for the great work.