__ Summary and Contributions__: This paper proposes to model permutation-invariant functions using recurrent models and introduces a permutation-invariance regularizer. This is in contrast to the traditional approach toward learning set functions, which impose permutation invariance in the model by design, as opposed to using a regularizer. As motivation, this paper proves that there exist functions which are more efficiently represented by recurrent models, which only have to learn a "local aggregation" rather than a global set function.
Post Author Feedback:
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I appreciate the authors' response to my questions, which address my low-level questions. The main weakness of the paper is still the experimental section; in addition to limited tasks, reviewer 4 brings up additional simple baselines such as using alternative pooling operations. I still like the main idea of the paper, and am keeping my score of 6.

__ Strengths__: - The main observation in this paper is quite nice
- The paper is well motivated, and the methodology makes sense

__ Weaknesses__: - The experiments are all synthetic and somewhat toy

__ Correctness__: The method is correct

__ Clarity__: The paper is well written and easy to understand

__ Relation to Prior Work__: Previous work is adequately covered

__ Reproducibility__: Yes

__ Additional Feedback__: Typos and notation:
- Definition 3 seems to be missing the dependence on $\mathcal{D}$. Is $(x_1, \dots, x_i)$ meant to be a sequence in the support of D?
- Above equation (3.4): I think you want to define $\tilde{\rho}(z) = \rho(nv_0 + z(v_1 - v_0))$
- Equation (4.4): I think $x_1, x_2$ are drawn from ${\cal S}$ instead of ${\cal D}$
- Line 252: missing a + in this equation?
Additional comments:
- Some of the experimental results seem hardly believable and could be elaborated on. In particular, the "sum" task being the hardest task is difficult to interpret. First, why is it harder than "unique sum", even when the sequence length is less than 10 (the number of possible distinct inputs)? Second, how can an MLP based model such as DeepSets fail at this task for n=15, when there are clearly trivial functions that can compute this?
- It would be nice to see the results of the other baselines on "half range" just for comparison

__ Summary and Contributions__: The authors propose to learn a permutation invariant RNN model through regularization rather than inductive bias, by including a loss term that encourages the RNN output to be invariant to transpositions.

__ Strengths__: The observation that sequential symmetric functions are more efficiently learnable with RNNs than DeepSets is an insightful one, and most of the synthetic experiments seem to confirm the efficacy of the method.

__ Weaknesses__: I'm somewhat concerned about other elements of the experiment section. The DeepSets paper included two baselines for the point cloud experiment, with the larger model performing substantially better (up to about 90% accuracy). It seems the authors use the smaller model in their experiment.
This mainly raises the question of if the RNN model would be able to scale up as well as the DeepSets model. As this is the only experiment on real data, it's very hard to judge the proposed method without that evaluation.

__ Correctness__: The claims and methodology appear correct.

__ Clarity__: The paper is mostly well-written with some slightly confusing elements, for example the notation for subsequences in Definition 2 and the lack of a definition for P in Definition 3.

__ Relation to Prior Work__: The relation to other works in symmetric neural networks is clear.

__ Reproducibility__: No

__ Additional Feedback__: Post-Rebuttal:
The author response suggests the focus of the paper is the representational result in Theorem 4, but the paper seems to be claiming that the proposed RNN model is competitive with DeepSets. Based on the difficulties pointed out by reviewer 4 with using the RNN model, I think that claim requires a stronger experimental section to demonstrate that the representational difference (1) appears in practice and (2) is significant enough to merit the difficulty of a recurrent model. Therefore, I will keep my score the same.

__ Summary and Contributions__:
The paper proposes a regularization method to adapt RNNs to the case of sets as input, by making the RNNs more invariant to input permutation. This can serve as a simpler and more parameter-efficient alternative to Deep Sets (Zaheer et al.).
The paper illustrates an example where a permutation invariant function can be represented by an RNN with O(1) parameters while Deep Sets representation requires at least Omega(log n) parameters, where n is the sequence/set length.
The effectiveness of the proposed regularizer is validated on synthetic tasks (e.g., parity of sequence of 0/1, arithmetic tasks on sequence on integers, point clouds). The proposed method shows similar performance to baselines (DeepSets, Janossy pooling) on short sequences (length 5-10) and beter performance on longer sequences (length 15).

__ Strengths__:
1. The idea of Subset-Invariant-Regularizer (SIRE) is a simple and clever way to ensure invariance. Imposing the stronger condition of subset invariance (instead of just permutation invariance), it leads to a much simpler condition of the RNN being invariant to the order of inputs in a pair. This simpler condition leads to a regularizer that's easier to approximate.
2. The example (Section 3) showing a case where RNN being more parameter efficient than Deep Sets is insightful and illustrative. The parity function is permutation invariant, and can be represented by an RNN with O(1) parameters while Deep Sets representation requires at least Omega(log n) parameters, where n is the sequence length. I enjoyed reading this section.
3. The proposed method (RNN + regularization) outperforms Deep Sets and Janossy pooling on longer sequences (length 15), on synthetic tasks. The proposed method also shows slight improvement in a toy task (half-range) where partial invariance is useful.

__ Weaknesses__:
1. All tasks are toy/synthetic, and there is no experiment result on more realistic tasks. Section 6.4 mentions "human activity recognition" and ECG readings. It would be interesting to see if SIRE helps there. That would make the results stronger.
2. Lack of details/reproducibility. The procedure to approximate SIRE by sampling is not described in details. This might make it hard to reproduce the results.
In particular, in the definition of SIRE (eq 4.4), one is to sample s from S_{D, \Theta}. However, S_{D, \Theta} is a set, not a distribution. Should one sample from the uniform distribution on this set (which I think is hard)? Or any distribution would work? How do different distribution affect the result? In practice, how many samples should it be averaged over? Is this tuned with cross-validation, or does the same hyperparam work across experiments?

__ Correctness__:
The claims and empirical methodology are correct. I have checked the proofs in the appendix (they are short).

__ Clarity__:
The paper is easy to read.
Some minor questions/comments on clarity:
1. The motivation for the method (end of page 1): it's not clear what the problems with previous approaches are. Is that they use more parameters than necessary?
2. Definition 3 is not clear. The variables i and t are unbounded. I think this is for any i and any t?

__ Relation to Prior Work__:
Differences to prior show are clearly discussed. There is sufficient novelty in the proposed method, to the best of my knowledge.

__ Reproducibility__: No

__ Additional Feedback__:
Questions/Suggestions:
1. What is the complexity / runtime penalty of sampling s in estimating SIRE?
2. [Minor] It would be interesting to work out what architecture of RNN (e.g. constraints on weight matrices and nonlinearity) will yield this invariance as defined in eq. 4.4. For example in the case of x and s always being 0, 1. This would give more intuition on the this subset-invariance regularizer.
3. [Minor] In Theorem 5, it shoudl be "if and only if"?
4. [Minor] In section 6.2, SIRE performs well up to sequence of length 15. How far can one push the sequence length until it no longer works?
5. [Minor] In section 6.4, I don't have great intuition what kind of "semi"-invariance is compatible with SIRE. So it's only invariant to *some* permutation, but what kind?
Suggestion for additional citation:
- Wagstaff, Fuchs, Engelcke, Posner, and Osborne. On the Limitations of Representing Functions on Sets. ICML 2019.
They show the function in Deep Sets has to be highly discontinuous, similar in spirit to Theorem 4.
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Update after authors' rebuttal: Thanks for clarifying the sampling procedure of SIRE. As remarked by other reviewers, the argument in the paper (RNN enjoying representation advantage compared to DeepSets) could be made stronger by empirical results comparing to DeepSets on larger and more realistic datasets.

__ Summary and Contributions__: This paper proposes a regularization scheme for RNN to be close to permutation invariant. Unlike the previous works that are permutation invariant by design, this work proposes to regularize RNNs towards permutation invariancy. A permutation invariant promoting regularizer is proposed based on an observation that if an RNN is commutative for a step it would be permutation-invariant for arbitrary length sequences.

__ Strengths__: - This work adds an interesting new direction to the literature regarding the permutation-invariant neural networks / set-input neural networks.
- Paper is generally well written with helpful illustrative examples.

__ Weaknesses__: - The main motivation to use RNN for permutation-invariant tasks is not fully justified. See below.
- The experiments are not through. See below.

__ Correctness__: The main argument to derive the regularizer seems to be correct.

__ Clarity__: Yes, the paper is well written and easy to follow.

__ Relation to Prior Work__: The paper is doing a good job of reviewing the literature and clearly states the difference from the previous works.

__ Reproducibility__: Yes

__ Additional Feedback__: Although I find the idea of regularizing RNN with the proposed regularizer interesting, I’m still unsure of why we should really consider using RNN instead of existing permutation invariant architectures by design (e.g., Deep Sets, Set Transformers, …).
The authors are demonstrating the parameter efficiency of RNN compared to Deep Set architecture using the parity example. This argument is valid only when we constrain Deep Set to use the sum pooling operation. One can actually consider various types of pooling operation, for instance featurewise min, max, or even featurewise XOR operation. Deep Sets with featurewise XOR pooling would be as efficient as (or even with fewer parameters) as RNNs.
As for the experiments, as far as I understand from the article the Deep Sets are using only the sum pooling operator. Again, this may not be a fair comparison, for instance the reference [18] considers various pooling operations such as mean/max/min. One can also consider a concatenation of all those pooling operations.
One critical downside of RNN, in my opinion, is its sequential nature in computation. Unlike the Deep Sets (implemented with feedforward neural nets) or set transformers that can process elements of sets in parallel, RNN requires sequential computation that cannot be parallelized. This can be problematic for both training and inference for long sequences. For instance, consider the amortized clustering example presented in [18], where the size of sets typically scales to several hundred to thousand. For such data, training should suffer from a typical problem that might happen for an RNN being trained with long sequences, and the inference would take much longer time than the Deep Sets or set transformers.
Also, this paper only considers permutation invariancy, but permutation equivariancy is also an import property one might ask for. Is it still straightforward to build a permutation equivariant network using RNN by using the proposed regularizer?
In summary, I see no strong reason to consider RNN instead of existing permutation invariant networks for the problems requiring permutation invariancy. It would be good for the authors to present more examples of the problems requiring semi-permutation invariancy, as in section 6.4.
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Post author feedback
I thank authors for their clarification. However, their feedback didn't really resolve the issue I raised. In order to claim that the proposed method does better than existing permutation-invariant by design methods without having to carefully design pooling functions, more experimental results are required (unless there is a strong theoretical guarantee). At least, the proposed method should be compared to DeepSets nontrivial pooling functions (max/min/xor, ... or concat of them (which I expect to be quite strong baseline). Otherwise, considering the sequential nature of RNN, I still don't see a strong reason to consider the proposed method as an alternative to existing ones. I keep my score as is.