__ Summary and Contributions__: The authors prove that sparse transformers (transformers in which the potential connections in the attention layer are limited) can 'universally approximate' functions over sequences of finite length - under certain asusmptions over which connections are allowed to remain. Previously, universal approximability has only been shown for 'fully connected' transformers. They present some experiments demonstrating the strength of sparse transformers on various tasks, comparing between different sparsity patterns and to dense transformers.

__ Strengths__: The claim that even (certain) sparse transformers may still universally approximate sequence-to-sequence functions (over sequences of finite length) is non-trivial. Indeed without it, one might intuitively guess that sparsity would reduce the expressive power of a transformer, and see this as an argument against using them. As sparse transformers have less connections, they require less computations to use than a 'regular' transformer, (all other hyperparameters of the networks being equal), making them prefereable in that sense. Hence it is valuable to see that they can be used without concern of losing power.

__ Weaknesses__: 1. Reading this paper, you are inclined to believe that (certain) sparse transformers can be both smaller than (due to their sparsity) and as powerful as (due to the result of this work) 'standard' transformers. However, on second look I have noticed that the claims and constructed transformers, at least as presented in the main body of the work, do not discuss the number of layers that the transformers need in order to achieve this result. This makes it harder to make meaningful conclusions about sparse transformers: what if transformers can be as powerful when the number of connections in the attention has been reduced from O(n^2) (full) to O(n), but this requires a factor of O(n) more layers? A clear discussion of the number of layers used in the constructions, in the main body of the paper, will be really valuable in order to understand whether we should be 'transitioning' to sparse transformers or not. (A result in either direction would be meaningful, and not detracting from the result itself.)
2. The proof of the claim is mostly presented in the (very long) appendix. Reading the paper alone, unfortunately, is not enough to fully convey the method. However, I am not sure if this can really be remedied given the page limit.

__ Correctness__: As so much is left in the appendix, I was unable to truly check the correctness of this paper. With respect to the parts presented in the main section: the formal framework presented for discussion of sparse transformers is neat and well constructed, the assumptions (with one comment below) under which the claim works are clearly presented and generally reasonable, and the main claim and proof sketch are well presented and plausible.

__ Clarity__: The writing in the main paper is generally clear, although it would benefit from expansions/intuitions/corrections as follows:
1. An intuitive discussion of the purpose of the assumptions, especially assumption (1.2) (the meaning of the assumptions is quite clear, but the reason for their necessity is not).
2. A discussion, presented in the main body of the paper, of how many layers are used in the constructions for the proofs.
3. In step 2, ReLU is replaced with piecewise functions that have 3 pieces, despite ReLU itself having only 2 pieces. Some discussion explaining the intuition behind this would be very useful.
4. If there is any additional space, I think a general expansion on what is happening in step 2 in the proof, and of paragraph 259-267, would be very helpful.
Additionally, a couple of sentences are unclear:
1. Line 297 - 'one sparse transformer block' - is the meaning that only one of the layers in the transformer is sparse? Or the transformer has only 1 layer? Or that each transformer has only one sparsity pattern?
2. Line 265 - 'every other token' in the sequence can sound like 'one token yes, one token no, yes, no, etc', though I think the meaning is "all the other tokens", maybe change?
Note: I have not checked the clarity of the supplementary material, which the paper heavily relies on.

__ Relation to Prior Work__: This work extends a previous result -- about dense transformers -- to apply also to sparse transformers. The differences and similarities between the two works are quite clear and also explicitly presented by the authors.
The work mentions, but does not really compare to another theoretical transformer-expressiveness work, which is a shame: in the paper 'Theoretical limitations of self-attention in neural sequence models', Hahn shows (for instance) that a transformer with hard attention is unable to recognise parity for arbitrary length strings, and that for some tasks the accuracy of a transformer with soft attention will reduce as strings become longer. As a knee jerk reaction, one might think this contradicts with the findings of this work, but in fact it does not: the claims in this work are for approximating functions over sequences of a finite given length, and so any one approximation need only be precise enough for the domain it is over, and is not concerned with increasing input length. I think noting this clearly in the work would be valuable.
On a different note, I think that whenever someone works so closely with transformers, and especially attention, it makes sense to cite the original works presenting attention, i.e.:
Effective Approaches to Attention-based Neural Machine Translation (Luong et al)
Neural Machine Translation by Jointly Learning to Align and Translate (Bahdanau et al)

__ Reproducibility__: Yes

__ Additional Feedback__: In addition to the comments detailed in the above section, I would appreciate the authors sharing any insight they have on the following experimental results:
1. In figure 1b, why does the BLEU score of the strided-multihead transformer increase as the sparsity increases?
2. (This is discussed minimally, but more may be interesting): why does the star-sparsity fail so strongly on the copying task, despite being theoretically as capable as the others? I wonder if this ties in with the discussion of number of layers needed to maintain the universal expressivity once the attention is sparse (mentioned also in 'weaknesses'). Insights the authors may have from the proof construction would be helpful here. (It would be nice if the theory could account for star's relative weakness on this task).
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I appreciate the clarifications in the author response, I think they should be brought to the main paper too. In particular, the authors should make it clearer how many layers (approximately, if necessary) are needed to construct the approximation, as a function of the various parameters of the construction eg the sequence length n, and other parameters (delta,d,...). Following some discussion, it may also be good to clarify that the universal sequence to sequence approximation is between sequences of same length (i.e. that the discussion is on the encoder).

__ Summary and Contributions__: Raised my score after reading the rebuttal.
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This paper provides a theoretical foundation for sparse transformers: a sparse transformer with O(n) attention connections in each layer can universally approximate any seq-to-seq model. The author extends the previous theory of universal approximation theory of dense transformers to the sparse transformers. In particular, they study the case that sparse attention satisfies several conditions.

__ Strengths__: This paper shows several conditions that sparse transformers needs in order to have the same expressive power as the dense transformers. These conditions may further help future researches on designing sparse transformers.
They also empirically compare the performance of the different sparse attention patterns.

__ Weaknesses__: It is a straight-forward extension of [28]. The result is not superising, since the model is allowed to have aritrary depth. The connection between any 2 positions can happen in deep layers. Since the number of layers in sparse transformers are usually very small (10~20) compared with the length of the setences (hundreds of tokens), why sparse transformers still perform similarly to the dense transformer is unclear. It would be more interesting to investigate the model expressive power when the depth is limited. Or how many layers does the transformer need to have the universal approximation ability.
If these strong conditions are required for a sparse transfomer to have the same approximation ability as the dense transformer, it would be better to theoretically prove that these conditions are required or empirically show that these conditions can not be broken. Although in real data, the sparse model may work without satifying these conditions. Some experiments with hand-crafted data can also help with showing that these conditions are necessary. Otherwise, I just feel that these assumptions/conditions are just too strong and make the results less interesting.
Can the analysis also extend to encoder-decoder attention?

__ Correctness__: Yes.

__ Clarity__: Yes.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The work claims to have generalized the approximation power of Transformer to the case of sparse attention patterns.
These patterns include some of the existing sparse Transformers.

__ Strengths__: The claimed result is new and important to the machine learning community.

__ Weaknesses__: The result is clear and important, but the hardness of the result and the proof remains further explanation.
The main point of the proof of the main result (Theorem 1) is to make an additional assumption (Assumption 1.3) on sparse attention which helps the authors to prove Lemma 7 on 'contextual mappings' of a token.
As claimed, the key finding of Lemma 7 is that these mappings really reflect the context of a token, and thus different for a token in different sequences/sentences (property 2 in Lemma 7). Moreover, in some direction u, projections of each token are all distinct (property 1 in Lemma 7).
Once these contextual mappings are constructed, authors use several position-wise feedforward networks to reach the target function f (Lemma 8).
I have the following two concerns on the proof.
1) To my understanding, property 2 in Lemma 7 characterizes a context instead of property 1. However, only property 1 is used in the proof? If so, this could be misleading because the so called "contextual mapping" is not really used in the proof, which makes the notion irrelevant to the result.
2) This problem is more difficult, but will undermine the difficulty of the work.
Without loss of generality, in the proof of Lemma 8, let us discuss the case when d=1. In this case property 2 in lemma 7 is equivalent to $g_c(H)_ k$ being all distinct (as a real number), which I denote by $y_k$.
The main point of Lemma 8 is to use a stack of position-wise feedforward layers to map y_k to $ \bar{f}(\boldsymbol{H}-\boldsymbol{E})_ {k}$, which I denote by z_k.
The problem is, To map $(z_k)_ k$ to $(y_k)_ k$ through a stack of token-wise feedforward layers, why is it necessary to have $z_k$ being distinct for different k?

__ Correctness__: The mathematical claim is correct, but the proof requires more writing and explanation.

__ Clarity__: The results are clearly stated, and the overall idea of the proof is well explained.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The paper introduces general of sparse attention. Using this framework the authors prove that Transformers with sparse attention are universal approximators under some natural and pretty weak conditions on sparsity patterns.

__ Strengths__: The class of models for which the proof applies includes popular in applications variants of sparse attentions such as Strided and Fixed patterns.
The paper shows that it's possible to express any function using sparse attention with O(n) complexity. The interesting part is that one can do so without changing the width of the layers as the input length grows.

__ Weaknesses__: The experiment section is not clearly connected to the theoretical part. For instance, that would be interesting to see whether breaking any of the assumptions of theorem causes significant degradation of the quality, e.g., breaking any clause in assumption 1 for the random connectivity pattern or for some handcrafted ones. The same for assumption 2, e.g., is that true that replacing SM with a hard top-k hard significantly degenerate the performance.
Most of the experiments are done for autoregressive case (LM and MT). It seems to break assumption 1.3, as token at first position will never see the whole sequence. If the theory doesn't apply to unidirectional transformers, maybe it's better to switch to naturally bidirectional models such as BERT.

__ Correctness__: The theoretic claims are valid. Empirical methodology is not clear, see above.

__ Clarity__: The paper is well written except for the experiment section. E.g., in G2 it mentioned that masking is used to enforce causality, but there is no mention of masking in G1. So it's not clear how's prediction done in this case.
For all masked experiments, it's not clear how STAR attention is adopted. From (6) it follows that every position attends to the last element that is not available in the masked. While a reader can guess that, that'd be great to have more clear explanation.
For G3, I assume an encoder-decoder architecture is used. Is sparse attention for both parts or only decoder?

__ Relation to Prior Work__: I believe the paper will also benefit from references to other universal approximators, e.g., RNNs and ResNets. Right now paper only compares with Dense Transformers.

__ Reproducibility__: Yes

__ Additional Feedback__: