Summary and Contributions: Thanks to the authors for an informative rebuttal. The rebuttal addressed my concerns to a satisfactory level: so I'd like to increase my evaluation by +1 and think this paper should be accepted. For the ``faster'' comment, I strongly encourage the authors to discuss not only the # of updates but also the actual wall-clock time (which I believe should be generally proportional to the # of updates?). I am sorry that I didn't fully appreciate the carefully documented appendices in my initial review. For the final version, I strongly recommend that the authors refer to the appendices for important remarks to make in the main paper. ++++++ The paper theoretically analyzes how self-attention helps mitigate the problem of gradient vanishing in training an RNN and proposes a screening mechanism that enables sparse self-attention that doesn't suffer quadratically increasing memory.
Strengths: The work provides sound theoretical analysis on how self-attention can help resolve the gradient vanishing problem and how sparse self-attention can prevent gradient vanishing while still maintaining constant memory usage. Full proofs with extensive details are provided in the supplementary material. The proposed screening mechanism turns out to be effective on several tasks, achieving compelling results while still being fast and memory-efficient. Empirical analysis also demonstrates that the gradient, using this mechanism, doesn't explode or vanish.
Weaknesses: The authors didn't spell out the relation between \kappa and d: higher \kappa tends to have smaller d. This relation is discussed in the paragraph of line-173 but not reflected in the formula of theorem-2. In experiments, the authors mentioned the proposed model is ``faster'' to train but didn't give any quantitative results. It seems useful to also show such quantitative results just as they do for the memory usage analysis.
Correctness: The theorems and proofs look correct to me. The method looks reasonable.
Clarity: The paper is overall well written, except that the mathematical notations seem a bit messy. E.g., the letter s is overloaded many times which makes it hard to follow---s can be neural state (eqn-2), index of i (proposition-1), depth s (definitions), etc. E.g., BPC in table-2 is undefined.
Relation to Prior Work: The paper clearly discusses how it is related to previous contributions.
Summary and Contributions: This paper presents a formal treatment of gradient propagation in attentive recurrent neural networks, in which the gradient norm is asymptotically quantified based on attention sparsity (k) and maximal dependency length (d). To balance between computational complexity (small k) and good gradient propagation (small d), a relevancy screening mechanism is introduced, retaining only top relevant timesteps for attention. The method is tested on synthetic memorization tasks, Penn Tree Bank, and MiniGrid reinforcement learning environment, demonstrating promising results, especially on generalization capability.
Strengths: - The theoretical analysis is interesting, providing a solid foundation for future theoretical works in attention mechanism - The method is simple, yet effective for the considering tasks
Weaknesses: - The literature review is incomplete, lacking detailed comparisons to relevant prior works - The empirical results are not strong - The method is only verified on simple/synthetic tasks
Correctness: Seems correct. I have not checked the theory clearly
Clarity: Overall yes
Relation to Prior Work: Not really
Additional Feedback: - Line 145, how can Theorem 1 be related to the early attention mechanism ? As the attention weights are computed adaptively, it is unlikely that they are uniform. - Memory-augmented neural networks (MANN [2,3]) are naturally sparse self-attentive RNNs (k is the number of memory slots). MANNs learn to store relevant hidden states to a fixed-size memory, which seems to have the same purpose as relevancy screening mechanism. What is the advantage of the proposed method over MANNs? How are MANNs related to the Theorem 2? - The paper neglects prior works that also aim to quantify gradient propagation in RNNs and attentive models [4,5]. How are the paper’s theoretical findings different from , wherein gradient norm of self-attentive RNN is also quantified? A clear comparison is necessary to highlight the theoretical contribution of the paper. - Using attention weights as a relevant indicator is explored before under reinforcement learning context . A clear comparison is necessary to highlight the novelty of the proposed relevancy screening mechanism. - Fig.1, values on the axis are too small - Line 260, Table 4 is not about Copy task - Table 3 is confusing. For example, in pMNIST, relLSTM underperforms expRNN yet both are bold. - In experiments, the authors should include stronger baselines to prove the benefit of the approach. For example, MANNs in  performs well on pMNIST tasks. In RL task, it is better to have a strong baseline such as .  Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural Machine Translation by Jointly Learning to Align and Translate. arXiv e-prints, art. arXiv:1409.0473, Sep 2014.  Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014.  Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka Grabska- Barwin ́ska, Sergio Gómez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, et al. Hybrid computing using a neural network with dynamic external memory. Nature, 538 (7626):471, 2016.  Shiyu Chang, Yang Zhang, Wei Han, Mo Yu, Xiaoxiao Guo, Wei Tan, Xiaodong Cui, Michael Witbrock, Mark A. Hasegawa-Johnson, and Thomas S. Huang. "Dilated recurrent neural networks." In Advances in Neural Information Processing Systems, pp. 77-87. 2017.  Hung Le, Truyen Tran, and Svetha Venkatesh. "Learning to remember more with less memorization." arXiv preprint arXiv:1901.01347 (2019).  Chia-Chun Hung, Timothy Lillicrap, Josh Abramson, Yan Wu, Mehdi Mirza, Federico Carnevale, Arun Ahuja, and Greg Wayne. "Optimizing agent behavior over long time scales by transporting value." Nature communications 10, no. 1 (2019): 1-12. ========= Update after rebuttal: While I feel satisfied with the authors' response regarding the theoretical contribution of the paper, I find the algorithm and experiments less convincing. The authors haven't answered my question on the novelty of the proposed relevancy screening mechanism. Hence, I would like to keep my score as is.
Summary and Contributions: The paper tackle the gradient vanishing problem of RNNs by augmenting with a sparse attention mechanism that supported by a theoretical analysis. The sparsity is achieved by prioritizing memories that attended more often during previous time steps. The model is compared to other RNN based baselines on toy tasks, language modeling, and pMNIST.
Strengths: The paper is well motivated and that motivation is backed by a detailed theoretical analysis. The theoretical part clearly shows the benefit of attention in RNNs, but also go further by showing that sparse attention work better than dense one. In addition, the paper proposed a novel sparse attention mechanism inspired by brain that is simple to implement. More importantly, the method reduces both computational time and memory usage as demonstrated by the experiments.
Weaknesses: I have few minor comments about the novelty and the theory in the paper, but my main concern is with the experiments. - Novelty: augmenting RNNs with sparse attention to prevent gradient vanishing is not novel in itself . The only novel part of the model is relevancy screening, but why that approach is better than other sparsity methods has no grounding in the theoretical analysis in the paper. Actually, the theoretical analysis itself is based on the top-k approach from  instead of relevancy screening. - Theoretical part: it is well known that an attention mechanism would reduce gradient vanishing. It feels trivial to me as there is a direct connection for gradients to pass. As that connection is weighted by softmax attention weights that sum to one, it's not hard to see that having fewer things in the softmax (i.e. sparse attention) would increase the attention weights, thus improved gradient flow. - Experiments: I think the experiments in the paper are quite weak. The only tasks that are not toy are char-PTB and pMNIST. But even char-PTB is small when compared to commonly used LM benchmarks like text8, enwik8, wikitext103. In addition, there was no improvement over a vanilla LSTM on char-PTB. The improvement in pMNIST is also marginal. And this is without including Transformer based methods, which work much better than RNNs on such tasks.
Correctness: The theoretical claims make sense and seem correct to me, but I didn't check the proofs in the supplementary material. The experimental setups also look good to me.
Clarity: In general, the paper is well written and easy to understand. Minor comments: - There was not much about the proposed method in the introduction. Explaining "relevancy screening" a bit more would make it easier to understand in a limited time. - I think the brain related claims like "just like NN in the brain" , "our brain do ... attempts at mimicking this ..." can be bit toned-down as they're only inspirations. - Typo: Fig 2 is referred as Fig 6 in lines 319-324.
Relation to Prior Work: Yes. I just have few minor comments: - The method has a lot of similarity with , but when discussing  in the background section, the authors only said "naively sub-sample input streams for memory usage", which is too vague and unclear to me. The top-k sparsity from  is simple, but works well and used in many subsequent works. Instead, I think the authors can mention the computational complexity of  as discussed in Sec. 4 and 6. - It is true that self-attention has a quadratic complexity, but there is a simple way to make it linear by limiting the attention span to recent L tokens as done in Transformer-XL, which is worth mentioning in the background section.
Additional Feedback: === Post-rebuttal comment: After reading the rebuttal, I'm upgrading my score to 7.