__ Summary and Contributions__: The paper considers the issue of memorization in deep neural networks, and build upon the work of Feldman [12] in this area.
The authors propose a set of experiments to estimate the influence of training examples, and propose an efficient way to estimate influence from a subsampled training set.
Experiments on CIFAR and Imagenet show examples of memorised images, and the influence of memorised images on the test set accuracy.

__ Strengths__: The paper builds upon an interesting theory by Feldman[12]: natural data distributions are divided into tiny subclasses, and a classifier performs drastically better on a subclass when it has seen only one sample from it. Hence, the job of a neural network is to “memorise” the representative of each subclass.
This theory is very appealing, and I think that the experiments done in this paper are a good step towards the empirical validation of it.
The experiment reported in Figure 2 is particularly interesting, and shows the importance on the test set of memorised samples.
Overall, the paper is well motivated and reads very well.

__ Weaknesses__: The only weakness that I see in the current version of the paper is that more experiments could be conducted to strengthen the validation of the theory: in particular, the correlation between memorization and influence would be an interesting statistic to have.

__ Correctness__: The claims and method are correct to the best of my knowledge.

__ Clarity__: The paper is very well written.

__ Relation to Prior Work__: Prior work is referenced and discussed.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The paper proposes a scalable estimation method to approximate the memorization value and influence value which indicate whether the given data is memorized in the model or not. Because the influence value, which is a proxy for the leave-one-one influence, is not scalable, the authors propose a "closely-related" statistic, called influence-m, by keeping m random subsets instead of removing one example. Furthermore, the authors propose a scalable way to efficiently estimate the proposed statistic. It intuitively seems that the proposed method produces different statistic values compared to the naive monte carlo estimate, but they provide the upper bound of the variance to justify their method. In the experiment, by using the proposed method, they calculate the influence-m in ImageNet and CIFAR100 with ResNet50. In addition, a bunch of qualitative results provide an insight for some practitioners.

__ Strengths__: * They propose a scalable estimator and a scalable method to approximate the original one.
* They provide the qualitative result for the first time, implying that the memorization is needed for large real-world datasets such as ImageNet.
* The theoretical upper bound of the variance of the estimated value is provided and the proof for the upper bound is given as well.

__ Weaknesses__: * Some explanations are missing. First, the relationship between influence and influence-m is not discusses in detail in the paper. The phrase "closely-related" cannot explain anything. Second, the relationship between the standard deviation of estimated value and the time complexity is not properly stated in the paper.
* The paper needs a rudimentary experiment. To empirically demonstrate the effectiveness of the proposed method, the estimated value needs to be compared with the naive monte carlo estimation based on the definition. In other words, the experiment that shows the two different estimation methods output the same result is appropriate for this paper, instead of other experimental results reported in the paper. Unless I misunderstood the Appendix, the experiment about estimation consistency in the Appendix does not correspond to this case, but the case of comparing just two different training methods. That experiment could be fairly hard due to the poor scalability of monte carlo estimation, but it is necessary to empirically justify this word. The authors are strongly recommended to use a small but long tail dataset such as CIFAR10.
* Reorganizing the structure of paper is needed. Also, the Figure 2 requires a legend.

__ Correctness__: There is no flaw in the experiment.

__ Clarity__: The structure of the paper need to be reorganized. We suggest to change the title of paper to explicitly state what they do.

__ Relation to Prior Work__: Yes. The previous work provided the theoretical analysis on the relationship between a long tail data distribution and the generalization of deep learning. This paper resolves the scalability issue that the previous work did not consider, and applies the proposed technique to the real-world dataset.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This paper studies the memorization phenomenon in deep learning and extends the work of [12] in many ways. A computationally efficient algorithm is proposed that can measure the amount of memorization of a training example and the degree of influence of a single training example on a single test example. It also shows human-interpretable results that demonstrate very high influence scores are often caused by nearly identical image pairs in training and test datasets.

__ Strengths__: The proposed algorithm requires much less training runs than that of [12] by utilizing many random subsets of examples. Visual examination is helpful for understanding the effect of long tail.

__ Weaknesses__: I would like to see some clarification on the long tail theory. Assume we define a more general measure that measures the amount of memorization by A on a set of examples (x_{i_1},y_{i_1}),...,(x_{i_k},y_{i_k}) as follows:
mem(A,S,i_1,...,i_k):=Pr_{h<-A(S)}[h(x_{i_1})=h(x_{i_2})=...=h(x_{i_k})=y_{i_1})-Pr_{h<-A(S \setminus {i_1,...,i_k})}[h(x_{i_1})=h(x_{i_2})=...=h(x_{i_k})=y_{i_1}),
where the k examples have the same label, i.e., y_{i_1}=y_{i_2}=...=y_{i_k}, and x_{i_1},...,x_{i_k} form a subpopulation. If the value of mem(A,S,i_1,...,i_k) is high, perhaps we can still call this phenomenon "memorization." If so, then memorization phenomenon is not just limited to long tails. Then, it seems to me the claim in [12] that memorization is needed due to long tail may not be showing a bigger picture.
The paper mentions that very high influence scores are due to near duplicates in the training and test examples. Such artifacts caused by data collection methods do not seem to be natural. If it's truly long tail, then it would be highly unlikely to have a matching image in the test set for an outlier training example. Long tail theory seems to work only because many subpopulations contain two or more near duplicates that happen to be split between training and test examples when we randomly split the whole dataset. If so, practical significance of long tail theory would be diminished for more natural datasets that do not suffer from such artifact problems.

__ Correctness__: I have not verified the proof of Lemma 2.1, but its result seems to make sense. All empirical methodology seem to be OK. I haven't found any problem.

__ Clarity__: This paper is well written and very easy to understand.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: Can the analysis in page 7 on whether the last layer suffices for memorization be made more general? Can we study the effect of last 2 layers, last 3 layers, and so on to pinpoint where memorization occurs?
After obtaining examples with high memorization scores, perhaps we can re-run Algorithm 1 after replacing the labels of those examples by random ones to see the memorization effect more vividly.
Why does choosing m=0.7n achieve a good balance? Why not m=0.5n? Does m=0.7n achieve good balance for all datasets considered in this paper? More explanation would be helpful.
In Fig. 2, there are two colored curves, green and orange. I couldn't find explanations on which color is which.
154 the a -> the
312 log tailed -> long tailed

__ Summary and Contributions__: This paper seeks to experimentally validate a “Long Tail" Theory put forth in prior work [12] to explain memorization deep neural networks. It proposes a tractable approximation based on subsampling to an influence estimation scheme proposed in the same work. Via this method, “memorization” and “influence” values are calculated for each training instance for common image classification datasets (ImageNet, CIFAR100, MNIST). The results empirically validate the proposed theory – memorized instances tend to be atypical / rare / occur in the “long tail”, often have high marginal utility to generalization, and are hard to distinguish from outliers / mislabeled examples.

__ Strengths__: – The paper addresses an interesting problem of seeking to understand memorization in neural networks.
– The paper’s experiments are appropriately designed to investigate the previously proposed long tail theory
– Some of the empirical findings are quite insightful - for eg. the higher marginal utility of memorized examples, and the correlation between typicality and estimated memorization

__ Weaknesses__: – The only concern I have is that the applications of the proposed memorization / influence estimation function is unclear to me. Beyond provide some empirical evidence for the "long tail” theory, do the authors believe the proposed estimators to have other applications? The authors list interpretability and outlier detection, but the proposed method appears too computationally inefficient as compared with existing methods to those problems. Further, while the proposed method works at an instance level, have the authors thought of potential utility in auditing datasets / providing dataset-level insights?
– Deep networks are typically trained via mini batch SGD, and the order in which instances are seen can lead to different memorization / forgetting behaviors for eg. see Toneva et al, ICLR 2019. How does the proposed estimator account for order in which data is seen?

__ Correctness__: Yes.

__ Clarity__: The paper is mostly well written and easy to follow. Minor comments:
– It might be useful to explicitly state that the “long tail” referred to in this paper is different from the long tail studied in long tail recognition (learning from imbalanced data), which typically describes the label distribution.
– Figure 2: The two lines should be labeled / caption should clarify what each color corresponds to
– Typo on L154: extra a

__ Relation to Prior Work__: Yes. A few additional works that could be included in related work are:
– Gierhos et al, Shortcut Learning in Deep Neural Networks, 2020
– Toneva et al, An Empirical Study of Example Forgetting during Deep Neural Network Learning, ICLR 2019

__ Reproducibility__: Yes

__ Additional Feedback__: This paper presents an interesting analysis of memorization in deep neural networks. My only concern is the usefulness of the proposed algorithm, and would be interesting in seeing more discussion about potential applications.
POST REBUTTAL:
After reading the author response and other reviews, I am increasing my rating to accept. I think this is an interesting contribution towards understanding memorization in neural networks that will be of value to the community and stimulate future work.