Paper: Unraveling the Enigma of Double Descent: An In-depth Analysis through the Lens of Learned Feature Space (ICLR-2024)

Reviewer comment (Novelty):
One main weakness is the fact that a big part of the paper is devoted to reproducing experiments from [1]. In these experiments, the only significant difference is the fact that they report both accuracy and cross entropy loss.

Author response:
We did replicate the experiment setups from [1] as we stated in our Experiment Setup section. Nonetheless, our primary focus in research is on the K-NN prediction test carried out on data with noisy labels. Our conclusions are drawn through a comparative analysis involving the interplay between the double descent phenomenon and prediction accuracy across varying levels of introduced label noise. Embracing a foundational and validated experimental setup proves advantageous for facilitating straightforward comparisons and validating our findings.

Reviewer comment (Evaluation):
I fully agree with the author’s comment on section 3 regarding tSNE maps: “While this visualization may not accurately represent the intricate inherent structure of high-dimensional features, it aims to gain insights into the clustering and distribution of data points, thereby enhancing our understanding of the model’s internal representations.” While tSNE maps might give intuition about a phenomenon, they can also fabricate patterns and might not be sufficient evidence for a proposition.

Author response:
We agree that the t-SNE experiments lack persuasiveness and have only a loose connection to our primary hypothesis. We have now reallocated the experiments to the appendix.

Reviewer comment (Writing):
The writing is sometimes confusing. For instance, in section 3.3: “we calculate the prediction accuracy Kp of noisy labelled data of the original clean labels for each noisy labeled data point matching the label prediction of its k-nearest neighbors in the learned feature space”.

Author response:
We have also strengthened the descriptions of our methodology, which includes a detailed explanation of how our hypothesis is founded, why our experiment can be used to validate our hypothesis, and re-formulated our mathematical description.

Reviewer comment (Novelty):
Thus, I do not see substantial empirical nor theoretical contributions in this work.

Author response:
Our distinctive contribution lies in illuminating the influence of model parameters on the strategy and effectiveness of learning representations. Furthermore, our study unveils the process by which over-parameterized models discern and effectively "isolate" label noise within the feature space. These are previously undocumented phenomena that have not been extensively explored or discussed in the existing body of literature.

Reviewer comment (Evaluation):
The abstract of the paper states that the main contribution is to demonstrate that model-size double descent is happening only with label noise present in the dataset. Nevertheless, the experiment with CIFAR100 in the paper directly disproves this claim already.

Author response:
A1: In the abstract, we stated that "its occurrence is strongly influenced by the presence of noisy data.". Speaking of the CIFAR-10 experiments, "We suppose that this phenomenon may arise from the intrinsic noise and increased complexity associated with image recognition of CIFAR-10 in contrast to MNIST.". I believe we have already addressed and explained your first concern in our paper. 

Regarding A1, pervasive label noise has been present in commonly utilized datasets, with validation studies revealing higher (label) error rates in CIFAR-10 compared to MNIST~[2]. Furthermore, upon closer examination of the double descent plot for MNIST trained without label noise, a noticeable alteration in the shape of the test error curve is observed at the interpolation threshold — a discernible rise. We contend that it is not the case that double descent occurs on CIFAR-10 but not on MNIST; rather, the distinction lies in the level of the interpolation peak, and this difference is due to the different label error rates.

Reviewer comment (Theory):
It is claimed that double descent will not happen in correctly regularized networks, but no regularization is ever discussed in the paper itself, except for an unusual claim that overparameterization is a form of regularization, which is also never discussed in the paper.

Author response:
A2: We reach the conclusion that double descent should not occur in well-regularized models from referencing previous research and a derivation of our analysis of imperfect learners and noisy data. This conclusion line is now removed from our paper.

Reviewer comment (Evaluation):
Given the above, the only possible contribution left is explaining how interpolating networks deal with mislabeled examples. This is aimed to be explained through nearest neighbors classification on the representations for mislabeled examples—showing that this accuracy is very high for interpolating models. Nevertheless, only the MNIST experiment demonstrates high accuracy of these predictions together with high accuracy on the noisy task; in other experiments this accuracy is considerably low.

Author response:
A3: Our explanation of how interpolating networks deal with mislabeled examples is demonstrated through the aligned trend of test accuracy and the percentage ${\rm P}$ (previously denoted as ${\rm Kp}$), instead of a high value. Although the accuracy of predictions on CIFAR-10 is relatively modest, it's worth noting that the performance of CNNs and ResNets on this dataset is also suboptimal. Considering the challenge of a 10-class classification problem, achieving nearly 60\% accuracy is commendable, and the comparison between models of varying widths still yields conclusive results.

Speaking of A3, the trend in development is enough to make the conclusions and validate our hypothesis. Drawing a comparison between the actual test accuracy and prediction accuracy (expressed as a percentage ${\rm P}$) is meaningless when label noise is absent, as the context for a k-NN comparison becomes non-existent. Our objective is to discern the correlation between the enhanced double descent phenomenon and the model's interpolation strategy concerning introducing label noise when explicit label noise is introduced.

For A3, as the exact distribution of data and their representations is uncertain, the k-NN method can only serve as an estimation technique. For instance, the k-NN prediction may perform inadequately on outliers that lack nearby neighbors. The distribution of neighbors surrounding each noisy labeled data point will also influence the outcome, thereby impacting the neural network's interpolation result. The exact reason behind the different circumstances required further investigation. We have stated the property of the k-NN prediction more clearly in the Methodology section.

Reviewer comment (Evaluation):
The t-SNE visualizations can be simply explained by the ability of larger models to get more distinguished representations for different classes, but the positioning of the noisy labeled ones does not change much (see class 0 in fig. 2 for example). Thus, the last contribution is not significantly supported by the results.

Author response:
A4: We acknowledge that the t-SNE experiments lack persuasiveness and have only a loose connection to our primary hypothesis. We have now reallocated the experiments to the appendix.

Reviewer comment (Writing):
Please check the grammar and typos in the paper.

Author response:
Minor corrections regarding grammar, typos, and formatting of titles and citations have been adopted. Equation 1 has been reformulated to enhance clarity and understanding.

Reviewer comment (Writing):
Please improve the explanation of formula (1), as it is currently extremely convoluted.

Author response:
Minor corrections regarding grammar, typos, and formatting of titles and citations have been adopted. Equation 1 has been reformulated to enhance clarity and understanding.

Reviewer comment (Presentation):
The font of the title does not correspond to ICLR style.

Author response:
Minor corrections regarding grammar, typos, and formatting of titles and citations have been adopted. Equation 1 has been reformulated to enhance clarity and understanding.

Reviewer comment (Writing):
What is the main goal of the research done?

Author response:
A5: Our distinctive contribution lies in illuminating the influence of model parameters on the strategy and effectiveness of learning representations. Furthermore, our study unveils the process by which over-parameterized models discern and effectively "isolate" label noise within the feature space. These are previously undocumented phenomena that have not been extensively explored or discussed in the existing body of literature.

For A5/A6, the better generalization ability of over-parameterized models has been documented through empirical studies showcasing the occurrence of the double descent phenomenon. The primary objective of this study is to unravel the underlying mechanisms driving this phenomenon, particularly exploring its association with label noise in the form of noisy labeled data. We demonstrated that over-parameterized models, when contrasted with under-parameterized ones, exhibit a more precise probability of predicting correctly in the vicinity of mislabeled data points within the training set; While the exact training point are still predicted with the incorrect label by these fully-interpolate models. 
We posit that this phenomenon has not been previously documented and could serve to elucidate the internal mechanism of the double descent phenomenon.

Reviewer comment (Evaluation):
What is the observation made with respect to the selected metric (nearest neighbors) from the double descent plots?

Author response:
A6: In all of our experiments, the trajectory of the ${\rm P}$ curve (denoted as ${\rm Kp}$ previously) is in accordance with the pattern observed in the test accuracy curve, indicating a consistent alignment between the two. This alignment exists both before and after the interpolation threshold. This observation underscores a statistical correlation between the local structures within the learned representations of noisy training data and the overall performance in generalization. This validation serves to affirm the accuracy of our hypothesis.

For A5/A6, the better generalization ability of over-parameterized models has been documented through empirical studies showcasing the occurrence of the double descent phenomenon. The primary objective of this study is to unravel the underlying mechanisms driving this phenomenon, particularly exploring its association with label noise in the form of noisy labeled data. We demonstrated that over-parameterized models, when contrasted with under-parameterized ones, exhibit a more precise probability of predicting correctly in the vicinity of mislabeled data points within the training set; While the exact training point are still predicted with the incorrect label by these fully-interpolate models. 
We posit that this phenomenon has not been previously documented and could serve to elucidate the internal mechanism of the double descent phenomenon.

Addressing your last inquiry, it is indeed a valid question whether we should compare the noisy labeled data to its counterparts in the training set or the test set. We contend that this choice does not markedly impact the validity of our hypothesis because the neighbors only serve as a convenient means of characterizing local structures in the high-dimensional space. As supporting evidence, we conducted empirical studies in both scenarios (initial submission on the training set; and revision on the test set), and the observations consistently align. Nevertheless, the prediction accuracy on the training set typically exceeds that on the testing set, often reaching full interpolation, and how interpolating networks interpolate the training data is surely more interesting. We have reverted to the initial submission version, incorporating experiment results comparing the training set, and modified the methodology description.

Reviewer comment (Writing):
The methodology section of this paper (only 14 lines of text and only 6 lines of description of the actual methodology) is poorly described and lacks math-related descriptions, making it difficult to accurately understand the author's intention.

Author response:
Thank you for your kind advice, we have now improved the methodology with a more complete description of how our hypothesis is founded, why our experiment can be used to validate our hypothesis, and re-formulated our mathematical description. Some paragraph is cited below: 
``Based on the existence of benign over-parameterization, we assume that test images, akin to training images mislabeled, are more likely to be correctly classified by over-parameterized models. For instance, we anticipate that an optimal classifier should yield accurate predictions for unseen images, even if it was trained on similar images with an adversarial label. Thus, we further hypothesize that the closest neighbours of mislabeled training images are correctly classified. ... We calculate the percentage ${\rm P}$ of mislabeled training data, and the majority of its nearest neighbours in feature space are in the same class ..."

Reviewer comment (Experiments):
Is it optimal to use K-nearest neighbors here? The feature space is probably in a high-dimensional manifolds, have the authors considered more complex cases?

Author response:
The application of a K-nearest neighbors (KNN) algorithm in a high-dimensional feature space can be reasoned based on several considerations. In a high-dimensional feature space, the notion of proximity becomes more nuanced, and traditional distance measures may lose their discriminatory power. KNN, as a non-parametric and instance-based algorithm, relies on the concept of similarity between data points (we use cosine similarity as our measurement tool). While we are trying to infer and observe local structures, KNN remains an effective tool for exploratory analysis.

Reviewer comment (Experiments):
I don't think this article convinces me enough about the mechanism by which it reveals that over-parameterization enhances generalization. The mechanism between over-parameterization and generalization cannot be revealed in depth only by the loss curves and T-sne images under a small number of experiments.

Author response:
Our explanation of how interpolating networks deal with mislabeled examples is demonstrated through the trajectory of the P curve (denoted as Kp previously) in accordance with the pattern observed in the test accuracy curve, indicating a consistent alignment between the two. This alignment exists both before and after the interpolation threshold. This observation underscores a statistical correlation between the local structures within the learned representations of noisy training data and the overall performance in generalization. This validation serves to affirm the accuracy of our hypothesis.

Reviewer comment (Novelty):
It is well recognized that over-parameterization is beneficial, especially since recent large models have similarly benefited from over-parameterization, and the authors should discuss the advantages of this paper in the context of recent over-parameterization research.

Author response:
We have added discussions of recent over-parameterization research in the related works section. Since we haven't encountered comparable hypotheses in previous literature, our distinctive contribution within the realm of over-parameterization research lies in illuminating the influence of model parameters on the strategy and effectiveness of learning representations. Furthermore, our study unveils the process by which over-parameterized models discern and effectively "isolate" label noise within the feature space. These are previously undocumented phenomena that have not been extensively explored or discussed in the existing body of literature.

Reviewer comment (Experiments):
As a non-expert, I interpret the experiments as showing that double descent phenomena appear with label noise in large models. I acknowledge that this is an interesting observation, but it does not seem sufficient to justify conclusions such as double descent not appearing in well-regularized models. I can see the intuition, but, in my non-expert view, the experimental data seems insufficient.

Author response:
We reach the conclusion that double descent should not occur in well-regularized models from referencing previous research and a derivation of our analysis of imperfect learners and noisy data. This conclusion line is now removed from our paper.

Reviewer comment (Experiments):
The paper considers only a small selection of data sets and models. It is not clear to me if such is sufficient to draw conclusions.

Author response:
Secondly, the paper delves into a limited array of datasets and models. This deliberate selection is based on the recognition that larger and more intricate datasets inherently introduce additional noise, potentially interfering with the precision of our experiments. Our focus has been on conducting empirical studies within foundational experiment setups in deep learning. We assert that these chosen scenarios not only align with widely accepted practices but also serve as fundamental benchmarks for validating deep learning techniques.

Reviewer comment (Experiments):
The paper does not provide a theoretical explanation (which is fine, but then I had wished for a wider selection of experiments).

Author response:
We contend that the empirical studies conducted in our research encompass various fundamental experiment setups in deep learning, specifically addressing real-world classification problems. Artificial data may lack sufficient persuasiveness, as the intricacy and noise inherent in more challenging tasks, as well as state-of-the-art methods, can significantly impact the actual structures of the feature space, introducing numerous uncontrollable factors. Our distinctive contribution lies in illuminating the influence of model parameters on the strategy and effectiveness of learning representations. Furthermore, our study unveils the process by which over-parameterized models discern and effectively "isolate" label noise within the feature space. These are previously undocumented phenomena that have not been extensively explored or discussed in the existing body of literature.

Reviewer comment (Presentation):
I found it very difficult to read the t-SNE plots.

Author response:
We acknowledge that the t-SNE experiments lack persuasiveness and have only a loose connection to our primary hypothesis. We have now reallocated the experiments to the appendix.

Reviewer comment (Writing):
What is the 'wedge product' in Eq. 1?

Author response:
Minor corrections regarding grammar, typos, and formatting of titles and citations have been adopted. Equation 1 has been reformulated to enhance clarity and understanding.