How Classifiers Extract General Features for Downstream Tasks: An Asymptotic Analysis in Two-Layer Models

Attachment link: anonymous.4open.science/r/icmlrebuttal-3B6F

Thank you for recognizing the theory as standard and for positively evaluating the experiments. We understand that you wanted us to clarify the relationship between the relevant studies and ours in order to highlight the novelty.

Our work focuses on the feature transfer of networks trained via classifiers, particularly in settings akin to metric learning, which has previously been used for performance without understanding the underlying principles. In contrast to many previous studies that analyze on feature learning phenomenon of training data or test errors.

For theoretical contributions, we agree as you mentioned, “classification scenarios should inherently share similar insights, making results transferable”. However, this adoption is challenging due to difficulties in dealing with multiple non-identical and arbitrarily labeled class conditional distributions. Such modifications on the problem setup require the more generalized assumption on data distribution, i.e., a non-centered Sub-Gaussian. Thus we provide mathematical tools to analyze non-centered sub-Gaussian distribution with an Hermite expandable activation, which are novel in the existing works (Section L, M). These technical contributions generalize the previous data assumption, and we believe future research can build on ours.

First, we reviewed the requested studies (Wang23, Fu24, Damian22, Dandi24) and noted them by the first author and year.

Note on Damian 22, Wang 23, Fu 24

These studies analyze the sample complexity for neural networks to learn the internal structure while regressing a teacher in the form of $g(x^\top Ax)$ where this form becomes progressively more complex (here, $x$ comes from a centered distribution.). These studies focus on regression problems using a teacher, whereas we assume only arbitrarily assigned classification labels without using a teacher. At the same time, the transfer learning part analyzes the sample complexity for learning when function head $g$ is changed while maintaining the internal structure. In contrast, we conduct research on feature transfer when new distribution inputs are introduced without additional learning.

Note on Dandi24

Dandi24 analyzed the learned feature extractor using an equivalent model to characterize the test error for regression and analyze the spectral tail behavior of the covariance matrix of the feature extractor. This study dealt with the phenomenon where spikes appear in the spectrum, making the tail heavy-tailed. Similarly, we derive an equivalent model suited to our classification setup and analyze the characteristics of clustering error for unseen distributions (Section 4.1). Then, we analyze how the spike term of the feature extractor operates (Section 4.2).

Additionally, we have provided a comparison table in Table 2 of the above anonymous link, which we plan to include in the Appendix to offer further explanation to the readers. We sincerely appreciate the references you shared.

Secondly, regarding the existing research on classification and transfer learning, we respond as follows

We’ve already covered existing studies on feature transferability in the Additional Related Works section, which focuses on intuitive explanations of feature transfer (L662) and feature learning phenomena, Neural Collapse, mainly for training data (L672). Since you reviewed only the mathematical part of the Supplementary Material, we suggest reviewing Section A.

Some additional classification studies were not cited as they either differ from our framework or focus mainly on training data. However, we plan to discuss them in our paper for readers' understanding.

For example, there are papers addressing the alignment phenomenon between the network and the training data in neural network classification tasks (arxiv.org/abs/2307.12851), studies on the increased separability and cohesion of training data features in classification settings (arxiv.org/abs/2012.10424, arxiv.org/abs/1909.06930), and research showing that classifier networks outperform linear classifiers through feature learning (arxiv.org/abs/2206.01717, arxiv.org/abs/2202.07626, arxiv.org/abs/2102.11742).

Additionally, there are empirical studies on classifier transfer learning (not feature transfer like ours, which doesn’t require learning) (arxiv.org/abs/2212.12206) and theoretical studies (arxiv.org/abs/1809.10374, arxiv.org/abs/2006.11650).

Furthermore, there are theoretical papers studying optimization properties like implicit margin maximization (arxiv.org/abs/2110.13905, arxiv.org/abs/2305.11788) and interpolation (arxiv.org/abs/2012.02409, arxiv.org/abs/2306.09955).

Finally, we will standardize spike_L to s_L.

Once again, we truly appreciate your thoughtful review and constructive suggestions, and if you have any additional concerns, let us know. We will do our best to address your concerns.

Attachment link: anonymous.4open.science/r/icmlrebuttal-3B6F

Thank you for your comments. We also appreciate your feedback that the motivation of our work and the phenomenology are very interesting.

After reviewing your comments, we found that you suggested improvements for readability, understood the paper's core focus as theoretical, requested a discussion on its relation to certain references, and faced difficulties in comprehending the proof section.

Through this rebuttal, we aim to 1. inform you of the revisions made based on your suggestions, 2. clarify that our contributions beyond theory, 3. discuss the gap between our work and the suggested references, 4. assist in your understanding of our theory, and 5,6. rebuttal to some misunderstandings

1. We made the following revisions based on your feedback.

First, for readability, we fix typos you suggested. Second, for better clarity of theory, we included the main idea before the proof and the proof structure as in the above link's Table 3-8 and Fig. 4–8. Also, we placed the proofs of main Theorems and Propositions upfront, while moving the lemmas back. Third, we included references to studies that you suggested.

2. Contrary to comment on experiments that “I think this is not the main point of the paper.”, the results in Section 4 and experiments are the major contributions of our study.

This work offers intuition based on theory to data collection, with experiments on settings not explored in metric learning literature. It contributes by leveraging existing theories with empirical adaption, enhancing the understanding of feature transfer, and establish widely accepted intuitions (as in L662) on a rigorous foundation. We hope this is reflected in your evaluation.

3. As stated in the right column on L84, the research we present is not a direct extension of the feature learning literatures including your suggestions.

The studies you mentioned focus on teacher-student regression. On the other hand, our research analyzes feature transfer and clustering tasks in a classification setups. Therefore, we primarily cited baselines that are required for the proof. However, since the distribution we are dealing with is non-centered Sub-Gaussian, which generalizes to all data setups in your suggestions. Additionally, you can find the comparisons between the papers you mentioned and our approach in Table 2 of the attachment link.

4. It seems you missed the logical flow of the proof, so we summarize and clarify its implications as follows.

Lemmas (I.xx, J.xx) resemble those in Ba et al. (2022) and Moniri et al. (2024) which deal regression, but our classifier setting need to deal with multiple non-identical distributions. This requires novel proof techniques in Section I, J and a novel generalized distributions be dealt within Sections L and M. There, we develop mathematical tools for analyzing sub-Gaussian distributions and Hermite-expandable activations, extending previous data assumptions. We trust this enhances your comprehension of our contributions.

5. We address sub-Gaussian distributions rather than Gaussian mixture distributions, which is more general distribution set.

We did not use the word "mixture" in the text. Thus, could you clarify why you mentioned this in the Summary and in referencing Scientific Literature? After reviewing Demir & Dogan 2024, we found that it is contemporary work. They assume a Gaussian mixture for the data, propose a Conditional Gaussian Equivalence and stochastically approximate a 2-layer network. This might seems similar to our Theorem 3.3, but our proof does not require conditional approximates nor constructing a stochastically equivalent model since we only utilize Sub-Gaussian property. This provides an advantage in analyzing the features of new data in a simple form.

6. Regarding the rejection, we understand that it was due to a lack of clarity in the proof of the Theorem.

We would like to appreciate if you clarify which part was difficult to understand, leading to your statement, "I am not in a position to say the proofs are correct"? Your concrete suggestions mainly focused on typographical errors, the distinction between vectors and matrices, etc. We do not believe these significantly hinder the understanding. Also, We appreciate your understanding in novel analysis induce unfamiliar notations, unavoidably.

Answer for Question:

The learning rate is exactly 1. Normalization follows Moniri et al., 2024. The learning rate corresponds to the case where $\alpha = 0$ in Moniri et al., 2024 or $\Theta(\sqrt{n})$ in Cui et al., 2024 In this setup, we make $||G||, ||W_0||, ||W|| = O_p(1)$. We will add the assumption that $\eta = \Theta(1)$ for clarity.

Finally, the constant $c$ in Eq. 1 is #_cls. We will revise this.

Looking forward to your response and discussion. Thank you.

Attachment link: anonymous.4open.science/r/icmlrebuttal-3B6F

Your thoughtful feedback is a great encouragement and reaffirms our commitment to furthering this research. We understand that your primary concern lies in fairness during experimental validation.

Thus, the primary purpose of this rebuttal is to dispel any misunderstandings that this setup is unfair,and to clarify that fairness has indeed been ensured. Moreover, in response to your point about the difficulty of verifying fairness as W3, we included the experimental setup mentioned at Appendix N in the main text.

After carefully reviewing your comments, we respond to your concerns and propose the following revisions:

First, We will address the concern on W1 about the unfairness of the claim on L411, i.e., Expr V. It seems to stem from the following sentences.

L355: we performed experiments on the whole classes ImageNet by sampling $100$ instances per class (say subsampled whole In1k).

L411: We find that adding classes from the entire ImageNet dataset during training, rather than including only related classes, does not significantly improve clustering

Appendix N: To balance the number of samples per class with those in the base fine-grained datasets, we extracted $82$, $58$, $5$, and $6$ samples per class for I(V), I(B), I(P), and I(C), respectively.

We sincerely apologize for the omission of some information in L355. To be more precise, in L355, we only specified the setup for Expr. VII, which used the subsampled whole In1k. In this case, as L355, we extracted $100$ instances per class from the entire ImageNet dataset. Since this experiment is related to the case of duplicate assignment and we use subsampled whole In1k alone, the situation you were concerned about did not occur. On the other hand, for Expr. V, we did not perform sampling in the same way. We already agree with your concerns, and we took them into account when designing the experiment. In Expr. V, to ensure fairness, we extracted $82$, $58$, $5$, and $6$ instances per class from the I+D case datasets and performed the experiment accordingly. We sincerely apologize for the omission of this detail when explaining the setup.

To ensure the accuracy of this experiment, we re-inspect the code for extracting $82$, $58$, $5$, and $6$ instances per class for the D+I case using subsampled whole In1k in Expr. V (Listing 1 in the above anonymous link) and then conducted the experiment again. As a result, we obtained nearly identical performance to the original results (with only minor performance variations due to seed differences, which had no impact on the claims). This can be confirmed in Figure 1 and Table 2 in the above anonymous link.

Secondly, you raised a concern on W2 that the ImageNet setup may not sufficiently support the claim that "adding semantically relevant classes to the training set leads to performance gains" However, there is a misunderstanding regarding this point. We presented this claim in the L90 and left column of L414, and the experiment that supports this claim is Expr. VI. We designed Expr. VI based on the same reasoning as yours, which is why we do not make the above claim by adding similar classes using a subset of ImageNet.

Consequently, as stated in L430 and as you suggested, in Expr. VI, we conducted experiments using $25$%, $50$%, $75$%, and $100$% of the domain dataset classes exclusively i.e. the four fine-grained datasets. As a result, we observed a trend where adding classes within a dataset led to performance improvements, which formed the basis of our claim. However, we have realized that this claim was not explicitly stated in the explanation of Expr. VI. To reduce any potential misunderstandings, we will explicitly add the statement "adding semantically relevant classes to the training set leads to performance gains" in L429, based on Expr. VI. Additionally, we will clarify that the domain datasets used are fine-grained datasets such as CUB, CAR, SOP, and ISC.

I hope this answer will help you understand this study better and please continue to rate it positively.

Thank you.