Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Model

Summary

The paper introduces an extension to the standard unconstrained features model in neural collapse theory by incorporating multiple layers above the unconstrained features. The study demonstrates that, under certain conditions and at optimality, all of the top layers exhibit all properties of neural collapse.

Strengths

-- The paper exhibits excellent writing and effectively surveys the existing literature. -- The paper contributes to a promising direction in the theory of neural collapse by presenting a significant extension, and it conducts a non-trivial theoretical analysis that offers compelling evidence for the occurrence of neural collapse at intermediate layers. This extension aligns with previous empirical findings, verifying the theoretical predictions made in earlier studies.

Weaknesses

-- Like other papers in the field, the rationale behind asserting that SGD converges to a solution satisfying neural collapse solely based on optimality is not entirely clear. It would greatly benefit the paper if the authors could provide further clarification and expand upon this aspect. -- While the paper focuses on the binary classification case, which is acceptable, it would be valuable to discuss the feasibility and potential challenges of extending the analysis to a multiclass setting. -- I believe that expanding on the implications and interpretations of the results would significantly improve the paper. It would be beneficial for the authors to discuss how these findings relate to aspects like generalization and what implications they have for our understanding of deep learning in general.

Questions

See weaknesses.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

There are no negative societal issues with this paper.

Summary

The paper studied Deep Neural Collapse (DNC) which extends the structure of Neural Collapse of the last layer to multi-layers in deep neural networks. Theoretically, this paper generalizes the established unconstrained features model to the deep unconstrained features model (DUFM) of multi-layer non-linear models for binary classification. Based on the deep unconstrained features model, they show the global optimum exhibits several extended properties. Empirical experiments demonstrate that the results of the optimized deep unconstrained features model and trained networks agree with the theory.

Strengths

1. The paper generalizes the Neural Collapse to Deep Neural Collapse which describes the structure of the feature other than the last layer. The proof based on the deep unconstrained features model for binary classification is clear and easy to follow. It gives a framework to find and prove the feature structure of other layers in DUFM. 2. The extensive experiments and numerical results shows that the theory successfully predicts the behavior of the deep unconstrained features model(DUFM) which exhibits proposed DNC properties. 3. The paper is well written, it is smooth to follow the whole logic and the idea of the method.

Weaknesses

1. Though the model and proof consider the Deep Neural Collapse which works for not only features of the last layer, the model and the proof are restrictive which works for binary classification and no bias case. Many proofs and derivations rely on binary classification like Lemma 5 - Lemma 7 which is the key to giving the conditional optimum of previous layers. 2. In addition, the DUFM adds a regularization term for the first layer feature which is not fit with the real training of neural networks to some extent. 3. Compared to the numerical results of DUFM, the experiments for real training of neural networks are limited to the 3-layer Resnet20. If there are more results for the training of other neural networks it will make the experiments more convincible. 4. There are some typos in the proof, like L517 in the appendix where I think the $\sigma_1^2$ should be $s_{L, i}^2$.

Questions

1. Most previous research[1,2, 3] focus on models with bias while in [3] they also study the unconstrained features model without bias and show the features exhibit OF rather than STF under UFM with bias. So I am just wondering if it is possible to extend the DUFM to the bias case and show the STF structure in DNC2? 2. There are some experiments results for 3-layer Resnet-20 in the paper which also show DNC, but in [4] the results seem to show that the Neural Collapse process gradually across the layers of the neural network like the measure of NC1 decreases exponentially. I think this may be caused by the gap between the DUFM and real trained networks. Are there more experiments on real trained networks and whether they exhibit DNCs? If so, why is there such a difference between the results in this paper and the results in [4]. If not, is that mean there is a gap and needs other explanations for such a phenomenon? 3. All experiments based on DUFM in this paper add the regularization term of the first layer feature. By the derivation of the DUFM, it seems that possible to add a regularization term on the intermediate layer feature, for example, add $||H_5||_2$ and optimize $W_l, l \ge 5$, are there any experiment results can demonstrate in this case how the DNC performs? Reference: [1] Zhu, Z., Ding, T., Zhou, J., Li, X., You, C., Sulam, J., & Qu, Q. (2021). A geometric analysis of neural collapse with unconstrained features. Advances in Neural Information Processing Systems, 34, 29820-29834. [2] Fang, C., He, H., Long, Q., & Su, W. J. (2021). Exploring deep neural networks via layer-peeled model: Minority collapse in imbalanced training. Proceedings of the National Academy of Sciences, 118(43), e2103091118. [3] Tirer, T., & Bruna, J. (2022, June). Extended unconstrained features model for exploring deep neural collapse. In International Conference on Machine Learning (pp. 21478-21505). PMLR. [4]He, H., & Su, W. J. (2022). A law of data separation in deep learning. arXiv preprint arXiv:2210.17020.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

4 excellent

Contribution

3 good

Limitations

The framework and the proof relies on the binary classification and no bias case which make the DUFM not so correlated to the real trained networks.

Summary

The paper proposes a novel approach to investigating deep neural collapse in deep neural networks by presenting a generalization of the analytical framework for neural collapse (NC) to multiple non-linear layers. The paper introduces the deep unconstrained features model to demonstrate that the unique global optimum of the MSE loss for binary classification exhibits all the properties typical of deep neural collapse. However, the paper only considers the MSE loss without a bias term for binary classification, which may limit the applicability of deep neural collapse for other losses. Specifically, the theoretical conclusions may only be provably optimal in some cases.

Strengths

- The paper presents a generalization of the unconstrained features model to multiple non-linear layers, filling a gap in the existing literature. - The paper contributes to a better understanding of deep neural collapse in binary classifcation deep neural networks, which has important implications for the design and optimization of deep learning models.

Weaknesses

- The theoretical analysis only involves binary classification, and it is unclear whether the proposed approach can be extended to multi-class classification problems, especially for a large number of classes that would change the final form (DNC2) of the resulting deep neural collapse. - The paper only considers the MSE loss without a (last-layer) bias term, which may limit the applicability of deep neural collapse for other losses. Specifically, benifiting from the MSE loss without the bias term, the unique global solution can exhibit close-formed results, which may further allow the derivation of the proof to be greatly simplified. - while neural collapse induces a simplex equiangular tight frame [1], which may not hold for other loss functions or if a bias term is included [2]. - The paper lacks extensive experiments to validate deep neural collapse. Additional experiments on a variety of datasets (especially for a large number of classes) and deep neural networks would help to better understand and confirm the prevalence of deep neural collapse. - The paper could benefit from a more thorough discussion of the practical implications of the proposed approach for the design and optimization of deep learning models. [1] Vardan Papyan, X. Y. Han, and David L Donoho. Prevalence of neural collapse during the terminal phase of deep learning training. In *Proceedings of the National Academy of Sciences (PNAS)*, volume 117, 2020. [2] Jinxin Zhou, Xiao Li, Tianyu Ding, Chong You, Qing Qu, and Zhihui Zhu. On the optimization landscape of neural collapse under mse loss: Global optimality with unconstrained features. In *International Conference on Machine Learning (ICML)*, 2022.

Questions

see weaknesses.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

NA

Summary

This paper introduces the concept of deep neural collapse (DNC), extending the understanding of neural collapse to earlier layers of deep neural networks. It proposes the deep unconstrained features model (DUFM) as a theoretical framework to analyze DNC. The authors demonstrate that for binary classification, DNC is a globally optimal solution in the DUFM, providing rigorous evidence of its occurrence in multiple layers. Numerical experiments confirm that gradient descent efficiently finds solutions in line with the theory, exhibiting neural collapse in multiple layers. The paper concludes by highlighting open problems for future research, such as generalizing the analysis to multiple classes, exploring gradient dynamics in the DUFM, and understanding the impact of biases and loss functions on the collapsed solutions.

Strengths

$\bf (1)\ \textbf{Strong Technical Analysis}:$ A key strength of this paper is its strong and rigorous technical analysis. The authors provide a solid theoretical framework, specifically the deep unconstrained features model (DUFM), to analyze and understand the phenomenon of deep neural collapse (DNC). Theorems and empirical evidence are presented to support their findings, enhancing the credibility and reliability of the results. $\bf (2)\ \textbf{Extension to Multiple Non-linear Layers:}$ This paper addresses a crucial gap in the existing literature by extending the analysis of neural collapse beyond the last layer to multiple non-linear layers. By generalizing the unconstrained features model, the authors provide insights into deep neural collapse (DNC) occurring in deeper layers of neural networks. This extension contributes to a more comprehensive understanding of the behavior of deep networks during training. $\bf (3)\ \textbf{Empirical Validation:}$ The paper provides empirical evidence to validate the theoretical framework. The authors demonstrate that by optimizing deep unconstrained feature models using gradient descent, the resulting solutions align well with the theoretical predictions. This empirical validation reinforces the credibility of the proposed modeling principles and their applicability in real-world scenarios. $\bf (4)\ \textbf{Relevance and Contribution to the Field:}$ The paper addresses a significant topic of interest in the field of deep learning by focusing on the phenomenon of neural collapse in deep neural networks. By providing theoretical insights and empirical evidence, the paper contributes to the existing body of knowledge and expands our understanding of the behavior of deep networks during training.

Weaknesses

$\bf (1)\ \textbf{Theorem 3 cannot explain the progressive neural collapse:}$ In Ref [8], the authors observe that each layer roughly improves a certain measure of data separation by an equal multiplicative factor. A similar observation is depicted in Figure 1 of the current paper. However, Theorem 3 contradicts this observation by demonstrating that for all layers $l \ge 2$, the within-class variability of features becomes zero, and the features form an orthogonal matrix. This discrepancy suggests that there may be additional factors or dynamics at play in the behavior of the layers beyond what is captured by the initial observation. Further investigation is necessary to reconcile these contrasting findings and gain a more comprehensive understanding of the behavior and properties of features in deep neural networks. $\bf (2) \textbf{Role of activations:}$ One weakness of the paper is the limited discussion or exploration of the specific role and impact of activations in the context of deep neural collapse (DNC). While the paper acknowledges that DNC1 and DNC2 hold for layer l ≥ 2 before and after activations, it does not provide an in-depth analysis or explanation of how the activations influence or interact with DNC. $\bf (3) \textbf{Limited Scope:}$ The paper primarily focuses on deep neural collapse (DNC) in the context of binary classification. While this specific focus allows for in-depth analysis, it may limit the generalizability of the findings to other problem domains or network architectures. Further exploration of DNC in more diverse settings could strengthen the paper's applicability. $\bf Typos:$ (1) At line 122, it should be $W_L \in \mathbb{R}^{K\times d_L}$.

Questions

$\bf 1.$ The results in this paper are based on the L-layer deep unconstrained features model. This omits all the structures in the data input. A natural question is whether the analysis can be extended to the case where the data input has some special structures, such as the case where the data is whitened, i.e., assuming $H_1H_1^T=I$ in Definition 1. $\bf 2.$ What is the main technical difficulty in analyzing multiple-class cases? $\bf 3.$ What is the role of the activations in deep neural networks? Why do DNC1 and DNC2 hold for layer l ≥ 2 before and after activations?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

Not applicable