Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: This is an extremely well written (articulate and readable) and diligently prepared (completely
by the book) manuscript. For example, it draws carefully on more than 100 other manuscripts.
This alone makes the manuscript valuable to other people, as it teaches how work can
be exposited and papers can be written.

The results are interesting and instructive, and in some way connect
these questions to some recent work in non-convex optimization.
The practical benefit of this work -- tens of percentage points speedups --
are meaningful.


main_review: Originality: yes this is original.
Quality: this is very well prepared as an academic work.
Clarity: this is exceptionally clear.
Significance: this paper shows that in this field,
theory can eventually make sense 
of phenomena which have been discovered
and carefully documented empirically. The
message of science progressing in this way is
worth spreading far and wide. Generally speaking theorists
working in this area have not been concerned with explaining empirically
uncovered phenomena. So this example may lead to a new
synergy.
limitations_and_societal_impact: In a few cases the paper betrays its authors' obvious theoretical 
origin. Theorists still believe that deep learning is somehow
in a problematic situation because of its reliance on empiricism and
phenomenology and the "mysteries" unearthed by empirical science.
This is simply a parochial viewpoint. It would be better for theorists to recognize that
the state of the art in deep learning is an empirical triumph and that 
it's better for theory to contribute to systematizing the various lessons that 
are being discovered by active machine learning researchers.
review_title: null

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: Neural collapse is an intriguing phenomenon that has been observed in neural network classifiers as the training error goes to 0 where the last layer of the neural network has collapsed input points to corners of a simplex, thereby minimizing variability, and maximizing separability. While this phenomenon seems well studied the authors extend it by studying the critical points of this optimization geometrically by relating it to a convex problem. They show in Thrm 3.2 that the critical points are either global minimal or saddle points with negative curvature which can be escaped by a minimizer. 

main_review: This work builds incremental, but important advances on what seems to be well-investigated theory regarding deep neural networks trained for K-class classification via a cross entropy loss, neural collapse. Neural Collapse was first reported in [1, 16] Key parts of the theory include the unconstrained feature model which involves analyzing only the last layer , and relating the non-convex optimization to a convex problem to characterize the minima. These parts of the theory have all been introduced and used in various works such as [23] and [30] respectively. To that this work extends these ideas a bit. They show that using [30] one can characterize the critical points as having at least one direction of negative curvature, by constructing a negative curvature direction. They then show that this that sufficiently powerful neural networks can escape these saddle points and reach a global optimum, and that this is independent of the optimization algorithm. They also propose an architectural choice of d=K in the last dimension, which is used frequently regardless. A more interesting idea is the fixing of the classifier. 

The writing of the paper however seems list-like and with a piece of theory and minor ideas for practice. Instead I would consider first explaining the previous theory fully, particularly the unconstrained feature model, which as the authors say in the supplement may be an alternative condition to the "infinitely wide condition of the NTK" for casting the neural network as a linear model in an alternative feature space. What is the intuition behind the unconstrained feature model? Does it imply that those layers are not necessary or that they can be trained randomly? Some experiments here would help as well, changing the width for example of previous layers. 

Second, the authors need to provide the background of [30] to address their geometric idea, which is in the title but not covered in the main paper. I believe this is the main contribution of the paper. This is what makes this idea practical and apply to modern DL training methods. Perhaps an additional metric would be to actually estimate the curvature of the landscape using Olivier-Ricci curvature or other data estimators of curvature would emphasize this points. See: https://www.sciencedirect.com/science/article/pii/S1631073X07004414?via%3Dihub.

Finally, I think that d=K is somewhat obvious as an architectural choice, particularly given one hot encodings commonly used in classifications. I would rather understand more about fixing the last layer. Does the exact simplex used matter? How is it enforced?
limitations_and_societal_impact: Yes
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper studies collapse phenomena of representations from neural classification models trained beyond zero training error with cross entropy. The theoretical contribution of the paper is an analysis of such phenomena in the so called unconstrained feature model. In this setting, the authors derive the global loss minimizers, and show that these can be obtained via gradient based optimization, as the critical points of the loss function are benign. These theoretical results are underpinned by experiments, which investigate the collapse phenomena for a ResNet-18 architecture when trained via different optimization algorithms or with randomly labeled data.
The theoretical results also indicate, that in practice, the last linear layer of a neural network can be fixed a priory and of low dimension.

main_review: 
## Overall assessment:

This is an interesting paper which promotes the understanding of loss minimizers and loss optimization in the supervised classification setting. There are several issues regarding the assumptions and claims made in the paper, but I think these can be cleared with a revision. Furthermore, its connection to related work is described kind of superficially in the paper. In general, this is a good submission and I am willing to raise my score, once the major issues are resolved.



### Significance/Originality:

Inspired by the observed neural collapse phenomenon in Papyan et al, there is a recent stream of research, which studies such phenomena from a theoretical perspective. Mainly, by deriving the loss minimizers under various assumptions in the unconstrained feature model. The first theoretical result in this paper is quite similar to these works. 
The second theoretical result provides a real addition to the literature. There, in contrast to previous work, the authors do analyze the critical points in the unconstrained feature model and thus take into account that neural networks are usually trained via gradient based optimization.
In summary, the paper promotes our understanding of the neural collapse phenomenon.
---

While previous work relies on norm constraints on the representations, the authors argue, that weight decay yields an implicit L2-penalty on the representations. This is a quite interesting and original assumption. The authors argue that the assumption is meaningful, but unfortunately, this was not discussed more in detail, see Quality.

---
From the paper itself, it is difficult for readers to assess its role in the literature. This is because the authors tend to simultaneously cite multiple references which are only summarized as a whole. As a consequence, most references and how they relate to the submitted work are not discussed properly, leading to superficial related work sections. In the following paragraphs, I will precise this with some examples from the text.

For example, the section related to existing landscape analysis consists mainly of the statement that the authors' "result has much broader implications than the previous results [31, 32, 35, 36, 38, 39, 85]" (line 244), however the contribution of these earlier works, and thus the reason for the, (according to the authors) limited implications of these works remains unclear. Addendum: There is a section in the supplementary material which discusses these works in more detail. The existence of this section and its most important parts should be highlighted in the main text.

The relationship to low rank-matrix recovery section suffers from similar problems. The authors write "As discussed in Section 1, it has been recently shown that the strict saddle property holds for a wide range of nonconvex problems in machine learning [66–79], including low-rank matrix recovery [74, 76, 80–83]" (line 233). (I did not find this in section 1, maybe the authors could specify the exact location of the discussion in section 1.) \
First of all, I don't think that there is much benefit in simply listing machine learning papers which study critical points of some functions. The authors should either expand on the relationship of these references to the submitted work, or remove them.\
Furthermore, this list ([66–79]) is far from complete.  To name a few additional references, critical points were also studied for residual networks, see [5'] and [6']. There is also the work by Kawaguchi, which the authors already cite (reference [32]), but not in this context. \
In fact, judging from the titles of [66–79], this list seems limited to the subarea of low rank matrix optimization and is due to mainly the same group of authors. If the point of including these references is to illustrate the existence of strict saddle points in many machine learning problems, then it is counterproductive to limit the references to a particular subfield. But as already stated, the main problem is not the lack of the references, but the superficial treatment of the provided ones.\
I am aware, that the works [66–79] are also presented in the supplementary material (starting at line 725). But even there, only the application areas of these works are named, but how these works relate to each other and the submitted work is not discussed. For example, it remains unclear, whether there is a central paper which introduced the required mathematics and the others are just applications of it, or if these works all build on each other and, if yes, how and to which extend. At the end of this paragraph in the supplementary material, two papers are highlighted; "In fact, our proof techniques are inspired by recent results on low-rank matrix recovery [73, 76]". But again, what these results entail, and how they inspired the authors' proof techniques remains unspecified.
---

I also want to highlight some related work not mentioned in the paper.

Assuming that the representations and classifier weights are constrained on a sphere and discarding bias terms, Wang et al.[1'] already showed in 2017 that if the representations are already collapsed, then the class means need to be maximally separated in order to minimize the cross entropy loss.

Closely related is also recent work [2'], which shows that in the unconstrained feature model loss minimization leads to neural collapse for models trained with cross entropy or with a supervised contrastive loss function. They assumed representations on a sphere, but allowed for weight decay. Similar to this paper, they also experimented on randomly labeled data and with fixed classifier weights. It would be worth adding a discussion about it.

Furthermore, there is [4'], in which deep neural networks are studied using a convex analytic framework. Therein, it is shown that the weight matrices align with the previous layers at optimality. This serves as an explanation of neural collapse (see Corollary 4.3 in the reference). It would be interesting to discuss similarities and differences between [4'] and this work, as convexity is central to both.


### Clarity:

The paper is well structured (global optima result -> local critical points result -> experiments) and easy to read. Good job. 
In particular, I like the bullet points after the problem formulation and theorems, which discuss the assumptions of the theorems or point to related work (As written above, the related work sections need more details).\
I also think that it was a good idea to graphically compare the works [24-26] on neural collapse in table 1. (minor remark to the authors: insert some additional white space between the caption and the table to improve readability.)
---

The proofs in the supplementary material are a bit difficult to follow. \
In general, the proof of theorem 3.1 seems to be more involved than the proof of similar results, for example in [2'], [3'].  For the reader it remains unclear, why this is the case and I would appreciate if the authors comment on this in their response to the review.\
In particular, I do not understand, why Lemma D.5 is in the form of "for any $c_1>0$, the cross entropy loss can be lower bounded by...".  As consequence,  all other intermediate results building on lemma D.5 are also of this form, which makes the proofs hard to read.
I might be wrong, but to me it seems, that $c_1$ is introduced artificially, but never really used, as one is only interested in the equality case between lines 945 and 946 anyways, and this does not depend on $c_1$.

### Quality:

The paper includes well designed experiments.\
Experiment 4.1 establishes, that neural collapse occurs independently from the optimization algorithm. Experiment 4.2 studies the unconstrained feature model assumption in greater detail. As one might already suspect, it clearly demonstrates that increasing the model capacity (measured by the width of the convolutional layers) leads to more simplex-ETF like representations. Thus confirming the unconstrained feature model assumption for highly overparametrized networks.
I really liked that the authors introduced metrics (NC1-NC4) for measuring each of the respective neural collapse properties separately.
---

However, I am not convinced, that the experiments demonstrate that the input "plays minimal influence" (line 310) for reaching neural collapse. This is simply not studied in the experiments, as they all operate on the same input data.

---
While I do not doubt the results, the experiments would definitely benefit from more statistical power. So far, only the results of a single run (in each setup) is presented, as acknowledged by the authors in the checklist. However, reporting learning curves does not automatically prohibit to report error bars. For example, one could report mean learning curves and shade the error regions.

---
The theory appears to be correct. However, I have to admit, that I was not able to check all of it thoroughly.

---
The main weakness of the paper is that its fundamental assumption (separating it from other works using the unconstrained feature model) is not justified enough.

The theory relies on the assumption, that the representations are subject to an L2-penalty in the unconstrained feature model. As the authors argue, it seems plausible, that this assumption is  meaningful, i.e. it provides a valid approximation of the representations of a neural network. However, on a closer look, it appears that this assumption is made only so that a connection to a convex optimization problem can be established via Lemma C.3. From this perspective, the validity of the this assumption needs to be discussed in greater details and to be confirmed in experiments.\
For example (other strategies are possible as well), the proof of theorem D.1 could be adapted such that optimal norms of the representations can be predicted (compare Corollary 1 in [2']). Thus, if one knows how the regularization strength $\lambda_H$ depends on the model depth and the weight decay parameter used in practice, this prediction on the norms could then be tested in experiments.

---
The authors state that they make the assumption of balanced class multiplicities in the training set due to simplicity (line 140). However, this is misleading, as this assumption is crucial for the theory (e.g. the proof of Lemma D.3 uses this assumption). If the class multiplicities are unbalanced, the loss minimizer is not a simplex ETF.


### Minor remarks:

the links within the paper do not work 

line 289: typo quantity -> quantify

The most left (and maybe also the most right) plots in figures 3 and 4 would probably benefit from a log scale. As of now, the most right plot in figure 4 is unreadable.\
In figure 4 (most left) the values t on the y-axis seems to be labeled incorrectly. What is meant there? t (i.e., as it is now), e^t, 10^t, or something else?\
I might have missed it, but please mention the width of a standard resnet-18 model for comparison.

line 185: "The requirement that d≥K is  necessary  for  Theorem  3.1  to  hold,  simply  because K vectors  in R^d cannot  form  a K-Simplex ETF if K > d."\
I don't think this is true. K vectors can still form a K-Simplex ETF if K=d+1, for example an equilateral triangle in R^2.

**References:** \
[1'] Wang et al., NormFace:l2hypersphere embedding for face verification, ACM Multimedia, 2017\
[2'] Graf et al., Dissecting Supervised Contrastive Learning, ICML, 2021\
[3'] Lu and Steinerberger, Neural collapse with cross-entropy loss, arxiv, 2020\
[4'] Ergen and Pilanci, Revealing the Structure of Deep Neural Networks via Convex Duality, ICML 2021\
[5'] Yun et al., Are deep ResNets provably better than linear predictors?, NeurIPS 2019\
[6'] Hardt and Ma, Identity Matters in Deep Learning, ICLR 2017

limitations_and_societal_impact: I agree with the authors, that there is no negative societal impact to be expected from this work. As acknowledged by the authors, the work is limited to the unconstrained feature model.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper provides a landscape analysis for neural collapse with regularization analysis. The theoretical results are nice, thus I voted an accept decision. But reviewer has a concern about the novelty of the proof and the implications of the theoretical results.
main_review: this is a nice paper analysis the landscape of the neural collapse model, The main concern reviewer have are following
[1] The novelty of no strictly bad local minimum:
The analysis is actually the same as the landscape analysis of the matrix decomposition model in my mind. The paper deal with the problem as a convex_function(WH)+\lambda ||W||^2_F +\lambda ||H||^2_F where the nonconvexity is introduced by WH. I wonder whether the landscape results are the same as the previous matrix decomposition paper

[2] The novelty of the proves of the unconstrained model. I wonder can we use the Lagrange dual relationship turns the global minimum of the unconstrained one to a constrained one. In practice, no one use regularization on features, I still can't see the benefit and novelty to turn the constrained model to a unconstrained one.

[3] The implications of the theoretical results are separated from the theoretical results in the paper. The "new method" you can do that even after the first neural collapse paper and the "new" methods can also be understood as changing a loss function[using target feature as loss] and this leads to question 5.

[4] From the theoretical result, can we say the layer before the last layer is also neural collapse? I wonder will the proof stays true while we inductive on the depth.
limitations_and_societal_impact: see detailed review
review_title: null

Meta-review: This paper provides an interesting well presented study of neural collapse, mostly focusing on solid theoretical results, but also demonstrating their empirical implication. All reviewers agree the paper should be accepted, commending the high quality of the presentation in the paper. Further, it is clear from the reviews and the paper that this is an important problem, and the results are sufficiently significant to be of interest to a wide audience at NeurIPS (as evidence, two reviewers marked the paper as top 50% of accepted NeurIPS papers with high or absolute confidence). Therefore, I recommend acceptance as a spotlight.

To the authors: please read carefully the reviewer comments and follow up on your responses with appropriate revisions to the paper.
Decision: Accept (Spotlight)