Title: Flatness is Necessary, Neural Collapse is Not: Rethinking Generalization via Grokking
Venue: NeurIPS 2025
Accepted: yes
Keywords: Generalization

flatness

neural collapse
Abstract: Neural collapse, i.e., the emergence of highly symmetric, class-wise clustered representations, is frequently observed in deep networks and is often assumed to reflect or enable generalization. In parallel, flatness of the loss landscape has been theoretically and empirically linked to generalization. Yet, the causal role of either phenomenon remains unclear: Are they prerequisites for generalization, or merely by-products of training dynamics? We disentangle these questions using grokking, a training regime in which memorization precedes generalization, allowing us to temporally separate generalization from training dynamics and we find that while both neural collapse and relative flatness emerge near the onset of generalization, only flatness consistently predicts it. Models encouraged to collapse or prevented from collapsing generalize equally well, whereas models regularized away from flat solutions exhibit delayed generalization, resembling grokking, even in architectures and datasets where it does not typically occur. Furthermore, we show theoretically that neural collapse leads to relative flatness under classical assumptions, explaining their empirical co-occurrence. Our results support the view that relative flatness is a potentially necessary and more fundamental property for generalization, and demonstrate how grokking can serve as a powerful probe for isolating its geometric underpinnings.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies NN generalization through the lens of neural collapse, grokking, and landscape flatness perspective. While all three concepts were previously observed to have links to the model's generalization, the paper offers a novel perspective on how these three concepts are related to each other and can potentially be used to diagnose the model's generalization. The work highlights that these concepts should rather be treated as diagnostic tools and not directly as quantities one would like to maximize in order to increase the model's performance.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n1. The whole work focuses on a single, well-defined topic. Given that the text is clearly written and structured, it's an easy-to-follow argument.\n2. Generalization is one of the most fundamental concepts of ML; thus, the motivation to better understand its roots or limitations is always well justified. \n3. The idea to combine seemingly related concepts (Neural Collapse, Grokking, Flatness), which are usually studied in a separate way, is an interesting approach worth studying. I'd like to point out that an interesting work with a similar approach that would be worth citing has been published at the recent ICML conference: https://arxiv.org/abs/2502.21009\n\n## Weaknesses\n1. ### Insufficient experiments on grokking, NC, and flatness \nWhile the paper's claim to \"disentangle the [Neural Collapse and flatness] contributions to the model performance by leveraging the grokking setting\" is an interesting idea, the only experiment that combines these three concepts is presented in Figure 1 and performed on a modular arithmetic task. I don't fully agree with the authors that Figure 4 represents a \"delayed generalization\" setting, as initially, both training and validation accuracy grow. Thus, while interesting, I'm not convinced that the hypothesis about grokking/NC/flatness relationship is supported by sufficient experiments. \n2. ### Imprecise claims\na) From the very beginning, the work claims the connection between NC and other phenomena. While authors note that NC consists of 4 different conditions (Section 3), in reality, the authors always check a simplified NCC condition (Definition 3.1) and argue that \"NCC measure correlates with the angular separation of means, **thus capturing all the characteristics of NC**\". First, the correlation is again evaluated only on a single, rather simple experiment (CIFAR-10, ResNet-18), which is not sufficient in my opinion. Second, I don't see how NCC + angular separation imply **all the characteristics of NC**, could the authors please elaborate on this? If that's true, it needs a justification; otherwise, I ask the authors to ease the tone on the connection of NC, as this is not a fully supported claim. \nb) As written above, I do not fully agree with the author's statement that the experiments presented in Figure 4 present the grokking phenomenon, which is related to \"delay generalization\". In these experiments, the generalization occurs at the beginning, but due to the regularization, it is diminished. \n3. ### Neural Collapse is not Necessary (Section 5)\nI'm not sure about the rationale of this experiment. I haven't found any literature suggesting that Neural Collapse is required to achieve generalization. In fact, NC appears only after TPT (Terminal Phase of Training) on restricted setups with no augmentations and prolonged training, and we know that these factors are not necessary to achieve strong generalization for the models, so I'm confused about what the point of this argument is.\n\nQUESTIONS:\n1. Why does the validation loss grow in almost all of your experiments? I'm referring to the Figures presented in the Appendix.\n\n2. Why did the authors use the ResNet-18 architecture without the recommended modifications required for CIFAR-10? Without these modifications, the model achieves poor performance due to pathologically low image dimensionality across the layers when trained on CIFAR-10 (32x32), and these experiments are not representative of any sensible setting. \n\n3. How to interpret the relative sharpness measure? What values are fine (we expect strong generalization), what values are not (the model should not generalize)? The explanation (last paragraph from the 6th section) does a poor job, and the Figures from the appendix do not provide any clarifying information. The scale is different across the plots, and it looks like regularizing **against the flatness** does not really harm this measure or maybe it's just a scale and it's hard to observe the difference?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies a fundamental research question of deep learning generalization by disentangling two widely studied properties: Neural Collapse and relative flatness. To explore their individual contributions, the authors begin with a symbolic prediction task, analyzing how training loss, val loss, flatness and the distance towards neural collapse evolve during the entire training process.\n\nTo isolate the effects of each property, the authors design controlled environments (1) penalize the neural collapse effect during the training (2) penalize the relative flatness during the training. The results reveal that generalization remains strong when neural collapse is suppressed, whereas forcing sharpness leads to a significant drop in validation performance. The performance degradation is reverted after unplug the flatness regularization.\n\nOverall, the experimental findings largely  support the authors' hypothesis that relative flatness plays a more crucial role than neural collapse in driving generalization.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths.\n- This paper is very well-written and easy to follow\n- This paper investigate an important problem that will be interesting for broader Neurips audience \n- This paper provides a well rounded discussion, which well communicates the conslusion, assumption, and limitation of this work.\n\nWeakness.\n- Potential Overstatement of Prior Work: While prior literature has observed that neural collapse (NC) often emerges at the terminal stage of training, few works directly associate NC with generalization to unseen test data. Some references cited as supporting the “common conjecture” (e.g., Line 30) either do not make such an association or explicitly caution against it. For example, Zhou et al. (2022) state that “... it does not have a direct implication for the features associated with test data as well as the generalization performance.” This mischaracterization of prior work undermines the contribution, as it builds upon a correlation between NC and generalization that is not well established.\n\n- On continue with the previous point, by only revealing correlation between relative flatness and generalization along, the novelty of this work seems insufficient. The relationship between flatness and generalization has already been explored extensively in prior works.\n\n- Limited Scale of Empirical Validation: The experimental evidence is primarily drawn from small-scale datasets (e.g., CIFAR-10, ImageNet-100). While these are useful for proof-of-concept, they limit the generalizability of the conclusions, especially considering the paper aims to make claims about the behavior of modern deep networks. Validation on larger-scale or more realistic settings would strengthen the empirical support for the paper’s claims.\n\nQUESTIONS:\nIn addition to the weaknesses part, I also have the following questions:\n\n1. How to understand that the val performance is still lower after unplug flatness regularization and trained for many iterations?\n\n2. One key assumption made in the paper is that “labels are locally constant.” Could the authors clarify this assumption? For instance, ImageNet is known to contain a non-trivial amount of label noise. How does such label noise impact the reliability of the conclusions, especially with respect to generalization performance and the connection to flatness?\n\nI am likely to adjust my rating based on the authors' rebuttal.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
This work explores the causal roles of flatness and neural collapse deep network generalization. The authors empirically show that loss landscape flatness is crucial for generalization, whereas neural collapse is not. They further prove that neural collapse implies relative flatness, explaining their observed correlation.

STRENGTHS AND WEAKNESSES:
**Strength.**
1. Understanding the relationship between neural collapse and generalization is important.
2. The paper is well-written and easy to follow.
3. The work provides a theoretical analysis of the relationship between neural collapse and flatness.

**Weakness.**
1. The idea that neural collapse is not a prerequisite of generalization is not novel. Section 2.1 of https://arxiv.org/abs/2202.08384 discusses this and provides a natural example.
2. The empirical validation in Section 6 is limited. The two experiments on CIFAR-10 and ImageNet-100 are insufficient to estabilish the strong claim like "flatness is necessary for generalization".
3. The intuition behind behind Proposition 5.2 is unclear. The authors should provide an intuitive explaination for why neural collapse leads to flatness.

QUESTIONS:
In Figure 9, relative flatness spikes at epoch 250, which coincides with the removal of the flatness regularizer. Why does this spike occur, as one might expect the relative flatness to decrease?

LIMITATIONS:
The experiments in these paper are limited to single architecture, two datasets, and computer vision tasks. The claims require support from more comprehensive experiments across various architectures, datasets, and tasks, including Natural Language Processing (NLP).
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates two phenomena associated with generalization—flatness and neural collapse. Neural collapse refers to when the representations of the penultimate layer in a network collapse onto the class centroid representations. Flatness simply refers to the notion that solutions in parameter space with low curvature tend to generalize well. By clearly defining a measure for both neural collapse clustering and relative flatness (a reparameterization invariant notion of flatness), the authors show theoretically and empirically that flatness is fundamental and necessary for generalization. In contrast, neural collapse does not causally lead to generalization, and instead it is a promoter of flatness but not the only way to achieve flatness.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nThis is work is notable for its clarity and rigor. The authors precisely scope their question, take a methodology that is technically sound and comprehensive, and the discussion is compelling and thorough. I commend the authors on a well-written and executed paper. The experiments are notable for going beyond correlational links between their measures and generalization. Causal interventions strongly support their claims and ought to be more common in the study of training dynamics and generalization through various measures. Ultimately, the significance of this paper is in its clarity of thought and rigor.\n\n**Weaknesses**\n\nBoth neural collapse and relative flatness are measures that do not only apply to the penultimate layer. Indeed, Hui et al. (2022) show that neural collapse cascades through the network layers. The authors clearly state that they wish to focus on the penultimate layer, but some discussion of whether their findings apply to earlier layers in the network would support their discussion more.\n\nQUESTIONS:\nIf space permits, a discussion of these findings with respect to the whole network would be an interesting addition.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-03"}