Double Descent Meets Out-of-Distribution Detection: Theoretical Insights and Empirical Analysis of the role of model complexity

A4. Concerns About NC Metrics and Generalization (Q1):

The link between the quality of OOD detection and Neural Collapse has been established in [1,2,3,4,5]; however, its connection to generalization quality remains unproven. We will revise the sentence, which was intended only to clarify the metric used and did not draw any conclusions. We acknowledge this oversight and greatly appreciate the reviewer bringing it to our attention. We encourage further feedback to help us address any similar issues.

A5. Clarification Needed for Remark 4.1 (Q2):

Thank you for highlighting the lack of clarity on that point. A lower value for the expected Out-of-Distribution (OOD) risk does not necessarily imply both better confidence on in-distribution (ID) data and lower confidence on OOD data simultaneously. However, it does indicate either better confidence on ID data or lower confidence on OOD data, or a combination of both.

In particular, the expected OOD risk is minimized when the classifier $\hat{f}(\cdot)$ is confident in its predictions over the in-distribution (ID) data and uncertain in its predictions over out-of-distribution (OOD) samples. Indeed, ideally, $\\hat{f}(\\mathbf{x})$ should be close to $1$ or $0$ for $\mathbf{x} \in P_{\mathcal{X},\mathcal{Y}}$ and $\hat{f}(\mathbf{x})$ should be close to $0.5$ for $\mathbf{x} \in P_{\mathcal{X},\mathcal{Y}}^\text{OOD}$. The more the classifier deviates from this ideal setup, the higher the OOD risk. Specifically, if the classifier becomes more uncertain on ID samples (with $\hat{f}(\mathbf{x})$ approaching $0.5$ for $\mathbf{x} \in P_{\mathcal{X},\mathcal{Y}}$) and more confident on OOD samples (with $\hat{f}(\mathbf{x})$ moving further from $0.5$ $\mathbf{x} \in P_{\mathcal{X},\mathcal{Y}}^\text{OOD}$), the OOD risk increases. This behavior is akin to confidence-based OOD detection methods, such as the Maximum softmax probability (MSP) or the MaxLogit metrics, which measure the confidence of the classifier's predictions.

References

[1] Liu, Litian, and Yao Qin. "Detecting out-of-distribution through the lens of neural collapse." arXiv preprint arXiv:2311.01479 (2023).

[2] Ben Ammar, M., Belkhir, N., Popescu, S., Manzanera, A., & Franchi, G. (2023). NECO: Neural Collapse Based Out-of-distribution Detection. ICLR 2024.

[3] Haas, Jarrod, William Yolland, and Bernhard Rabus. "Linking neural collapse and l2 normalization with improved out-of-distribution detection in deep neural networks." arXiv preprint arXiv:2209.08378 (2022).

[4] Zhang, Jiawei, et al. "EPA: Neural Collapse Inspired Robust Out-of-distribution Detector." ICASSP 2024-2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2024.

[5] Wu, Y., Yu, R., Cheng, X., He, Z., & Huang, X. (2024). Pursuing Feature Separation based on Neural Collapse for Out-of-Distribution Detection. arXiv preprint arXiv:2405.17816.

We thank Reviewer ZoGQ for their insightful comments and suggestions. Below, we address the concerns raised, organized into five key points, presented separately in two parts due to character limit restrictions:

A1. Motivation for the Paper (W1):

OOD detection is a rapidly growing field in deep learning, with most algorithms relying heavily on deep neural networks (DNNs). While the double descent phenomenon has been extensively studied for classification and regression tasks, its implications for OOD detection remain largely unexplored.

OOD detection differs fundamentally from classification, particularly in unsupervised settings, where it can be linked to the epistemic uncertainty of DNNs. Given that OOD detection primarily relies on classification-based DNNs, it is already indirectly influenced by the model’s generalization performance. This work aims to establish a more explicit link between OOD detection and the double descent phenomenon, filling a gap in the literature.

Our paper not only provides a theoretical framework to demonstrate this phenomenon but also empirically validates it across multiple datasets and DNN architectures, offering robust evidence of its significance in OOD detection.

A2. Motivation for the Expected OOD Risk Metric (W2):

To the best of our knowledge, no prior work introduces an "Expected OOD Risk Metric." Traditional OOD approaches typically focus on metrics like AUROC, AUPR, and FPR, optimizing OOD detection through various thresholding. Our proposed OOD risk metric, $R_\text{OOD}(\hat f)$ (5) (line 208), is linked to confidence-based OOD detection methods, such as Maximum Softmax Probability (MSP).

Concretely, $R_\text{OOD}(\hat{f})$ is minimized when the binary classifier $\hat{f}$ is confident for in-distribution (ID) samples ($\hat{f}(\mathbf{x}) \in \{1, 0\}, \mathbf{x} \in P_{\mathcal{X}, \mathcal{Y}}$) and uncertain for OOD samples ($\hat{f}(\mathbf{x}) = 0.5, \mathbf{x} \in P_{\mathcal{X}, \mathcal{Y}}^\text{OOD}$). Deviations from this behavior, such as uncertainty on ID samples ($\hat{f}(\mathbf{x}) \approx 0.5$) or confidence on OOD samples ($\hat{f}(\mathbf{x}) \neq 0.5$), lead to higher $R_\text{OOD}(\hat{f})$ values. This metric naturally extends to the multi-class case (Remark 4.2) using the infinity norm. In this context, $R_\text{OOD}(\hat{f})$ is minimized when logits are maximally confident on ID samples (only one logit is non-zero) and uniformly distributed on OOD samples. This setup describes the behavior of a perfectly calibrated classifier using the MSP method.

Regarding conclusions (1) and (2), these follow from the behavior of $z$, which is a function of $\mathbf{x}$. If $\hat{f}$ approximates $f^*$ well, $z$ behaves as:

where $\epsilon'$ is a small perturbation. Thus, $R_\text{OOD}(\hat{f})$ can simultaneously minimize both terms under the assumption that $\hat{f}$ is a good approximation of $f^*$.

A3. Experimental Limitations with the Width (W3 + Q3):

We are unsure of the exact limitation referenced by the reviewer. Double descent links model capacity/complexity to generalization performance. By definition, model depth relates to the model complexity, as deeper models are inherently more complex (at equal widths). Therefore, as the model complexity depends on the depth of the neural network, we can assume that the depth of the neural network also plays a role in the double descent phenomenon. To the best of our knowledge, we do not know works that considered varying the depth to study the double descent phenomenon. Our findings support the hypothesis that deeper models may achieve the interpolation threshold at lower widths, providing an avenue for future exploration.

Thanks for the authors' rebuttal. Based on their clarifications, I am inclined to raise my score. However, I still have one major question.

Neural Collapse (NC) is a phenomenon observed in model optimization, but does a lower NC value necessarily imply better performance? While some literature may suggest this conclusion, I remain skeptical. When introducing Neural Collapse, the authors are advised to avoid directly associating lower NC values with better performance. Instead, NC could be presented as a metric, which would provide a more neutral and rigorous framing than the current description.

For example, the statement in lines 327–329,"Neural Collapse: we report NC metrics, where lower values imply better performance in terms of OOD detection," might be reconsidered for greater clarity and precision.

We would like to thank reviewer UnVj for their insightful comments and suggestions. We address the concerns raised in the following four points, presented separately in two parts due to character limit restrictions:

A1. Significance of Theoretical Results (Q1):

We understand the reviewer’s concern regarding the significance of Theorem 2. However, we would like to emphasize that, in Belkin et al. (2019), the authors focused on a regression problem where the outputs were noisy responses of a linear function. In contrast, our work extends these results to binary classification and out-of-distribution (OOD) detection problems. To adapt the regression problem to binary classification, we introduce an activation function $\Phi(\cdot)$, which maps the outputs of a linear regression model to the probability space of the labels. This is typically achieved using functions like softmax or sigmoid for $\Phi(\cdot)$. The challenge lies in incorporating this non-linear function, which leads to inequalities in our results (Theorems 1 and 2) under Assumption 4.1, as opposed to the equalities presented in Belkin et al.'s theorem, which did not include this assumption. We hope this explanation clarifies the theoretical contribution of our work.

A2. Assumptions on OOD Distribution (Q2):

Our objective in both Theorem 1 and Theorem 2 is to show that the expected risk and the expected OOD risk exhibit a double descent descent phenomenon that we associate with an infinite peak when the number of samples $n$ is close to the number of features $p$ as noted in Remark 4.5 and Remark 4.6.

To this end, Assumption 4.1 was introduced in Theorem 1 to ensure that the matrix $\Sigma$ is non-singular with $\lambda_{\min}(\Sigma)>0$. In particular, $\lambda_{\min}(\Sigma)>0$ guarantees that the lower bound of the expected risk is lower bounded by $c(n, p, \sigma)$. Since the expected risk is also upper bounded by $c(n, p, \sigma)$ and that $c(n, p, \sigma)=\infty$ when$n-1 \leq p \leq n+1$, we can deduce from the squeeze theorem that $E_X[R_\text{OOD}(\hat f)]=\infty$ when $n-1 \leq p \leq n+1$.

In Theorem 2, we still want to obtain a lower bound for the expected OOD risk that depends on $c(n, p, \sigma)$ to show an infinite peak when $n$ is close to $p$. Assumption 4.1 is sufficient to guarantee this result since the lower bounds exists even if $\lambda_{\min}(\Sigma_{OOD})=0$. This means that, regardless of the degree of distributional shift between the in-distribution (ID) and OOD distributions, an infinite peak occurs when $n-1 \leq p \leq n+1$.

It is true that when $p < n-1$ or $p>n+1$, the degree of distributional shift between ID and OOD distributions influences how close the lower and upper bounds are to the expected OOD risk. However, our primary objective was not to provide accurate proxies for the expected risk and the expected OOD risk, but rather to highlight the existence of an infinite peak when $n$ is close to $p$ that we associate with the double descent phenomenon.

A3. Sensitivity of NC1 Metric to Changes in Layer Dimensions (Q3):

In our study, we sought to explore the correlation between Neural Collapse (NC) and model overparameterization. As part of this, we varied the width of the model to measure NC, in accordance with the double descent framework. As with the double descent tests, the size of the last-layer dimension increases dynamically alongside the intermediate layers. Altering only the intermediate layers could disrupt the architecture. While it might seem that the NC1 metric is sensitive to changes in the dimension $d$, empirical results from [1] suggest otherwise. Specifically, they showed that the overall model capacity, rather than the dimension $d$ of the last layer, is more important. In Section 4.3 of [1], two models with the same capacity (ResNet-18) were trained on CIFAR-10, one with a standard last-layer size of $d=512$ and the other with a smaller $d=10$ but the same number of classes ($c=10$). Despite the difference in the size of the last layer, the NC measurement results were similar for both models, indicating that NC1 is more influenced by the learning dynamics across the entire architecture, with the model’s overall capacity being the key factor. Thus, we believe it is fair to compare NC1 values of different models in terms of their generalization, irrespective of minor differences in the last-layer size.

A4. Heuristic Connections Between Overparameterization and OOD Detection (Q4):

We appreciate the reviewer’s concern on this point. To clarify, all of our training datasets are balanced [2], as is common in the field, ensuring consistency with prior work [3,4,5,6,7]. This alignment with established theoretical and empirical studies facilitates a direct comparison with existing research. Furthermore, all the state-of-the-art OOD metrics we tested were developed with balanced datasets in mind, and imbalanced datasets were not considered in these methods. Although we did not address the case of imbalanced data in this study due to the substantial complexity involved, we acknowledge the relevance of this issue and will discuss it in our paper, particularly in Appendix section D.4. While we could conduct experiments on imbalanced datasets if required, we believe this would extend beyond the scope of our current work.

References

[1] Zhu, Z., Gong, Z., Wei, C., & Liu, J. (2021). "Geometric Analysis of Neural Collapse with Unconstrained Features." arXiv preprint arXiv:2105.02375.

[2] Alex Krizhevsky. Learning multiple layers of features from tiny images. University of Toronto, 05 2012

[3] D'Angelo, E., et al. (2021). Can Neural Nets Learn the Same Model Twice? Investigating Reproducibility and Double Descent from the Decision Boundary Perspective. NeurIPS 2021.

[4] Yang, Z., et al. (2020). Rethinking Bias-Variance Trade-off for Generalization of Neural Networks. ICML 2020.

[5] Unraveling the Enigma of Double Descent: An In-depth Analysis through the Lens of Learned Feature Space. ICLR 2024.

[6] Belkin, M., et al. (2019). Reconciling Modern Machine Learning Practice and the Bias-Variance Tradeoff. PNAS 2019.

[7] Nakkiran, P., et al. (2020). Deep Double Descent. ICLR 2020.

We want to thank the reviewer for his comment and his help. We highly appreciate being able to discuss with the reviewer.

A1

Thank you for pointing out the lack of clarity. Assumption 4.1 actually holds for any strictly monotonically non-decreasing function, which includes the logistic function. To enhance the clarity and readability, we have specifically restricted Assumption 4.1 to strictly monotonically non-decreasing activation function. The proof supporting this assumption is detailed in Lemma 1 in the Appendix.

A2

We appreciate the reviewer’s comment regarding the importance of high logit values. While we agree with this observation, it is important to clarify that Neural Collapse (NC) focuses on different aspects. Specifically:

• NC1 states that the variability within the latent space of each class diminishes, leading to class-wise collapse.
• NC2 describes how the centroids of different classes move farther apart.
• NC4 suggests that the last fully connected layer behaves equivalently to a nearest neighbor classifier for predicting classes. To the best of our knowledge, there is no direct connection between these principles and logit or softmax values. However, as we mentioned, multiple papers [1,2,3,4] have empirically demonstrated that NC can be leveraged to improve OOD detection. It is worth noting, though, that there is currently no theoretical proof supporting this connection—only empirical evidence, which remains open to debate. That is why we totally agree with the reviewers (ZoGQ, and UnVj ) on that point. In light of this, we have carefully reviewed and rephrased all instances in our paper where Neural Collapse is discussed, ensuring that our claims are accurate and avoid overstatement.

Thank you again for your thoughtful feedback, which has helped us improve our paper. We would be happy to address any additional comments or questions you may have.

References

[1] Zhang, J., Chen, Y., Jin, C., Zhu, L., & Gu, Y. (2024, April). EPA: Neural Collapse Inspired Robust Out-of-distribution Detector. In ICASSP 2024-2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (pp. 6515-6519). IEEE.

[2] Ammar, M. B., Belkhir, N., Popescu, S., Manzanera, A., & Franchi, G. (2023). NECO: NEural Collapse Based Out-of-distribution detection. arXiv preprint arXiv:2310.06823.

[3] Haas, Jarrod, William Yolland, and Bernhard T. Rabus. "Linking Neural Collapse and L2 Normalization with Improved Out-of-Distribution Detection in Deep Neural Networks." Transactions on Machine Learning Research.

[4] Zhao, Zhilin, and Longbing Cao. "Dual representation learning for out-of-distribution detection." Transactions on Machine Learning Research (2023).

We would like to express our gratitude to reviewer eoHk for their insightful comments and suggestions. Below, we address the concerns raised in the following five points,presented separately in two parts due to character limit restrictions:

A1. Assumptions on the Model (W1):

We acknowledge the reviewer's concern regarding the model's simplicity. Theoretically demonstrating the existence of the double descent phenomenon remains a highly active and challenging area of research in machine learning. Most theoretical studies on the double descent phenomenon consider simple models, such as single hidden layer models, to operate within more mathematically tractable frameworks. In particular, this approach enables researchers to leverage powerful mathematical tools, including Random Matrix Theory, to gain better insights into the double descent phenomenon. In particular, [1,2,3,4,5] considered simplified models like linear regression or random features to obtain mathematical insights into the double descent phenomenon. To obtain similar theoretical insights into the OOD metrics and the double descent phenomenon, we have adopted a similar approach.

A2. Assumptions on the Data (W2+Q2):

The assumption of Gaussian-distributed data is common and often used in theoretical works, especially when studying phenomena like double descent and Random Matrix Theory [2,3,4,5]. This assumption allows for more tractable and accurate results, particularly in deriving the Wishart distribution for the Gram matrix of the data. While this assumption might not perfectly match real-world data, it enables us to derive meaningful bounds that depend solely on model complexity. This helps illuminate the double descent phenomenon.

Furthermore, in high-dimensional random projections, data often becomes isotropically distributed, even if the original data is not perfectly Gaussian. The Central Limit Theorem also suggests that sums of independent random variables will approximate a normal distribution as the number of terms increases. Therefore, high-dimensional aggregated data often closely approximates a Gaussian distribution. Empirical studies in deep learning also show that deep neural network embeddings often approximate isotropic Gaussian distributions, particularly in high-dimensional feature spaces.

A3. Validity of Nonlinear Approximation (W3):

We acknowledge the reviewer's observation about the lack of an important assumption in the paper regarding the nonlinear approximation. Specifically, we forgot to state that the optimal weight $\\hat{\\mathbf{w}}$ should not differ significantly from the optimized weight $\\mathbf{w}^*$. This assumption, while challenging to justify rigorously, is supported empirically in our study, where the difference between the two is found to be small. We will make this assumption explicit in the revised manuscript.

A4. Clarification of Equation 5 (Q1):

We appreciate the reviewer pointing out the ambiguity in Equation (5). The dependency of $z$ on $x$ was not sufficiently clear, and we have revised the manuscript to clarify this. Specifically, $z$ is dependent on whether $x$ is from the in-distribution (ID) or out-of-distribution (OOD) set. We now explicitly define $z$ as follows:

This definition ensures that the prediction function $f^*$ approaches 0.5 for OOD data and 1 for in-distribution data, which helps minimize the OOD risk when plugged into the risk function R_\\text{OOD}(\\hat f).

A5. Further Details on Remark 4.6 and Relation to OOD Detection (Q3):

Our objective in both Theorem 1 and Theorem 2 is to show that the expected risk and the expected OOD risk exhibit a double descent descent phenomenon that we associate with an infinite peak when the number of samples $n$ is close to the number of features $p$ as noted in Remark 4.5 and Remark 4.6.

Specifically, we aim to derive lower and upper bounds that depend solely on the model complexity (which is proportional to the number of samples n and the number of features p) and the level of noise $\\sigma$, through the function $c(n, p, \sigma)$. As $c(n, p, \\sigma)=\\infty$ when $n-1 \\leq p \\leq n+1$, we can deduce from the squeeze theorem that E_X[R_\\text{OOD}(\\hat f)]= \\infty when $n-1 \\leq p \\leq n+1$ in this regime.

We associate this infinite peak in the expected OOD risk with the double descent phenomenon. Since we have introduced the expected OOD risk as an OOD detection metric, to the best of our knowledge, this work is the first to mathematically describe a double descent phenomenon for an OOD metric.

References

[1] Francis Bach (2023).High-dimensional analysis of double descent for linear regression with random projections. SIAM Journal on Mathematics of Data Science

[2] Belkin, M., Hsu, D., Ma, S., & Mandal, S. (2019). Reconciling modern machine-learning practice and the classical bias–variance trade-off. Proceedings of the National Academy of Sciences, 116(32), 15849-15854.

[3] Louart, C., Liao, Z., & Couillet, R. (2018). A random matrix approach to neural networks. The Annals of Applied Probability, 28(2), 1190-1248.

[4] Liao, Z., Couillet, R., & Mahoney, M. W. (2020). A random matrix analysis of random fourier features: beyond the gaussian kernel, a precise phase transition, and the corresponding double descent. Advances in Neural Information Processing Systems, 33, 13939-13950.

[5] Belkin, M., Hsu, D., & Xu, J. (2020). Two models of double descent for weak features. SIAM Journal on Mathematics of Data Science, 2(4), 1167-1180.

I believe the Neural Tangent Kernel (NTK) does not assume that the data follows a Gaussian distribution. The connection between NTK training and Gaussian processes arises from the initialization of neurons using a Gaussian distribution, along with certain additional assumptions.

Also, NTK has been found to lack practical applicability to neural network training due to its strong assumptions. Specifically, it links neural network training to Gaussian processes (random feature learning), whereas practical training involves neurons learning features from input data. For example, see [1]. Could the author share your thoughts on this point?

[1] Feature Purification: How Adversarial Training Performs Robust Deep Learning

Let me clarify my second point. I am not asking you to compare your paper with the specific content in [1]. My point is that [1] argues that using Gaussian processes to analyze DNNs does not align with practical settings, whereas the framework proposed in [1] claims to address this issue. Could you share your thoughts on this?

Thank you for your clarification. We recognize that there may have been a misunderstanding arising from our initial response, in which we wrote: "Empirical studies in deep learning also show that deep neural network embeddings often approximate isotropic Gaussian distributions, particularly in high-dimensional feature spaces." We would like to correct this statement: we did not intend to imply that DNNs follow an isotropic Gaussian distribution or are Gaussian processes. This was a misstatement and we formally retract that point in our reply. To clarify, our paper does not rely on the assumption that DNNs follow a Gaussian distribution or a Gaussian Process. Instead, we assume that the input data follows an isotropic Gaussian distribution. As highlighted in our previous replies, such assumptions are common in the literature [1,2,3,4] and are necessary for deriving formal proofs/providing insights. We acknowledge the argument developed in [5] that using Gaussian processes to analyze DNNs may not align with practical settings, and we do not disagree with this point. However, we emphasize that our work does not depend on such assumptions regarding the DNNs themselves. We hope this response resolves any remaining confusion and are happy to continue this discussion further.

References

[1] Bach, F. (2017). Breaking the curse of dimensionality with convex neural networks. Journal of Machine Learning Research, 18(19), 1-53.

[2] Bach, F. (2024). High-dimensional analysis of double descent for linear regression with random projections. SIAM Journal on Mathematics of Data Science, 6(1), 26-50.

[3] d’Ascoli, S., Refinetti, M., Biroli, G., & Krzakala, F. (2020, November). Double trouble in double descent: Bias and variance (s) in the lazy regime. In International Conference on Machine Learning (pp. 2280-2290). PMLR.

[4] Belkin, M., Hsu, D., & Xu, J. (2020). Two models of double descent for weak features. SIAM Journal on Mathematics of Data Science, 2(4), 1167-1180.

[5] Allen-Zhu, Zeyuan, and Yuanzhi Li. "Feature purification: How adversarial training performs robust deep learning." 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2022.

Thank you for your response and for appreciating our answers.

We appreciate you pointing out the typo. Indeed, we intended to state "strictly monotonically increasing" rather than "monotonically increasing" in the assumption. The case where the activation function $\Phi(\cdot)$ is constant is not relevant to our problem. To the best of our knowledge, there is no constant activation function used in classification. We have updated the paper to correct this error.

We are unclear about your specific question regarding the difficulty in analyzing the case of multiple neurons. Could you please clarify whether you are referring to multiple classes with a single fully connected layer, or multiple fully connected layers? This clarification will help us address your concern more accurately.

Regarding your question about the empirical risk, we would like to clarify that our focus is solely on the population risk. Classically, in Machine Learning, the true objective is to minimize the population risk, while the empirical risk serves as an estimate using a finite number of samples [1]. If you are interested in connecting the empirical risk to the population risk, which is not the primary focus of this paper, concentration inequalities can be employed. In the case you also want to increase d with n the situation becomes more complex as highlighted for example in [2,3,4] or with the Marchenko-Pastur distribution. Nonetheless, it is still possible to obtain concentration inequalities and establish connections between the empirical and population risks [4,5].

References

[1] Bach, Francis. Learning theory from first principles. MIT press, 2024.

[2] Couillet, Romain, and Zhenyu Liao. Random matrix methods for machine learning. Cambridge University Press, 2022.

[3] Liao, Zhenyu, Romain Couillet, and Michael W. Mahoney. "A random matrix analysis of random fourier features: beyond the gaussian kernel, a precise phase transition, and the corresponding double descent." Advances in Neural Information Processing Systems 33 (2020): 13939-13950.

[4] Louart, C., Liao, Z., & Couillet, R. (2018). A random matrix approach to neural networks. The Annals of Applied Probability, 28(2), 1190-1248.

[5] Brellmann, David, et al. "On Double Descent in Reinforcement Learning with LSTD and Random Features." The Twelfth International Conference on Learning Representations.

Thanks for the authors' rebuttal. I will take a closer look into the corresponding proof later this week with your clarification of point two.

A quick comment on point 1 is that, since the data follows a Gaussian distribution (unbounded), for an activation function like the sigmoid, such an epsilon (>0) does not exist.

Edit: I believe this issue could be resolved using advanced tools. The key point is likely that the results are based on a stronger assumption but yield much weaker outcomes compared to the original conclusion. I need some time to read the paper again. In the meantime, I suggest that the author review the revised proof again, as it appears rushed and requires more time for refinement.

We thank reviewer WRDX for their insightful comments and suggestions. We address their concerns in the following two points, presented separately due to character limit restrictions:

A1. Label Noise and Real-World Applicability (W1):

We appreciate the reviewer’s comment on label noise. It is important to note that the use of label noise is a well-established practice in the community for studying double descent [1,2,3,4,5], and we incorporated it into our experiments to align with these prior works. Label noise serves to amplify the effects we are studying and ensures our findings are comparable to existing theoretical and empirical studies.

Nevertheless, we recognize the reviewer’s point about the value of noiseless experiments. In response, we have included additional experiments without label noise in Appendix Section D.5. Our observations indicate similar model behavior to previous studies, where error rates plateau around the interpolation threshold rather than peak, and we see a comparable trend in OOD detection performance. This reinforces the utility of label noise in clearly observing these phenomena.

Moreover, we would like to emphasize that adding label noise aligns with real-world application scenarios, where datasets are often corrupted by noise, as demonstrated in the table below.

A2. Connection Between Theoretical Results and Practical OOD Detection Methods (W2):

We appreciate the reviewer’s suggestion to clarify the connection between our theoretical results and practical OOD detection methods. In this regard, we would like to highlight that the OOD risk expression $R_\text{OOD}(\hat f)$ (equation 5 in line 208) is closely related to confidence-based OOD detection methods, such as the Maximum softmax probability (MSP) or the MaxLogit metrics. In the binary classification scenario, $R_\text{OOD}(\hat f)$ is minimized when the binary classifier $\hat{f}(\cdot)$ is confident in its predictions over the in-distribution (ID) data and uncertain in its predictions over out-of-distribution (OOD) samples. Ideally, \hat{f}(\mathbf{x}) should be close to $1$ or $0$ for $\mathbf{x} \in P_{\mathcal{X},\mathcal{Y}}$ and $\hat{f}(\mathbf{x})$ should be close to $0.5$ for $\mathbf{x} \in P_{\mathcal{X},\mathcal{Y}}^\text{OOD}$. The more the classifier deviates from this ideal setup, the higher the OOD risk. Specifically, if the classifier becomes more uncertain on ID samples (with $\hat{f}(\mathbf{x})$ approaching $0.5$ for $\mathbf{x} \in P_{\mathcal{X},\mathcal{Y}}$) and more confident on OOD samples (with $\hat{f}(\mathbf{x})$ moving further from $0.5$ $\mathbf{x} \in P_{\mathcal{X},\mathcal{Y}}^\text{OOD}$), the OOD risk increases. This behavior is akin to confidence-based OOD detection methods, such as the Maximum softmax probability (MSP) or the MaxLogit metrics, which measure the confidence of the classifier's predictions.

This link between the OOD risk and the confidence-based OOD detection methods extends to the multi-class setting as highlighted by the remark 4.2 with use of the infinity norm ($\lVert \cdot \rVert_\infty$). Indeed, in the multi-class scenario, the OOD risk expression is minimized when all logits except the predicted class are null for ID data predictions. This corresponds to a perfectly calibrated classifier where the MSP method would yield high confidence in the correct class. For OOD data, the ideal scenario is a uniform distribution of logits, indicating maximum uncertainty.

References

[1] Nakkiran, P., et al. (2020). Deep Double Descent. ICLR 2020.

[2] Belkin, M., et al. (2019). Reconciling Modern Machine Learning Practice and the Bias-Variance Tradeoff. PNAS 2019.

[3] D'Angelo, E., et al. (2021). Can Neural Nets Learn the Same Model Twice? Investigating Reproducibility and Double Descent from the Decision Boundary Perspective. NeurIPS 2021.

[4] Yang, Z., et al. (2020). Rethinking Bias-Variance Trade-off for Generalization of Neural Networks. ICML 2020.

[5] Unraveling the Enigma of Double Descent: An In-depth Analysis through the Lens of Learned Feature Space. ICLR 2024.

[6]Northcutt, C. G., et al. (2021). Pervasive Label Errors in Test Sets Destabilize Machine Learning Benchmarks. ICML 2021.

[7]Xiao, T., et al. (2015). Learning from Massive Noisy Labeled Data for Image Classification. CVPR 2015.

[8]Li, W., Wang, L., Li, W., Agustsson, E., & Van Gool, L. (2017). "WebVision Database: Visual Learning and Understanding from Web Data." arXiv preprint arXiv:1708.02862.

[9]Song, H., Kim, M., & Lee, J.-G. (2020). "Learning from Noisy Labels with Deep Neural Networks: A Survey." arXiv preprint arXiv:2007.08199.

[10]Kuznetsova, A., Rom, H., Alldrin, N., et al. (2020). "The Open Images Dataset V4: Unified Image Classification, Object Detection, and Visual Relationship Detection at Scale." International Journal of Computer Vision (IJCV).

Thanks to the authors for the rebuttal. Both of my questions were well addressed, and I have increased my score.

Dear Area Chair and Reviewers,

We sincerely thank you for your constructive feedback, which has greatly helped improve the quality of our paper. We have carefully addressed all the comments and performed additional experiments. The updated manuscript incorporates these changes, and we have ensured that the results, analysis, and references are emphasized throughout the revised submission with a different color (orange). Below, we summarize the key updates:

Clarification on the math formulas

• $z$ dependence on $\\mathbf{x}$ is highlighted in the main paper with its values clearly illustrated (Remarques 4.1 and 4.2).
• Our assumption on $\\hat{\\mathbf{w}}$ being close to $\\mathbf{w}^*$ is integrated to the main manuscript (Assumption 4.2).

Significance of our theoretical framework

• The connection between our theoretical results and practical OOD detection methods is clarified, with added experiments in section D.4 highlighting this.
• The significance of our derived theorems is clarified, and the motivation behind its components is justified.
• Concerns regarding the pertinence and utility of our model and data assumptions have been addressed.
• Additional explanations have been provided for the reviewers regarding the connections between the different parts of our theoretical framework, and to clarify its various parts and components.

Motivation and experiments

• Experiments without label-noise have been added in section D.5. Its real-word nature and the utility of its integration are addressed.

• The concerns about neural collapse (NC) sensitivity to the last-layer dimension have been addressed.

• The motivations and implications of our theoretical and empirical work were explained.

• The utility behind our OOD risk formula is explained, with the added experiments in section D.4 illustrating its links to confidence-based OOD detection methods.

• Concerns regarding the possible experimental limitation from varying the model width are addressed.

Attached, you will find a revised version of the paper incorporating all changes in orange color. We hope these revisions fully address the concerns raised and clarify our contributions. Please feel free to let us know if there are further issues that require our attention. Thank you for your time and effort in reviewing our work.

Sincerely, Corresponding authors

Dear Area Chair and Reviewers,

We sincerely thank you for your constructive feedback. Thanks to you we have performed the following update :

1. Assumption 4.1

Based on your feedback, we have modified Assumption 4.1 to enhance clarity and readability. The former version of Assumption 4.1 about the nonsingular property of the matrix Sigma, which held for any strictly monotonically non-decreasing function, including the logistic function. To improve clarity, we have specifically restricted Assumption 4.1 to monotonically non-decreasing activation functions. The proof supporting this revised assumption is detailed in Lemma 4.1 in the Appendix.

2. Assumption 4.2

Using the new version of Assumption 4.1, we have been able to remove Assumption 4.2, introduced in our previous reply, which required $\\hat{w}$ to be close to $w^*$. With the revised Assumption 4.1 and the application of the mean-value theorem, we can now derive the results of Theorem 1 without relying on the Taylor expansion or assumptions about $\\hat{w}$ being close to $w^*$. Details can be found in the proof of Theorem 1 in the Appendix.

3. OOD Risk

We have performed experiments with the out-of-distribution (OOD) risk in a practical setting and observed that it also exhibits a double descent phenomenon in real-world scenarios. Figures can be found in Appendix D.

Thank you again for your time and effort in reviewing our work. Do you have any additional comments or suggestions?

Best regards,

Authors