Deep Classifiers with Label Noise Modeling and Distance Awareness

Abstract: Uncertainty estimation in deep learning has recently emerged as a crucial area of interest to advance reliability and robustness in safety-critical applications. While there have been many proposed methods that either focus on distance-aware model uncertainties for out-of-distribution detection or on input-dependent label uncertainties for in-distribution calibration, both of these types of uncertainty are often necessary. In this work, we propose the HetSNGP method for jointly modeling the model and data uncertainty. We show that our proposed model affords a favorable combination between these two complementary types of uncertainty and thus outperforms the baseline methods on some challenging out-of-distribution datasets, including CIFAR-100C, Imagenet-C, and Imagenet-A. Moreover, we propose HetSNGP Ensemble, an ensembled version of our method which adds an additional type of uncertainty and also outperforms other ensemble baselines.

Reviews and author responses:

 Thanks for your responses.

I think this work is well motivated.  But after carefully reading the response, I still have concerns for both technical and empirical contributions which are as follows.

+ The contribution of combining SNGP and heteroscedastic models into a single model may not be enough. As I aforementioned, algorithms in this paper seem to be similar to SNGP [1].
+ The authors have not clearly explained that why these two types of uncertainty are complementary, and why these two types of uncertainty can not be both achieved. 
+ I agree with the authors that not all tasks and datasets require both types of uncertainty. However, It is not rigorous to claim that the method achieves the best possible tradeoff mentioned in authors' responses. The reason is that before claiming that the method achieves the best possible tradeoff, the optimal goal (tradeoff ) tried to achieve has to be defined in advance. However, I failed to find any formal definition of the optimal goal (tradeoff ) which we want to achieve. Additionally, for results in all the tables, the performance improvement seems marginal.


Therefore, I will keep my original score. 

Best Regards,

Reviewer m1FZ 

[1]. Liu, Jeremiah Zhe, et al. "Simple and principled uncertainty estimation with deterministic deep learning via distance awareness." arXiv preprint arXiv:2006.10108 (2020).

 Thanks for your response.

Regarding the novelty of this work, as Reviewer m1FZ, wQfZ and tqBj agreed, it is not sufficient.

Given the experiment results with the authors' explanation, the assumptions do limit the scope of application of the method.

Therefore, I tend to keep the initial score.




This paper studies the combination between model uncertainty and data uncertainty based on spectral-normalized Gaussian process. Theoretical results show that heteroscedastic SNGP allows for joint modeling of model and data uncertainties. The paper then proposes an approximate inference scheme for efficient model training. Experiment results support these claims on synthetic datasets and typical classification benchmarks.  The first contribution, that jointly model two different types of uncertainty, as far as I know,  make the first step for noisy label. The second contribution, amounts to large-scale matrix computation approximation, makes the whole method be accessible. However，the contribution of the whole paper can not meet the requirement of ICRL as (1) limited novelty. The proposed method is a limited improvement of SNGP. (2) The computation approximation all come from the existing techniques [1,2,3]. (3) For joint uncertainty estimation setting, it is not very novel and easily to be solved. We can use methods in [1,3] to handle the data uncertainty, then use the method in [2] to handle model uncertainty.

(1)	Experimental results. Compared to the baseline model SNGP, the proposed method uses more complex method but achieve slight performance promotion even performance reduction. Only using the ensemble strategy can bring stable performance promotion (Table 4).  More discussion is needed.
(2)	For computation approximation, four types of approximations are used, the performance and time yield should be discussed in the experimental section. For practicable, the author can verified the algorithm (w/wo approximation) on small datasets.
(3)	In experimental section, the evaluation metrics are not clear. 

[1] Mark Collier, Basil Mustafa, Efi Kokiopoulou, Rodolphe Jenatton, and Jesse Berent. Correlated
input-dependent label noise in large-scale image classification. In Proceedings of the IEEE/CVF
Conference on Computer Vision and Pattern Recognition, pp. 1551–1560, 2021.
[2] Jeremiah Zhe Liu, Zi Lin, Shreyas Padhy, Dustin Tran, Tania Bedrax-Weiss, and Balaji Lakshminarayanan. Simple and principled uncertainty estimation with deterministic deep learning via
distance awareness. arXiv preprint arXiv:2006.10108, 2020
[3] Mark Collier, Basil Mustafa, Efi Kokiopoulou, Rodolphe Jenatton, and Jesse Berent. A simple
probabilistic method for deep classification under input-dependent label noise. arXiv preprint
arXiv:2003.06778, 2020.
 The jointly model data and model uncertainty seem some novel. However, the contribution is incremental when compared to SNGP. The experimental analysis is not sufficient and strong. Overall, I tend to reject this paper. 

 hanks for the responses. The authors' explanation about experiental results gap looks reasonable to me. I agree with peer reviewers' concern  about the novelty of SNGP. Thanks for adding more experimental results.  Therefore, I tend to raise the initial score.

 Thank you for your valuable time and feedback. We updated the manuscript to address your points (updated parts are highlighted in orange). We will address your individual points in the following.

**Re evaluation metrics**: We added a paragraph to Section 4 in the manuscript describing the evaluation metrics in more detail.

**Re experiments**: The main purpose of our experiments was to assess whether we can successfully combine the heteroscedastic and SNGP methods to trade off their respective complementary uncertainties. Naturally, not all tasks and datasets require both types of uncertainty, so we should not expect our method to always outperform the baselines. In fact, we should not expect the distance-aware OOD uncertainties to be useful on in-distribution data, such that in those settings, the heteroscedastic method's performance should be considered an upper bound on the performance of our combined method.  However, in tasks that require heteroscedastic in-distribution uncertainties, we would expect that our method (just like the heteroscedastic method) should outperform the SNGP. Similarly, in tasks that require distance-aware OOD uncertainties, we would expect that our method (just like the SNGP) should outperform the heteroscedastic method. In the tasks where both types of uncertainty are needed (usually on OOD datasets that themselves contain heteroscedastic label noise), we might hope that our method outperforms both baselines. Indeed, we see all these confirmed in the experiments and thus achieve the best possible tradeoff. For instance on CIFAR (Tab. 1), Imagenet (Tab. 2), and Imagenet21k (Tab. 4), the heteroscedastic and HetSNGP methods both outperform the SNGP in-distribution (no OOD uncertainty needed), while the HetSNGP outperforms both baselines on the OOD data (both types of uncertainty needed). We clarified this in the manuscript in Section 4.

**Re baselines and runtimes**: We have added two baselines from the epistemic uncertainty literature MC Dropout [4] and MIMO [5] for the Imagenet experiment, together with their respective inference costs. These methods are known to improve neural network calibration and we evaluate these methods and our method on expected calibration error (ECE) as well accuracy and log-likelihood. See results in Tables 6,  7 & 8, Appendix A.4. Not only those baselines give us additional points of comparison but they also offer new insights that we have now summarized in Table 9. In particular, the strong performance of MIMO alone (both for in- and out-of-distribution related tasks) seems to indicate that the ensembling component in capturing model uncertainty is essential. As a result, in addition to our current set of results, we plan for the camera-ready version of the paper to compare with HetSNGP + MIMO that appears to be a method gathering all the properties of interest (see Table 9, Appendix A.4).

 
[1] Immer, A., Korzepa, M., & Bauer, M. (2021, March). Improving predictions of Bayesian neural nets via local linearization. In International Conference on Artificial Intelligence and Statistics (pp. 703-711). PMLR.

[2] Immer, A., Bauer, M., Fortuin, V., Rätsch, G., & Khan, M. E. (2021). Scalable marginal likelihood estimation for model selection in deep learning. arXiv preprint arXiv:2104.04975.

[3] Daxberger, E., Kristiadi, A., Immer, A., Eschenhagen, R., Bauer, M., & Hennig, P. (2021). Laplace Redux--Effortless Bayesian Deep Learning. arXiv preprint arXiv:2106.14806.

[4] Gal, Y., & Ghahramani, Z. (2016, June). Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning (pp. 1050-1059). PMLR.

[5] Havasi, M., Jenatton, R., Fort, S., Liu, J. Z., Snoek, J., Lakshminarayanan, B., ... & Tran, D. (2020). Training independent subnetworks for robust prediction. arXiv preprint arXiv:2010.06610.

 Thank you for your valuable time and feedback. We updated the manuscript to address your points (updated parts are highlighted in orange). We will address your individual points in the following.

**Re novelty**: While both the SNGP and heteroscedastic model have doubtlessly existed before, successfully combining them into a single model was by no means trivial. Indeed, we had to come up with the proposed hierarchical architecture, because other designs were not able to robustly disentangle the different sources or uncertainty. When using other architectures, for instance, using the heteroscedastic uncertainty directly in the likelihood of the GP or using two additive random variables with respective GP and heteroscedastic distributions, the two models will compete for the explained uncertainty and the heteroscedastic prediction network will typically try to explain as much uncertainty as possible. Only in this hierarchical model setting, where the SNGP adds uncertainty to the mean first and the heteroscedastic uncertainty is added in the second step, can we robustly achieve a correct assignment of the different uncertainties using gradient-based optimization. Moreover, as you correctly noted, effectively training the model requires a number of computational approximations that have to be carefully combined to ensure that the model is both trainable as well as computationally feasible. We updated the manuscript in Section 3.4 to make these points more clear.

**Re assumptions and approximations**: It is true that our model is based on some assumptions. However, the choice of a hierarchical generative model is by no means arbitrary, but is necessary to ensure a robust disentanglement of the two complementary types of uncertainty. Moreover, while the inference requires many different computational approximations, they are all well-understood approximations that have been used in many related works. Moreover, most of them (low-rank, RFF, Monte Carlo) contain a tuneable hyperparameter that can make them as exact as needed with increased computation. They can therefore be tuned for each combination of data set and computational budget. Only the Laplace approximation is intrinsically non-exact, since it only fits a local mode of the posterior, but it has been shown in recent work [1,2,3] that it can work well in many practical settings. We have clarified these points in Section 3.3 of the manuscript.

**Re experiments**: The main purpose of our experiments was to assess whether we can successfully combine the heteroscedastic and SNGP methods to trade off their respective complementary uncertainties. Naturally, not all tasks and datasets require both types of uncertainty, so we should not expect our method to always outperform the baselines. In fact, we should not expect the distance-aware OOD uncertainties to be useful on in-distribution data, such that in those settings, the heteroscedastic method's performance should be considered an upper bound on the performance of our combined method.  However, in tasks that require heteroscedastic in-distribution uncertainties, we would expect that our method (just like the heteroscedastic method) should outperform the SNGP. Similarly, in tasks that require distance-aware OOD uncertainties, we would expect that our method (just like the SNGP) should outperform the heteroscedastic method. In the tasks where both types of uncertainty are needed (usually on OOD datasets that themselves contain heteroscedastic label noise), we might hope that our method outperforms both baselines. Indeed, we see all these confirmed in the experiments and thus achieve the best possible tradeoff. For instance on CIFAR (Tab. 1), Imagenet (Tab. 2), and Imagenet21k (Tab. 4), the heteroscedastic and HetSNGP methods both outperform the SNGP in-distribution (no OOD uncertainty needed), while the HetSNGP outperforms both baselines on the OOD data (both types of uncertainty needed). We clarified this in the manuscript in Section 4.

**Re baselines**: We have added two baselines from the epistemic uncertainty literature MC Dropout [4] and MIMO [5] for the Imagenet experiment, together with their respective inference costs. These methods are known to improve neural network calibration and we evaluate these methods and our method on expected calibration error (ECE) as well accuracy and log-likelihood. See results in Tables 6,  7 & 8, Appendix A.4. Not only those baselines give us additional points of comparison but they also offer new insights that we have now summarized in Table 9. In particular, the strong performance of MIMO alone (both for in- and out-of-distribution related tasks) seems to indicate that the ensembling component in capturing model uncertainty is essential. As a result, in addition to our current set of results, we plan for the camera-ready version of the paper to compare with HetSNGP + MIMO that appears to be a method gathering all the properties of interest (see Table 9, Appendix A.4).

 Thank you for your valuable time and feedback. We updated the manuscript to address your points (updated parts are highlighted in orange). We will address your individual points in the following.

**Re assumptions and approximations**: It is true that our model is based on some assumptions. However, the choice of a hierarchical generative model is by no means arbitrary, but is necessary to ensure a robust disentanglement of the two complementary types of uncertainty. Moreover, while the inference requires many different computational approximations, they are all well-understood approximations that have been used in many related works. Moreover, most of them (low-rank, RFF, Monte Carlo) contain a tuneable hyperparameter that can make them as exact as needed with increased computation. They can therefore be tuned for each combination of data set and computational budget. Only the Laplace approximation is intrinsically non-exact, since it only fits a local mode of the posterior, but it has been shown in recent work [1,2,3] that it can work well in many practical settings. We have clarified these points in Section 3.3 of the manuscript.

**Re baselines**: We have added two baselines from the epistemic uncertainty literature MC Dropout [4] and MIMO [5] for the Imagenet experiment, together with their respective inference costs. These methods are known to improve neural network calibration and we evaluate these methods and our method on expected calibration error (ECE) as well accuracy and log-likelihood. See results in Tables 6,  7 & 8, Appendix A.4. Not only those baselines give us additional points of comparison but they also offer new insights that we have now summarized in Table 9. In particular, the strong performance of MIMO alone (both for in- and out-of-distribution related tasks) seems to indicate that the ensembling component in capturing model uncertainty is essential. As a result, in addition to our current set of results, we plan for the camera-ready version of the paper to compare with HetSNGP + MIMO that appears to be a method gathering all the properties of interest (see Table 9, Appendix A.4).

[1] Immer, A., Korzepa, M., & Bauer, M. (2021, March). Improving predictions of Bayesian neural nets via local linearization. In International Conference on Artificial Intelligence and Statistics (pp. 703-711). PMLR.

[2] Immer, A., Bauer, M., Fortuin, V., Rätsch, G., & Khan, M. E. (2021). Scalable marginal likelihood estimation for model selection in deep learning. arXiv preprint arXiv:2104.04975.

[3] Daxberger, E., Kristiadi, A., Immer, A., Eschenhagen, R., Bauer, M., & Hennig, P. (2021). Laplace Redux--Effortless Bayesian Deep Learning. arXiv preprint arXiv:2106.14806.

[4] Gal, Y., & Ghahramani, Z. (2016, June). Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning (pp. 1050-1059). PMLR.

[5] Havasi, M., Jenatton, R., Fort, S., Liu, J. Z., Snoek, J., Lakshminarayanan, B., ... & Tran, D. (2020). Training independent subnetworks for robust prediction. arXiv preprint arXiv:2010.06610.

 Thank you for your valuable time and feedback. We updated the manuscript to address your points (updated parts are highlighted in orange). We will address your individual points in the following.

**Re novelty**: While both the SNGP and heteroscedastic model have doubtlessly existed before, successfully combining them into a single model was by no means trivial. Indeed, we had to come up with the proposed hierarchical architecture, because other designs were not able to robustly disentangle the different sources or uncertainty. When using other architectures, for instance, using the heteroscedastic uncertainty directly in the likelihood of the GP or using two additive random variables with respective GP and heteroscedastic distributions, the two models will compete for the explained uncertainty and the heteroscedastic prediction network will typically try to explain as much uncertainty as possible. Only in this hierarchical model setting, where the SNGP adds uncertainty to the mean first and the heteroscedastic uncertainty is added in the second step, can we robustly achieve a correct assignment of the different uncertainties using gradient-based optimization. Moreover, as you correctly noted, effectively training the model requires a number of computational approximations that have to be carefully combined to ensure that the model is both trainable as well as computationally feasible. We updated the manuscript in Section 3.4 to make these points more clear.

**Re assumptions and approximations**: It is true that our model is based on some assumptions. However, the choice of a hierarchical generative model is by no means arbitrary, but is necessary to ensure a robust disentanglement of the two complementary types of uncertainty. Moreover, while the inference requires many different computational approximations, they are all well-understood approximations that have been used in many related works. Moreover, most of them (low-rank, RFF, Monte Carlo) contain a tuneable hyperparameter that can make them as exact as needed with increased computation. They can therefore be tuned for each combination of data set and computational budget. Only the Laplace approximation is intrinsically non-exact, since it only fits a local mode of the posterior, but it has been shown in recent work [1,2,3] that it can work well in many practical settings. We have clarified these points in Section 3.3 of the manuscript.

**Re experiments**: The main purpose of our experiments was to assess whether we can successfully combine the heteroscedastic and SNGP methods to trade off their respective complementary uncertainties. Naturally, not all tasks and datasets require both types of uncertainty, so we should not expect our method to always outperform the baselines. In fact, we should not expect the distance-aware OOD uncertainties to be useful on in-distribution data, such that in those settings, the heteroscedastic method's performance should be considered an upper bound on the performance of our combined method.  However, in tasks that require heteroscedastic in-distribution uncertainties, we would expect that our method (just like the heteroscedastic method) should outperform the SNGP. Similarly, in tasks that require distance-aware OOD uncertainties, we would expect that our method (just like the SNGP) should outperform the heteroscedastic method. In the tasks where both types of uncertainty are needed (usually on OOD datasets that themselves contain heteroscedastic label noise), we might hope that our method outperforms both baselines. Indeed, we see all these confirmed in the experiments and thus achieve the best possible tradeoff. For instance on CIFAR (Tab. 1), Imagenet (Tab. 2), and Imagenet21k (Tab. 4), the heteroscedastic and HetSNGP methods both outperform the SNGP in-distribution (no OOD uncertainty needed), while the HetSNGP outperforms both baselines on the OOD data (both types of uncertainty needed). We clarified this in the manuscript in Section 4.
Re clarity: Thank you for your feedback regarding the clarity of the writing. We hope that the revised manuscript addresses this issue. Please let us know if anything is still unclear.

[1] Immer, A., Korzepa, M., & Bauer, M. (2021, March). Improving predictions of Bayesian neural nets via local linearization. In International Conference on Artificial Intelligence and Statistics (pp. 703-711). PMLR.

[2] Immer, A., Bauer, M., Fortuin, V., Rätsch, G., & Khan, M. E. (2021). Scalable marginal likelihood estimation for model selection in deep learning. arXiv preprint arXiv:2104.04975.

[3] Daxberger, E., Kristiadi, A., Immer, A., Eschenhagen, R., Bauer, M., & Hennig, P. (2021). Laplace Redux--Effortless Bayesian Deep Learning. arXiv preprint arXiv:2106.14806.

The paper proposed a new model HetSNGP that captures distance-aware model uncertainties as well as heteroscedastic data uncertainties. HetSNGP combines these two types of uncertainty in a beneficial way, allowing for out-of-distribution detection and generalisation as well as in-distribution performance and calibration on a variety of benchmark datasets. Strengths:
This paper is well-motivated. Uncertainty estimation in deep learning has lately been a hot topic for improving reliability and resilience in safety-critical applications. This paper combines the complementary benefits of these two types of uncertainty modeling and achieves a good calibration performance.

Concerns:
1.	The proposed method Is a combination of well-known techniques. The novelty of this paper Is limited.
2.	The proposed method Is a little bit too heuristic. There are many assumptions which are not sufficiently justified. This may limit the scope of application of the method.
3.	The experiment results are weak. The proposed method cannot surpass baseline consistently.
4.	This paper is hard to understand. 
 Overall, I think this is a well-motivated work but the novelty and contributes are limited.

This paper presents a new method for model and data uncertainty estimation in deep neural networks combining the heteroscedastic method (Collier et al. 2020) and the Spectral-Normalised Gaussian Process (SNGP) method (Liu et al. 2020). It is shown that the two methods are complementary and their combination outperforms state-of-the-art methods on Out-Of-Distribution (OOD) detection on common image classification benchmarks. 
 The paper is well written and the scientific contributions are clearly formulated and experimentally validated.
The experimental results look quite convincing and show the benefits of this combined uncertainty model.

There are many technical details that are quite hard to understand and that seem to originate from other papers from the literature, notably in Section 3.3 (computational approximations). 
This makes the paper difficult to follow.
Also, it seems that the experimental evaluation is missing a comparison with some more existing methods for uncertainty estimation and OOD (like the ones mentioned in Section 5, "Related Work").

Overall, I still think the paper is interesting and of good quality.
 The proposed approach is scientifically sound, the presentation is of good quality. 
But the experimental results may lack some SOTA comparison.

The authors propose a method that can jointly learn the model uncertainty for out-of-distribution detection and data uncertainty for in-distribution calibration. The effectiveness of their method has been validated on different out-of-distribution datasets. strengths
+ How to build a model can jointly mode the model and data uncertainty is an interesting problem.
+ A method is proposed that has a good calibration performance. It can be used for out-of-distribution detection. 

weakness
+ The technical contribution of this paper may not be enough. Specifically, algorithms in this paper seem to be similar to SNGP [1]. I think that the major difference between the two methods comes from the different data generative processes.
+ The assumption used in this paper might be strong. The authors propose a hierarchical model of two latent random variables which are sampled from two Gaussian distributions. It is better to add more explanations that how realistic this assumption is. I think that the proposed method could be more useful if this assumption can be mitigated. Additionally, different approximations are made to ensure the tractability of distributions. Although some of them are commonly used, it is better to add more discussions to show that these approximations will not lead to large biases. 
+ The experimental advantage of the proposed method is not large. Table 2, Table 3, and Table 4 show that the performance of the proposed method and the baseline method SNGP are quite similar.  
The baselines considered are not that inclusive. I understand this problem is novel and there hasn't been fruitful literature. But I think it is better to add more calibration methods as baselines.

[1]. Liu, Jeremiah Zhe, et al. "Simple and principled uncertainty estimation with deterministic deep learning via distance awareness." arXiv preprint arXiv:2006.10108 (2020). Overall, I think this work is well motivated. But I have some concerns for both technical and empirical novelties.

Meta-review: This paper studies the combination between model uncertainty and data uncertainty based on the spectral-normalized Gaussian process. Empirical results show the effectiveness of the proposed method. Overall, the paper is well-motivated and well-written. However, there are several concerns about the paper. (1) The novelty is marginal. The contribution of combining SNGP and heteroscedastic models into a single model may not be enough. (2) More analyses and insights are needed on why the mentioned two types of uncertainty are complementary. (3) More recent state-of-the-art methods on classification with noisy labels are suggested to be included to interest the readers. There are diverse scores. However, no one wants to champion the paper. We believe that the paper will be a strong one by addressing the concerns.

Decision: Reject