Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes addresses the issue of asymptotic over-confidence of Bayesian ReLU neural networks. The idea is to add a Gaussian process with a cubic spline kernel to the GP, which ensures quick (cubic) growth of uncertainty as we move away from the training data. The authors also extend the method by computing the GP on top of the latent features extracted by the BNN, which improves uncertainty near the data. Finally, the authors provide experiments, showing promising results for calibration and OOD detection with the proposed method.
main_review: Originality: The paper is based on earlier work [1, 2], but uses a novel idea of applying GPs to increase the rate of uncertainty growth in BNNs. The idea of applying GPs on top of intermediate layers for OOD detection is novel and interesting, but related to the paper [3] which I suggest the authors include in the related work.


Quality: The paper overall high quality. The theory is well-supported by empirical experiments. The methods are simple, but do their job reasonably well.

Experiments in the paper are sufficient, but not outstanding. In particular, the authors do not compare to any OOD detection baselines, such as [3]. If this paper was presented as a practical method for OOD detection, I would argue that the experiments are insufficient, but I think they are reasonable given the positioning of the paper.

One downside of the paper (shared with [2]) is that the term BNN seems to be equated with a neural network with a Gaussian distribution over the parameters. In particular, I don't think anything in the paper depends on the distribution being related to the actual BNN posterior in any way. It would also be interesting to see experiments with more precise BNN baselines than last-layer Laplace approximation, e.g. [4]. 

Could you please comment on whether or not everything in the paper applies to regular ReLU NNs and not just BNNs?


Clarity: The paper is well-written.


Significance: I think the significance of this work is somewhat limited, it is unclear how important the asymptotic overconfidence of BNNs is, and how pronounced it is in actual BNNs (i.e. with more precise posterior approximation). The paper is nonetheless interesting and thought provoking. The idea of training a GP on the latent representations seems interesting and useful.


[1] Matthias Hein, Maksym Andriushchenko, and Julian Bitterwolf. Why ReLU Networks Yield High confidence Predictions Far Away from the Training Data and How to Mitigate the Problem

[2] Agustinus Kristiadi, Matthias Hein, and Philipp Hennig. Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks.

[3] A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks
Kimin Lee, Kibok Lee, Honglak Lee, Jinwoo Shin

[4] What Are Bayesian Neural Network Posteriors Really Like?
Pavel Izmailov, Sharad Vikram, Matthew D. Hoffman, Andrew Gordon Wilson
limitations_and_societal_impact: The authors discussed relevant limitations of their work.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes a method to overcome the issue of neural network overconfidence, particularly focussing on the area far away from the training data. The motivation is that Bayesian neural networks (BNNs) can deal with overconfidence near the training data, but even they struggle in the areas far away. The proposed method is based on enriching a BNN with a GP that is obtained as an approximation of a NN with infinite number of ReLUs units. Although the method is justified with this theoretical approach, in practice it means a simple post-hoc procedure with a low computational overhead. 
main_review: Update after rebuttal: I have read the authors' response and other review. As per discussion below, I would like to thank the authors for their clarification of my concerns and I am increasing the score of the paper. 

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Originality: The proposed idea is interesting and appears to be novel. The related work is adequately cited and it is clearly indicated how the proposed work differs from it.

Quality: The work is technically sound. The proposed method is theoretically grounded and theoretical analysis is provided. The method is also empirically tested rather extensively.
Some claims though are not justified. For example, the main motivation that the quadratic variance growth is not enough to give appropriate confidence far away from the data, while the cubic growth is enough is never justified. 
Theoretical derivations are rather loose, such as approximations are specified only as $\approx$ without any further details.

Clarity: The paper is mostly well written and easy to follow. There are some not major issues listed below.

Significance: The paper addresses an important issue of overconfidence of neural networks and proposes an interesting approach for this. I believe the idea may be interesting for a wide audience and can be used both in practice and as a base to build upon especially in the area of NN and GP connections.

Conclusion: Despite some weak points mentioned the paper seems to be a valid contribution to the community, therefore I am voting for acceptance of the paper.

Specific comments/suggestions:
1.	Lines 3-8. “This issue arises since the output variance of a BNN with finitely many features is quadratic in the distance from the data region. Meanwhile, Bayesian linear models with ReLU features converge, in the infinite-width limit, to a particular Gaussian process (GP) with a variance that grows cubically so that no asymptotic overconfidence can occur.” – it is not obvious why quadratic is not enough, and cubic is enough.
2.	Line 9. “it can be used concretely” – “can be used in practice”?
3.	Lines 19-20. “the confidence of BNNs can be bounded away from one” – unclear
4.	Lines 41-42. “And due to the second property, the extended BNNs exhibit super-quadratic output variance growth, and thus are guaranteed to predict with uniform confidence away from the data.” – the idea is still not obvious. At least a reference to prove it is missing.
5.	Lines 62-64. “maximum a posteriori (MAP) estimate \theta_{MAP} = arg max_{\theta} log p(\theta| D). One can also treat \theta as a random variable”. If you call it MAP, you already treat \theta as a random variable.
6.	Integral in line 67. One does not have to use softmax. 
7.	Line 123-125. “An infinite-ReLU GP with the cubic spline kernel has cubic variance growth, which, combined with the probit approximation, yields the uniform confidence in the limit.” – The third time – why?

Minor points:
1.	Line 1. “ReLU nets” -> “ReLU neural networks”
2.	Line 17. “overconfident issue” -> “overconfidence issue”
3.	E.g., line 17. “Kristiadi et al. [2]” – a mix of reference styles




limitations_and_societal_impact: The authors adequately addressed some limitations of the work such as the trade off between ECE and OOD detection. The paper does not seem to have any straightforward potential negative societal impact.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The author(s) achieve cubic growth (as a function of distance from the training data) in uncertainty for ReLU BNNs. They construct the double-sided cubic spline (DSCS) kernel which sums over 1-D input kernels to ensure that departure for any dimension increases uncertainty. The authors then use a Gaussian Process with this DCSC kernel to model ReLU BNN residuals. This treatment, for well-trained BNNs, results in a GP posterior that is approximately the the ReLU BNN's posterior plus the residual's prior. This observation provides the ability to apply the proposed after training the BNN as usual. Theorem 3 proves that their method in the limit produces a uniform distribution over class labels (maximum uncertainty).
main_review: ### Originality:
- I am unfamiliar with Meinke and Hein (2020), but this work does seem differentiated after a cursory glance. However, Meinke and Hein (2020) seems to be more generally applicable.

### Quality:
- I did not carefully review the proofs, but propositions and theorem reasoning make intuitive sense.
- Figure 6 does well at showing the proposed methods achieve the lowest confidence without a noticeable drop in accuracy.

### Clarity: 
- Submission is well written.
- Figures are very small and without captions.

### Significance: 
- Reducing over-confident predictions by classification networks is a very important field of study and this work address such for BNN. 
limitations_and_societal_impact: The authors discuss that their methods are limited to ReLU BNNs and not non-Bayesian ReLU NN. I agree with the author's assessment that societal impact is minimal.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This method analyses overconfidence in ReLU BNNs and introduces an 'infinite-feature' extension that corrects overconfidence far away from training data. The authors propose a method to post-hoc correct trained overconfident ReLU BNNs by adding a mean zero GP to the trained BNN, which has approximately zero variance on the training data, but has variance that increase cubically with distance from the training data, which is enough to guarantee uniform confidence predictions. The method is also adds minimal computational cost to standard evaluations of BNN predictive variance (which require multiple forward passes). Experimental evidence is provided both on 2d visualisations and standard image classification tasks which demonstrate improved performance over standard baselines.

*Update after rebuttal*
I am raising my score to a 7 due to the additional comparisons to baselines that the authors are including
main_review: *originality* This work builds on previous works looking at correcting overconfidence in ReLU BNNs. Whereas other works were only able to introduce methods that show that asymptotically away from the training data they will not have certain confidence, this submission provable has uniform confidence (theorem 3), which is more desirable.

*quality* The paper is well written and proposes a sensible idea for an important question (overconfidence on ood in BNNs), which is both theoretically and experimentally shown to improve performance. It's also relatively cheap to implement, which is a plus.

*clarity* The paper is well written and clear, especially with regards to flow of its arguments and motivation for the problem they consider.

*significance* Overconfidence and poor uncertainty estimates in BNNs is certainly an issue which I believe is of interest to the wider NeurIPS community. This paper introduces a solid contribution to correct this issue. I have doubts whether this method would become a commonly used post-hoc technique, especially with regards to the difficulty in tuning the introduced $\sigma$ parameter.
limitations_and_societal_impact: - As unavoidable in a lot of these uncertainty methods is the problem of tuning hyperparameters and indeed it seems like this method also has such difficulties as described in sec 3.4 (even the proposed L_OOD in fig 4d doesn't look great). I appreciate the authors for their honesty and clarity in this limitation.
- It's not clear that the choices of using the hidden layer activations in section 3.3 to better model uncertainty in deep networks is well motivated. You could pick a lot of things here, e.g. the neural tangent kernel features or restrict to the later layers where we have 'richer' features. It would be good to see some ablations here as to how this choice affects the inference of your methods.
- It would improve the paper to see some experimental comparison, in e.g. Table 1, to more recent methods in uncertainty quantification for DNNs e.g. DUQ https://arxiv.org/abs/2003.02037 or SNGP https://arxiv.org/abs/2006.10108, especially as they only require a single forward pass at test time.
review_title: null

Meta-review: As pointed out by all the reviewers, this paper seems to be addressing an important problem of over-confidence in Bayesian ReLU neural networks. The paper is well-written, the problem is well motivated and the proposed solution is well-supported with theoretical and empirical work. 
Decision: Accept (Spotlight)