Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposes $\Delta$-UQ, a novel uncertainty estimation technique with a single neural network. The main idea is to ensemble predictions computed from inputs transformed by random anchors; to justify the use of ensembling by random anchoring, the authors analyze the Neural Tangent Kernel (NTK) of the predictions evaluated at inputs shifted by anchors and show that the NTK indeed changes according to the different anchors. Then this observation is further elaborated to introduce a single neural network model, $\Delta$-UQ, where the prediction is just computed from a single neural network but using different random anchors for each input. The proposed method is demonstrated to outperform existing uncertainty quantification methods for deep learning on various benchmark tasks.
strengths_and_weaknesses: Strength
- To the best of my knowledge, the idea of ensembling from multiple shifts (anchors) is novel, and it is also good to see that the proposed method works well with a single neural network. 
- The experimental results are promising and convincing.

Weakness
- The main argument based on NTK does not correspond to the actual settings on which the neural networks are trained.
- It is rather unclear how the single neural network argument is justified (see below questions).
- No experimental results on the most standard task - image classification & uncertainty quantification for in distribution data.
questions: - Probably the most important part of the paper is the argument where the use of a single neural network on lifted space (the joint space $(c, x-c)$); but at least for me, it is unclear how this is justified. As far as I understand, what the paper is showing is as follows:
1) Derive the NTK for the shifted inputs (eq (2)) and linearized the infinite width limit of the prediction (eq (5)) in the original space ($(x-c)$).
2) Derive the same NTK and predictions on the joint space $(c, x-c)$ and show that those are similar to eq (2) and eq (5).
So what I get from this is that the NTK and predictions computed with $(x-c)$ and $(c, x-c)$ for a "specific" $c$ have similar forms; how this leads to the justification of "using a single $f$ but with multiple $(c, x-c)$ can simulate ensembling of multiple models trained with different $c$s? I'm pretty sure I'm missing something here.

- At least from the NTK argument there seems to be no restriction on the choice of $c$s, but the actual implementation of $\Delta$-UQ relies on setting $c$s as shuffled training set; wouldn't the algorithm work if $c$s are drawn from arbitrary distributions such as uniform or Gaussian distributions? 

- If I understood correctly if we set $c$s to be shuffled batches during training, for the inference, we should reload the training set and draw $c$s from it to compute predictions (eq (6)). This can actually be quite impractical especially when the size of the training set is large (e.g., ImageNet). Can you comment on this?

- The main argument (NTK) assumes infinitely wide neural networks; does the proposed method work well for narrow neural networks?

- Does the proposed method work seamlessly with batch norm?
limitations: - The NTK argument is driven by a very restricted setting; equation (1) is only for the 2-layer MLP. I think that the same argument would not extend to the NTKs for arbitrary feedforward neural networks.
- Similarly, the linearization argument in equation (5) does not work for the loss function other than $L_2$ loss. The most apparent example would be classification models trained with cross-entropies, which seem to be commonly used settings for the experiments.
- There still is a gap between what the theory is assuming and the actual settings with which the neural networks are trained; e.g., NTK initialization and small learning rate. At least for me, the NTK argument does not seem very convincing to justify the empirical success of $\Delta$-UQ.
- While the experimental results on OOD detection tasks or corrupted data classification are quite convincing, there is no result on the standard image classification tasks for in-distribution data missing; how does $\Delta$-UQ perform on image classification benchmarks such as CIFAR and ImageNet in terms of accuracy, negative log-likelihoods, Brier scores, or expected calibration errors? Does it still outperform deep ensembles?

P. S. 
There is an important relevant work worth mentioning - Teye et al., Bayesian uncertainty estimation for batch normalized deep networks, ICML 2018. Here, the authors propose a single neural network uncertainty quantification method where the epistemic uncertainty is derived by randomness in shift and scaling vectors for batch norm, and the prediction is done similarly to this paper where they randomly draw mini-batches from the training set, compute batch statistics, and ensemble the predictions computed from multiple batch statistics.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This work presents $\Delta$-UQ, a simple approach to quantify deep neural network model's predictive uncertainty. The method proceeds by train a single model with constant-shift data augmentation, and produce predictive uncertainty by averaging the prediction over multiple versions of constant-shifted testing examples.

Authors conducted NTK analysis of the neural model under constant shifts, showing that at asymptotic limit, the constant-shifted model indeed lead to a different kernel. They also conducted comprehensive evaluation of the method under outlier rejection, calibration under distribution shifts, and sequential optimization, showing improved performance across all scenarios.


strengths_and_weaknesses: **Strength**

 * A novel discovery that stochastic data centering is sufficient in producing effective predictive uncertainty.
 * A clear NTK analysis that strong motivates the approach, and elucidates it's impact on model's output uncertainty.
 * The method is simple to implement, and performs strongly on realistic benchmarks.

**Weakness**

I am largely happy with the paper's novelty, clarity, and evaluation, and find the finding significant. I only have a minor suggestion on conducting some ablation studies to further elucidate the contribution of different components, which I listed in **Question**


questions:  * Effectiveness of training-time augmentation.
   The model training procedure effectively applies a constant-shift data augmentation during training, and the inference procedure effectively applies a test-time augmentation. To this end, there exists training-time augmentation methods (e.g., AugMix) that only applies training-time augmentation, illustrating good performance. Therefore it might be interesting to decouple the contribution of the two component of the methods by studying their ablations. For example, what is the performance of a model that only receives constant-shift augmentation at training time, and at test time we only evaluate it using $[0, x]$?
limitations: The only limitation I can see is the author can consider testing it on more data modalities (e.g., UCI datasets). Given the simplicity and the generality of this approach, not illustrating its usefulness on modalities where high-quality data augmentation is not available seems to be an missing opportunity. With that said, I believe the experiments in the current paper is already sufficient to justify publication.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The authors propose to learn an ensemble of networks sharing the same initialisations on data set shifted by a constant. The shifting constant is augmented to the model's input.

The authors provide experiments on ImageNet dataset using ResNet50, experiments with sequential optimisation and GAN.
strengths_and_weaknesses: The authors propose to learn an ensemble of networks sharing the same initialisations on data set shifted by a constant. The shifting constant is augmented to the model's input.

The authors provide experiments on ImageNet dataset using ResNet50, experiments with sequential optimisation and GAN.


Like authors say, the proposed method is simple. This idea has not been considered in the literature to the best of my knowledge, but in spirit is similar to data augmentation. In a more generic way, one could learn an ensemble of networks with different data augmentations, and one would expect this increases variability of networks' predictions.

As the proposed method is simple, this paper attempts to put emphasis on the experimental side.
While the experiments are interesting, I think the results are not clearly demonstrating that the proposed method is superior. 

First, I'd like to see the likelihoods of the different algorithms compared. While the method might have better properties in terms of epistemic uncertainty, likelihood on held-out data remains a standard way to compare quality of predictions.

Second, the results in a sequential optimisation experiment are mostly not statistically significant (the error bars are overlapping). It's well-known that BNNs are heavily underfitting the data and don't work well for sequential optimisation, yet it's performance in the table is sometimes close to $\Delta UQ$ (e.g. Hartmann).

The paper is mostly clear, but leaves some important experimental details not sufficiently discussed (e.g. selection of sampling distribution of c and if it is tuned).
questions: 
Fig 1: It appears that the $\Delta - UQ$ is more overconfident on left hand side than the second fit?

Does the proposed method achieve better likelihood than standard deep ensembles (the results reported in Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles Lakshminarayanan et al.)

It is not clear at all how authors sample c, ad if this has been tuned on the actual test sets of the experiment (were the experiments run multiple times?)

limitations: The authors do not discuss if the method performs better or worse in terms of standard metrics and focus only on uncertainty metrics. The authors do not provide examples or insights were the method can fail.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The authors propose $\Delta-$UQ, an approach inspired by the observation that difference in predictions from neural networks trained on constant-bias shifted data can act as strong indicators of epistemic uncertainty. For a random bias (anchor), it is shown that an "anchor ensemble" can be approximated using a single model. When evaluated on Bayesian optimization benchmarks, the authors demonstrate competitve performance hinting that the uncertainties learned are meaningful.
strengths_and_weaknesses: Strengths:

* The authors take a different perspective to ensemble construction. Instead of working with samples of neural network parameters, they work with randomized constant-shifted datasets. The ability construct this ensemble using a single model is very practically appealing.

Weaknesses:

The authors aim to solve for well-calibrated estimate of epistemic uncertainty, which would imply uncertainty in the model definitely by parameters $\theta$. However, at test time, the authors seem to marginalize over $c$ for a given $\theta^\star$ instead of marginalizing over $\theta$. There is nothing inherently wrong with this, but it is a subtly different notion of uncertainty than what the competing methods provide. In that sense, the methods may not be entirely comparable.

Alongside the calibration metrics, it will be important to take a look at the performance numbers. Often times, calibration and predictive performance can be at odds and it will be helpful for the reader to understand what kind of numerical trade-off to expect.

Overall, I think this is technically sound and a promising direction to investigate further.

Minor:

* The GAN experiments could use a more elaborate description
questions: * Is there a consistency result for $\Delta-$UQ that says that in the limit of data, uncertainty goes to zero? This would only be true for epistemic uncertainty, which appears to be the focus of uncertainty quantification in the main text. If not, then I think the distinction of aleatoric/epistemic should just be removed from the main text since it does not really add value and probably just talk about the merits of predictive uncertainty that the method provides.

* As noted earlier, can we look at the raw performance numbers for the experiments in Table 1?
limitations: Yes.
review_title: null

Meta-review: metareview: The paper proposes a method that allows single model uncertainty estimation by training a model with a random data augmentation. The proposed approach is simple and scalable. It is comparable to or better than deep ensemble in terms of NLL, ECE, and Brier score. The application to sequential optimization tasks presented in the paper looks interesting. All reviewers support accepting this paper. While there could be more theoretical support, I think this paper would be of wide interest to the NeurIPS community. if possible, accept as Spotlight.
confidence: Certain
Decision: Accept