Single Model Uncertainty Estimation via Stochastic Data Centering
Authors: Jayaraman J. Thiagarajan
Rushil Anirudh
Vivek Narayanaswamy
Peer-timo Bremer
Venue: NeurIPS 2022 Accept (2022)
Keywords: uncertainty quantification
calibration
active learning
deep ensembles
sequential optimization

Abstract:   We are interested in estimating the uncertainties of deep neural networks, which play an important role in many scientific and engineering problems. In this paper, we present a striking new finding that an ensemble of neural networks with the same weight initialization, trained on datasets that are shifted by a constant bias gives rise to slightly inconsistent trained models, where the differences in predictions are a strong indicator of epistemic uncertainties. Using the neural tangent kernel (NTK), we demonstrate that this phenomena occurs in part because the NTK is not shift-invariant. Since this is achieved via a trivial input transformation, we show that this behavior can therefore be approximated by training a single neural network -- using a technique that we call $\Delta-$UQ -- that estimates uncertainty around prediction by marginalizing out the effect of the biases during inference. We show that $\Delta-$UQ's uncertainty estimates are superior to many of the current methods on a variety of benchmarks-- outlier rejection, calibration under distribution shift, and sequential design optimization of black box functions. Code for $\Delta-$UQ can be accessed at github.com/LLNL/DeltaUQ


Single Model Uncertainty Estimation via Stochastic
Data Centering
Jayaraman Thiagarajan" Lawrence Livermore National Laboratory jjayaramelIni
Rushil Anirudh"
Lawrence Livermore National Laboratory anirudh1 helnnl gov
Vivek Narayanaswamy
Peer-Timo Bremer
Arizona State University
Lawrence Livermore National Laboratory bremer5611m gov
vnaray29Gasu, edu
Abstract
We are interested in estimating the uncertainties of deep neural networks, which play an important role in many scientific and engineering problems. In this paper, we present striking new finding that ensemble of neural networks with the same weight initialization, trained on datasets that are shifted by constant bias gives rise to slightly inconsistent trained models, where the differences in predictions are strong indicator of epistemic uncertainties. Using the neural tangent kernel (NTK), we demonstrate that this phenomena occurs part because the NTK is not shift-invariant. Since this is achieved via trivial input transformation, we show that this behavior can therefore be approximated by training single neural network using technique that we call A A- that estimates uncertainty around prediction by marginalizing out the effect of the biases during inference. show that A A-UQ is uncertainty estimates are superior to many of the current methods on a variety benchmarks- outlier rejection, calibration under distribution shift, and sequential design optimization of black box functions. Code for A A-UQ can be accessed at Wuthu..ommLLLN//Detttlt
Introduction
Accurately estimating uncertainties in a deep neural network (DNN) is active area of research due to its implications in a wide range of scientific and engineering problems. Broadly, there are two kinds of uncertainties that are often considered (a) aleatoric: uncertainties the data generating process that are typically irreducible, and (b) epistemic: uncertainties of the model that can be reduced by observing more data, which is the focus of our work. Some of the popular techniques for the latter include Bayesian methods [1-4] that use prior on the network weights, Monte Carlo approximations such as MC Dropout [5] that approximate sampling from the posterior, and empirical methods such as Deep Ensembles (DEns) [6]. In particular, DEns trains an ensemble of neural networks with different initializations, such that the uncertainty estimate on a test sample is given by the inconsistency between predictions from the member models. In practice, DEns has been found to often outperform other related methods [7, 6, 8), but it comes the cost training several DNNs to obtain reliable uncertainties (typically 10 20), which is severe computational bottleneck when it comes to modern DNNs especially on large scale datasets. light of this, there is increased interest in developing single model estimators that still produce high quality uncertainties. There has been some promising work in this direction in the recent past, specific to deep classifiers [8] or regression models [9], and has been shown to perform comparably to DEns in some use-cases.
"equal contribution
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

In this work, we first begin by exploring an alternate approach for constructing deep ensembles instead of using multiple randomized weight initializations, we propose shift the training domain using random biases, such that every model in the ensemble is trained with data that has been shifted by a different constant bias c. Formally, for a labeled dataset {(x, y)}. the t model is trained to fit {(x - ck: }) where cg (referred as an anchor) is of the same size as x. Though this manipulation appears trivial, the kernel induced by deep networks is not inherently shift invariant [10, I1], thus implying each DNN learns slightly different model due to the bias. This leads to one of our key observations:
Anchor Ensemble: When an ensemble of DNNs, with the same fixed initialization, is trained on a dataset shifted by random constant biases, the variation across the ensemble's 's predictions is a strong indicator of model uncertainty.
Based analysis with the neural tangent kernel (NTK) [10], a random variable, the corresponding kernel is stochastic such that for each c, the model converges to slightly different NTK. Consequently, this anchoring-based ensembling provides different approach to DEns for sampling solutions from the hypothesis space. However, interestingly, since the anchor ensembling emerges primarily from the act of a trivial data transformation, it lends itself be approximated easily using a single DNN:
we show that when the anchor is made
A U-UQ: For randomly chosen anchor c, 1 can be approximated using single
DNN trained on the dataset transformed as {x. v) - {[., - -c.,). y). 2
During inference, we obtain multiple predictions for a given sample by varying the choice of the anchor, such that the standard deviation of the predictions is our estimate for uncertainty. Without affecting the performance of the model, we find that A-UQ produces meaningful epistemic uncer- tainties that we validate in a variety of applications: outlier rejection, calibration under distribution shifts on ImageNet, and sequential optimization of a large suite of black-box functions. We observe that A-UQ consistently outperforms existing uncertainty estimates, while also being efficient to train as single model estimator. With just few simple changes, one can easily modify the training of any existing DNN to support A A-UQ estimation.
2 Background and Related Work
Denote training data as D = (xx;, y:) where x, E X' and y: E V, to train a neural network f(0) H with randomly initialized weights 9,, such that a loss function L is minimized, i.e., arg ming (x[(x;) 0), v)- Here, H denotes the hypothesis space of potential solutions for fitting the observed data. Given prior distribution on the weights p(0), we can define the posterior over 0 as P()(D) and subsequently, infer the posterior predictive distribution for a test sample (x1, Wt). This can be used to quantify the uncertainty around the prediction as p(ye Xx,, D) So p(y Xx, (O)))))
A challenge for neural networks, however, is that the posterior p(0) is computationally intractable. This has motivated the use of Bayesian Neural Networks (BNNs) [3] using different approximations to the posterior including Monte-Carlo Dropout [5]. variational inference [12, 4], and sampling methods such as Markov chain Monte Carlo [3. 13]. In parallel, it has been empirically shown that Deep Ensembles (DEns) [6] often tend to outperform Bayesian methods in terms of model calibration performance, even under challenging distribution shifts [7]. The success of DEns has been attributed to its ability to sample differrnnt functional modes [14] from the hypothesis space, and thus approximate the posterior predictive distribution [1]. However, a critical limitation of DEns is the need to train multiple models (typically 10 -00) in order to obtain well calibrated uncertainties, which can be impractical for complex model architectures that have become commonplace today.
Characterizing the behavior of deep uncertainty estimators has mostly been done using empirical evaluation based on model calibration or out-of-distribution detection, but the recent advances the neural tangent kernel (NTK) theory [10, 15-17] provide convenient framework for more rigorous analysis. The basic idea of NTK is that, when the width of a neural network tends to infinity and the learning rate of stochastic gradient descent (SGD) tends to zero, the function fx; 0) converges to
solution obtained by kernel regression using the NTK defined as Kx.x, = Eg
0)) 00)
80
80
When the samples Xy E 8d-1, i.c. points on the hypersphere and have unit norm, the NTK for
2

---

simple 2 layer ReLU MLP can be simplified as dot product kernel [15-17]]:
X,,,, = HINTK (x. xz)
--m x x 2xt
- cos (xos=((x]) x,))
The NTK framework and its extensions that enable a posterior interpretation in the infinite limit have been used to study deep ensembles [2]. Previous work [18-20] has also shown the existence of distinct yet related kernel, referred as the neural network Gaussian process (NNGP) kernel, where the initialization tends to GP in infinite width limit.
3 Uncertainty estimation with A A-UQ
As discussed above, most existing frameworks sample the hypothesis space either through different random initializations of (0o), perturbing the weight space after training, using Bayesian methods by placing a prior on p(0). Here, we propose to use a new strategy that involves injecting multiple trivial biases into the training data and analyze the resulting models using the NTK framework.
3.1 Anchor Ensembles: ensembling by injecting trivial biases
Let examine the scenario where we shift entire dataset (both train and validation) using constant bias, c, to obtain a new dataset De using which we train the model fe- Since we always choose from the training distribution at random, this has the effect of zero-centering the dataset around different training points. Let J Je....., Sc. different c's, then our goal to characterize the relationship between them as function of If the NTK induced by f is shift-invariant (for e.g., when Fourier features [11] are used), the shifts will make no difference, resulting in identical models fo, models like MLPs and CNNs are not inherently shift-invariant [17] we find that the models lead to an effective deep ensemble, wherein the variation across the predictions is strong indicator of epistemic uncertainties.
denote the set of models trained using
fee. However, since NTKs for
Effect of shifted training on NTK: We interested in understanding how (1) changes when the entire training domain is shifted by c- i.e. hNTK e)x(x, - c)). Without loss of generality, assume x; and X, are also made unit norm. To
for the sake of notational convenience, simplify the expansion, we use Taylor series expansion for the cos function: cos (u c) R cos (u) (u-(u-))²
Expanding (x; c) (x₃ - as x, cx, +X c) and letting (x: Xj c). obtain the expression for bNTK under a shifted domain as follows:
1
c. () cos -1 -xx - cTv)
(xi- (xx -c)
(x
xj
2x
c(x] xj - cv 2TT / (x, x, vv2
x; ( cos -1 2xxx x4)) (xx]xx)))
c. Tv(TT- cos cos-cos--- "x, x,))
2T
2x
= -K
T Nx,x3.
(
where we combine all terms dependent on into I'x, e, which also behaves as dot product kernel. From (2), we note that a trivial shift in the domain results in non-trivial shift in the NTK function itself. In other words, (2) outlines the effective NTK as a function of c. We also note that I does not affect the spectral properties of the original NTK, as we observe in Figure!
Let us now consider the prediction on a test sample x, in the limit as the inner layer widths grow to infinity. It has been shown that (c.f. [17, 16]):
fxo fo(x,) K XxxxXxX(oOOX Y).
By shift-ing inffit-ivariaeee, we refer to the domain shift induced because of the random bias, opposed to spatial hhift-invariance that is more commonly associated with convolutional operators.

---

10P
NTK Fourier spectra
NTK spectra of anchor ensemble
NTK spectra A-UQ
A
B
C
10
Skgalaeee
20*
10a
10a2
10
-1
-1/2
7/2
x/2
-X/2
x/2
Etoquaay
Doqqary
with positional embedaing
E
F
10f
10*
10f
10-
10-4
n/2
8/2
0
1/2
6/2
0
5/2
Freguncy
Supeay
Figure 1: Fourier spectrum an NTK an MLP model (A.D); spectra of an anchor ensemble (B, E); and NTK spectra using A A-UQ (C, F). Bottom row shows NTK spectra when inputs are passed through sinusoidal PE. We make two key observations a) trivial shifts in the input domain cause the effective NTK to be distinet function of the shift c, as in eqn. (2); and b) AUUQ achieves similar effect but with single model.
where X is the matrix of all training data samples. As before, we consider the case where the domain shifted by c. Using (3):
foo (xe - c) = fu(x, - c) - K -)) (x x
- Y)
fo(x - c) -KKxx
- - - -) XY
fu(xe - c) (Kxix Xe) Kxx+ + Σ (K (Kxx " (fo(X - c) Y)
m==
(4)
fo(x)) XxxXxx((f(X (fi(X) Y)
(5)
deterministic for fixed 0g
random due to
In (4), we utilize Woodbury's Inverse Identity [21] to expand the inverse of a sum of matrices. Next, in (5), we combine all the terms dependent on c into function g. Note, we also expand fo - c) using the Taylor series approximation to represent as a sum of fo(x) and other terms (details of this derivation can be found in the appendix).
For given initialization tg the first term is deterministic, and exactly the same as (3), since all other the terms are fixed. However, we see that the second term can vary based on the choice of c. Existing ensembling approaches [6] rely on the randomness of the initialization On to pick diverse solutions from the hypothesis space [2]. In contrast, equations (2) and (5) suggest that even for fixed 0g it is possible to make the NTK stochastic (in c) by shifting the entire input domain. To better understand how the NTK actually changes from these expressions, we study the spectral properties of a simple MLP network empirically, following the analysis in [1]].
Spectral properties of shifted NTKs. We compute the Fourier spectra using the same MLP on several shifted domains in Figure (BB.. The original spectrum for the MLP without any shift in the training domain is shown for comparison in I(A). As indicated by (2), we see that each individual shift leads a different NTK (as indicated by the spectra), either by flattening in the frequency domain. The same behavior persists even when we construct positional embeddings (PE), based sinusoidal functions, prior building the MLP model (I(D-E)). This leads to one of our main findings stated in and illustrated in Figure 3.
making narrower
3.2 A A-UQ rolling anchor ensembles into single model
Since different models in an anchoring-base ensemble are trained with the same initialization, we present new technique to approximate the epistemic uncertainties using a single neural network. More specifically, we perform a simple coordinate transformation by lifting the domain to higher dimension E {c,x c}. we refer to the residual by A x This transformation

---

Truel Function Estimate
Training Samp Devistion
Tue Function
Training Samples
Target
Target
-1
Anchoring-bawet Ensembles
UQ Training
0.5 10
2.0
2.5
1.
1.5
2.0
Input
Input
Figure 3: Comparing anchor ensembles and A A-UQ in function fitting with an MLP. As expected, we see that the disagroement between models in an anchor ensemble correlate strongly with the epistemic uncertainty, and that A UQU,, with single model, matches this behavior very closely.
allows the use of multiple representations (w.r.t. different anchors) for the same input sample i.., Sa (fcc, 3) /a((c2, (2))
Sa((c, -c,)), where Sa refers to the UQ model that takes the tuple ((C&, x - )x)) and predicts the target
It is easy to see that [; cc X. - cl x, xj c + +x; 2c). That is, the dot product of the transformed inputs takes the same form as before (except for scaling factor). Therefore, the expressions for the equivalent NTK for different anchor shifts, seen in (2), and the corresponding prediction on a test sample seen in (5) remain the same for A-UQ by setting similar form is also obtained in the more general case when different anchors are used for different samples (as shown the appendix), which leads our main claim stated in 2 that the A-UQ model achieves similar perturbations of the NTK as an anchor ensemble, based on the choice of c.
X; + x, 2c.
Training. During training, for every input x, choose anchor random sample from the training dataset. Subsequently, we obtain the coordinate transformation {[c, c] 3r using which train the model. With vector-valued data, this is implemented as simple concatenation. In the case of image data, we append the channels to create a 6 dimensional tensor (for a 3-channel RGB image). We show simple Py Torch snippet for training A A-UQ in Figure 2. Other than increasing the number of parameters in the first layer of the network, -U does not incur additional computational overheads.
Over the course of training, expectation, every train- ing pair gets combined with large number of anchors. Since the prediction on this training pair regardless of anchor choice must always be the same, this places implicit consistency the predictions that they must be similar matter which anchor chosen. This consis- tency trades-off with diversity of the kinds of functions that can be learned when compared with anchor semble, where the models are trained independently. This can be seen in the comparisons of the NTK spectra for A UQ with anchor ensembles in Figures ICC) and (F). In practice, however, we find that the diversity from this single model is still sufficiently large, estimate good quality uncertainties.
Figure 2: Mini-hatch training with A U-UQ.
for inputs, targets in trainloader:
Chuffle(inputs. xx Anchors
D inputs x% Delta
X_d torch. carch.cat(e cat((A, D] axissi)
y_d model(X_d) x% prediction
loss criterion (y_d, targets)
Inference. For a test sample Xt, we obtain the prediction from U- as the mean prediction across several randomly chosen anchors; and the standard deviation around these predictions is our estimate for the epistemic uncertainty. In other words, we marginalize out the effect of anchors to obtain the final prediction mean and uncertainty. Formally, the predictive distribution is given by p(ye ) = Josx P(ya Xxe, c, (c)ddd. In practice, for a trained A-UQ model specified as 0", we compute the sample mean and uncertainty around it as:
K
K
1
p(y((x))= =
f((ck.x 0"); (
Σ Km
(f((ce - -c,, - 22),
K
K
(6)
5

---

Discussion. In Figure 3, we show a ID regression example using 20 training examples along with the predicted mean and estimated uncertainties. As it can be seen, both the anchoring- based ensemble (left) and A 2- UQ training show higher epistemic uncertainties around regions with no training samples. the ensembles version, we train 20 different networks while in UQ we use just single network, where the uncertainty is obtained using all 20 anchors during inference.
We always use only a single anchor for an input during every training iteration, but multiple anchors during inference time. In theory, multiple anchors could be used during training as well, where the loss is imposed on the mean (obtained with multiple anchors). However, we find that simply using a single random anchor in each iteration achieves a similar effect, as it enforces consistency that the same training pair (x. y) when combined with many different anchors over the course of training as [c,,x; ci], [C2, X2 c2)...., [,,, X, ca. must all produce the same prediction, y/-
A-UQ relies on randomly drawn anchors for uncertainty estimation, which is similar to DEns [6]. that relies on the diversity of the base learners in an ensemble. Arguably, sampling a random set of anchors from the training distribution is simpler than sampling from the posterior pD) Furthermore, since every anchor realizes a slightly different function, using a small number of anchors (10 - 20) during inference is typically sufficient obtain high quality estimates as we show in our experiments. Finally, due to the nature of training with random anchors we also see that A UUUQ produces particularly effective uncertainty estimates when the training set size is small, and this proves to be very useful in applications such as sequential optimization.
4 Experiments
We validate our approach in this section using a variety of applications and benchmarks (a) first, consider the utility of epistemic uncertainties in object recognition problems where they have been successfully used for outlier rejection and calibrating models under distribution shifts. We show that A-UQ can be very effective even with large-scale datasets like ImageNet [22]; (b) next, we consider the challenging problem of sequential design optimization of black-box functions, where the goal is to maximize a scalar function of interest with the fewest number of sample evaluations. Using a Bayesian optimization setup, we show that the uncertainties obtained using UQ outperform many competitive methods across an extensive suite of black-box functions.
Outlier rejection. A popular application for epistemic uncertainties is in rejecting outliers since, by definition, they are in regions outside of the training distribution. As such, we expect the model to produce highly uncertain predictions for these images, which should help us design an effective OOD detector. To evaluate this hypothesis, we train a modified ResNet-50 [23] model on ImageNet that accepts 6 input channels (anchor, A) as outlined earlier. We train the model using standard hyperparameter settings except, training it longer for 120 epochs - our top-1 accuracy (76. matches that of a standard ResNet-50. Specifically for images, we found that corrupting the anchors with common transforms like random crops, Gaussian blurs, and color jitter improves performance. That is, instead of [c. x c), we use (T(c), x c) where TT a simple transform like Gaussian blur. The uncertainty for a test sample is given by the standard deviation of the logits obtained by varying the anchors. To obtain a scalar statistic, we compute the mean across all classes as the uncertainty score for that sample. We provide more details on the transformations used for corrupting the anchor, and the accuracies of our model across shifted variants more detail in the appendix.
We show the results for outlier rejection in 4b, where we follow the protocol established in [25], that uses a Gaussian blur of intensity 5 from ImageNet-C [26] as the outlier set, and the clean ImageNet validation data as inliers. We use the estimated uncertainty obtained with 10 anchors as in (6) as our score for outlier rejection and report commonly used metrics such as AUROC, Detection Accuracy (DTACC), and AUPR-in/out We note that, just the inconsistency of predictions obtained using A A-UQ outperforms many baselines including mean-field stochastic variational inference (SVI) [12, 4], SVI-AVUC [25], Monte Carlo dropout [5], and temp. scaling [24]. While this can be further improved by taking the mean prediction into account, similar to existing approaches for semantic novelty detection (with scores such as entropy, energy [27] etc.), our focus here is to evaluate the quality of uncertainty alone. In Figure 4a, we observe that A AUUQ 's uncertainty estimate changes smoothly the outliers become farther away (more severe intensity) from the training distribution.

---

Caussion Blur: Level 5
Caussion slur Level
Coussian Blu: Level
Method
AUROCT
DTACC1
AUPR-in/one
Garaxian Bur lesel
Gaussian Blur
ResNet-50 [2:3]
93.36
86,08
92.82 / 99.71
Donsity
iageetet
Temp-Scal [24]
93.71
86.47
93.21 09,11/94.01 994001
Deep Ens [6]
95.49
88.82
95.31 / 95.64
MC Dropout [5]
96.38
89,98
96,16 / 96,67
SVI [4]
96.40
90.03
95.97 / 99.83
A UQ (ours)
97.49
91.90
97.56 / 99.447
Estimated Uncertainly
(b) Uncertainties from A-UQ for outlier rejection
(a) Uncertainties change meaningfully as out- liers get more severe
Figure 4: Rejecting outliers with epistemic uncertainties: We evaluate A UQ on the benchmark introduced by [25] where use Gaussian Blur of level 5 intensity the outliers from the ImageNet validation set. At inference, uncertainties are estimated the sum of std. dev of predictions obtained with 10 anchors.
Table 1: Calibration under distribution shift: A ResNet-50 model that is tempered by uncertainties obtained from UQ (sec text) outperforms several competitive baselines averaged across 16 different corruptions of ImageNet-CC C highest severity level
Temp Scaling
LL-
SVI-
Metric
Vanilla
DEns
MCD
SVI
Ours
Dropout
AvUC
lower quartile
0.124
0.096
0.050
0.078
0.093
0.072
0.032
0.022
median
0.174
0.139
0.090
0.134
0.145
0.114
0.045
0.038
ECE
mean
0.194
0.160
0.088
0.153
0.161
0.119
0.054
0.044
upper quartile
0.274
0.236
0.126
0.219
0.236
0.172
0.070
0.063
lower quartile
4.635
4.53
4.035
4.699
4.563
4.322
4.164
4.014
median
5.115
4.993
4.624
5.093
5.034
4.853
4.823
4.617
NLL
5.234
5.091
4.604
5.553
5.201
4.865
4.707
4.352
upper quartile
6.292
6. 6.165
5.893
6.522
6.342
6.034
5.778
4.987
lower quartile
0.941
0.926
0.877
0.933
0.923
0.906
0.883
0.868
median
0.987
0.970
0,922
0.967
0.969
0,943
0.935
0,925
Brier +
0.964
0.945
0.888
0.961
0.947
0.922
0.900
0.887
upper quartile
1.052
1.027
0.989
1.025
1.025
1.013
0.985
0.949
Following our observation that our uncertainty estimates are effective in rejecting outliers in 4b, here we study they can be leveraged to calibrate ImageNet models under distribution shifts. To calibrate classifier, we simply scale the logits of the mean
b. Calibration under distribution shifts.
by the uncertainties as follows: Healib. 11 a), where ō the standard deviation estimated from (6), where once again for classification we simply compute the average of standard deviation across all the 1000 classes, followed by min-max normalization to [o, 1). This simple scaling of the mean reflects our prior belief highly certain prediction must remain unchanged, whereas an uncertain one gets tempered down. Note, this scaling is applied to logits from all classes and hence the accuracy of the mean remains unchanged before or after calibration. We evaluate how calibrated the predictions are using three commonly used metrics Calibration еттоr (ECE), negative log likelihood (NLL), and Brier Score. We use the same ResNet-50 classifier trained on ImageNet as before, and measure calibration for predictions on 16 different ImageNet-C corruptions at severity We list the corruptions and show examples of them in the appendix. We report the 25th (lower quartile), 50th (median) and th (upper quartile) along with the mean of the three metrics across 16 corruptions Table 1. Across all measures we see that A-UQ is able to calibrate models better, even in comparison to state-of-the-art approaches that use explicit calibration objective to adjust prediction probabilities, further validating the quality of its uncertainty estimates.
Sequential Optimization. Denoting sigh-dimensional function as f D - R. our goal to solve the following optimization problem: x* arg maxxeD [(x). Here, D refers to a bounded design space comprising D different parameters with their corresponding value ranges (Ea, ha), Vil 1... D.. The high computational or financial cost of evaluating I (invoking a simulator rumning experiment) motivates the additional objective of minimizing the number of evaluations.

---

Table Sequential optimization: We rigorously evaluate the performance different uncertainty estimators on a suite of black-box functions and report the AUC metric (1) averaged across multiple random seeds and trials. In cach case, we also indicate the number of initial samples and optimization steps.
Function
Dim.
Init.
Steps
GP
MCD
BNN
DEns
Ours
Multi Optima
25
0.51 + 0.2
0.45 = 0.16
0.64 + 0.12
0.28 + 0.17
0.73 + 0.09
Ackley
25
0.23 1: 0.08
0.76 0.03
0.71 +: 0.1
0.75 0.04
0.83 1: 0.03
Beale
0.64 0.31
0.55 0.22
0.27 0.17
0.81 0.03
0.85 1 0.04
Booth
2
26
0.39 = 0.21
0.55 = 0.14
0.3 = 0.2
0.68 = 0.06
0.79 + 0.04
Branin
25
0.35
0.28
0.28 0.19
0.22 = 0.14
0.46 + 0.1
0.67 + 0.06
Bukin
25
0.36
0.12
0.55 0.07
0.38 U.11
0.59 0.11
0.76 0.1
Camel
25
0.83
0.08
0.86 = 0.06
0.84 = 0.03
0.83 0.07
0.89 + 0.03
Dropwave
2
5
25
0.68 + 0.15
0.57 0.18
0.67 + 0.13
0.67 + 0.11
0.79 + 0.14
Griewank
25
0.83 1: 0.02
0.74 0.04
0.59 0.17
0.7 1: 0.14
0.86 1: 0.03
Holder
0.12 0.06
0.36 0.28
0.36 0.37
0.39 0.29
0.57 1. 0.07
Levi N.13
2
25
0.26 = 0.26
0.75 = 0.1
0.7 = U.1
0.6 + 0.11
0.87 : 0.07
Lovy
25
0.57 0.18
0.61 = 0.32
0.55 = 0.16
0.59 + 0.16
0.83 + 0.03
Hartmann
0.57 0.07
0.49 0.11
0.46 0.16
0.53 0.15
0.68 + 0.07
Ackley
10
25
0.17 1 0.02
0.06 0.04
0.1 1. 0.03
0.14 1 0.02
0.59 1 0.05
Griewank
4
10
25
0.37 + 0.08
0.47 0.06
0.30 = 0.05
0.43 1: 0.07
0.69 1 0.07
Levy
4
10
25
0.1 : 0.08
0.4 0.3
0.27 0.21
0.21 1: 0.1
0.62 0.2
Hartmann
0.15 :: 0.01
0.2 0.08
0.1 0.05
0.15 0.04
0.27 1 0.15
Ackley
10
50
0.06 + 0.09
0.09 = 0.13
0,08 = 0.12
0.11 + 0.05
0.36 + 0.09
Griewank
10
50
0.07 0.02
0.12 = 0.13
0.08 = 0.07
0.19 + 0.07
0.32 + 0.11
Levy
0.12 0.04
0.16 0.08
0.11 0.07
0.13 0.07
0.47 + 0.03
Avg. Rank
3.8
3.0
4.1
2.95
1.0
Given the high-dimensional nature of the design spaces, a simple brute-force search space- filling random sample designs [28] often require significantly large sample sizes to identify the optima, thus motivating the use of sequential optimization strategies. In particular, Bayesian Optimization (BO) techniques based on statistical surrogates (e.g., Gaussian processes) form an important class of solutions [29]. In a nutshell, given initial experiment design and their function evaluations, {x,, [(x:))]a, V- sequential optimization techniques incrementally select candidates to achieve the so-called exploration eetiiion-exlliiattoon trade-off using an appropriate acquisition function [30]. In this study, we use the popular expected improvement (EI) score to perform candidate selection.
Sctup: In this experiment, we consider large suite of black- box optimization functions with varying dimensionality (1 to 8) and complexity. We provide definitions and details of these functions used in our study in the appendix. We also perform an experiment with pre-trained generative model (GAN) trained on MNIST handwritten digits, wherein we perform optimization in the 100-D latent space Z such that thick- ness of the resulting digit is maximized: Σ, T(x, > 0), Vi, where Il denotes the identity function. We use the following baseline uncertainty estimation approaches in our study: (i) Gaussian processes (GP); (ii) Monte-Carlo dropout (MCD); (iii) Bayesian neural networks (BNN) trained via variational inferencing: and (iv) deep ensembles (DEns). For all neu- ral network surrogates, we computed positional embeddings (sinusoidal) of the raw parameter inputs prior to building a fully-connected network with 4 hidden layers each containing 128 neurons and ReLU activation. All methods were trained with the same set of hyperparameters: Adam optimizer learning rate and 500 epochs, except for BNN, which required 1000 epochs for convergence. With MCD, we used
0.
Evaluation Metric
E07
seen
0.6
0.5
AUC: 92.68
0.4
0.3
0.2
5
10 Iteration
15
Figure 5: Area under the curve (AUC) metric for evaluating sequential optimization performance.
8

---

50 forward passes at test time for each sample to obtain the uncertainties, Finally, with A A-UQ, we the number of anchors for inferencing as min(20, r2), where is the number samples in the observed dataset in any iteration.
AdMyy Fanction
Griewank
Endket
DIM
DIM
DIM
Figure 6: Convergence curves obtained with different uncertainty estimation methods: We show the best function value achieved for three different functions at dimensions 4 8 respectively (for random seed, trials). We find that A -UQ consistently outperforms other baselines. The effectiveness of our approach in producing meaningful uncertaintics at small sample sizes becomes more apparent as dimensionality increases.
Iatent
E Rd
Thideness
Thickness 251
Thicknem
Thictrane 260
Thlonees 276
GAN
synthesis
X
Init.
Steps
GP
MCD
BNN
DŻas
Preposed
Σ Ms 68
50
203.8 19.4
252.2 17.3
249.7 7 4: 16.5
266.6
Figure GAN-based optimization: A-UQ consistently produces images with higher function values (thickness) for the same sampling budget, when compared to existing baseline methods.
The DEns model was constructed using constituent members (increasing this did not provide any benefits), each trained with a different initialization. The number of initial samples and the number of steps in the sequential optimization were set to the same across methods. In each round of the Bayesian optimization, we used 10, 000 samples for initialization and 15 restarts (i.e., starting points for multistart acquisition function optimization), and finally one candidate (g = 1) was evaluated with the black-box function and added to the observed dataset. We performed experiments with 5 random seeds (different initializations). each for independent trials. Since the goal is to reach the global optima with the fewest number of samples, use the widely adopted area under the iteration best achieved function value curve to obtain holistic evaluation of different approaches (see Figure 5).
Results: From table 2, we find that A-UQ produces significantly higher AUC scores in comparison to existing baselines, across all benchmark functions. While MCD and DEns behave reasonably well in low dimensions, their performance suffers when we go to higher dimensions (see figure 6). Furthermore, we find that the performance of BNN generally lower due to the inherent small samples sizes that operate in. Finally, shown in figure 7, our approach consistently achieves

---

higher function values in the MNIST GAN-based optimization, thus validating the quality of the uncertainties produced via anchoring.
Broader Impact
We presented simple, scalable, and accurate single model uncertainty estimator that outperforms many existing techniques. Uncertainty quantification (UQ) plays a significant role in a variety of applications in science and engineering, from developing safeguards in critical applications such as healthcare, security, or finance to exploration of novel design spaces using sequential optimization. Our paper addresses both of these applications using demonstrative problems to showcase the potential capabilities. Our goal is to present a fundamental approach for uncertainty estimation that maybe applicable in different kinds of problems and settings. Due its fundamental nature, we do not foresee misuse directly. However, we note that our method does not mitigate biases that may come with the training data, which could potentially affect the quality of uncertainties themselves.
Acknowledgement
This work was performed under the auspices the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, Supported by the LDRD Program under projects 21 21-ERD-0288, 22-ERD-0066 with IM release number LLNL-JRNL-836221
Appendix
The appendix contains the following additional information: (a) expanded derivations, (b) Py code snippets for Á training & inference; (c) results on calibration in standard regression settings on the UCI benchmarks dataset; (d) results on accuracy of predictions on ImageNet and CIFAR- 10 and their corrupted variants; (c) calibration results on CIFARIO-C; (f) ablations on anchor-base training; (f) details on the design optimization experiment, including definitions for the benchmark functions and convergence plots.
References
[1] Andrew G Wilson and Pavel Izmailov. Bayesian deep learning and probabilistic perspective of generalization. Advances in neural information processing systems, 33:4697-4708, 2020. (Cited on 2,
[2] Bobby He, Balaji Lakshninarayanan, and Yee Whye Teh. Bayesian deep ensembles via the neural tangent kernel. Advances Neural Information Processing Systems, 33:1010-1022, 2020. (Cited on 3, 4)
[3] Radford M Neal. Bayesian learning for neural networks, volume 118. Springer Science & Business Media, 2012. (Cited on 2)
[4] Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. In International Conference on Machine Learning, pages 1613-1622. PMLR, 2015. (Cited on 1, 2, 6, 7)
[5] Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning, pages 1050-1059, PMLR, 2016. (Cited on 1, 7)
[6] Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in neural information processing systems, 30, 2017. (Cited on 2, 4, 6,
[7] Yaniv Ovadia, Emily Fertig, Jie Ren, Zachary Nado, David Sculley, Sebastian Nowozin, Joshua Dillon, Balaji Lakshminarayanan, and Jasper Snoek. Can you trust your model's uncertainty? evaluating predictive uncertainty under dataset shift. Advances neural information processing systems, 32, 2019. (Cited on 1,
10

---

wwwpppeeeeeeeeeeeeeeeeeeeeeeeeee [8] Joost Van Amersfoort, Lewis Smith, Yee Whye Teh, and Yarin Gal. Uncertainty estimation using single deep deterministic neural network. In International Conference on Machine Learning, pages 9690-9700. PMLR, 2020. (Cited on 1)
[9] Moksh Jain, Salem Lahlou, Hadi Nekoei, Victor Butoi, Paul Bertin, Jarrid Maksym Korablyov, and Yoshua Bengio. Deup: Direct epistemic uncertainty prediction. arXiv preprint 200xx:2000.08000,, 2021. (Cited on 1)
[10] Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems, 31, 2018.
(Cited on 2)
[11] Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Keill, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems, 33:7537-7547, 2020. (Cited on 2, 3,4)
[12] Alex Graves. Practical variational inference for neural networks. In Advances in neural information processing systems, pages 2356..356. Citeseer, 2011. (Cited on 2, 6)
[13] Max Welling and Yee W Teh. Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th international conference on machine learning (ICML-11), pages 681-688 Citeseer, 2011. (Cited on 2)
[14] Stanislav Fort, Huiyi Hu, and Balaji Lakshminarayanan. Deep ensembles: A loss landscape perspective. arXiv preprint arxi 191200777 2019. (Cited on 2)
[15] Sanjeev Arora, Simon Du, Wei Hu, Zhiyuan Li, and Ruosong Wang. Fine-graing analysis of op- timization and generalization for overparameterized two-layer neural networks. In International Conference on Machine Learning, pages 322-332. PMLR, 2019. (Cited on 2)
[16] Alberto Bietti and Julien Mairal. On the inductive bias of neural tangent kernels. Advances in Neural Information Processing Systems, 32, 2019. (Cited on 3)
[17] Jachoon Lee, Lechao Xiao, Samuel Schoenholz, Yasaman Bahri, Roman Novak, Jascha Sohl- Dickstein, and Jeffrey Pennington. Wide neural networks of any depth evolve as linear models under gradient descent. Advances in neural information processing systems, 32, 2019. (Cited on 2, 3)
[18] Jaehoon Lee, Jascha Sohl-dickstein, Jeffrey Pennington, Roman Novak, Sam Schoenholz, and Yasaman Bahri. Deep neural networks gaussian processes. In International Conference on Learning Representations, 2018. URL (Cited on
[19] Alexander G. de G. Matthews, Jiri Hron, Mark Rowland, Richard E. Turner, and Zoubin Ghahramani. Gaussian process behaviour in wide deep neural networks. In International Conference on Learning Representations, 2018. URL https: /twp://eeee//eeeee///eoe wttprriwwwee/fouum id=H1 -GgWC..
[20] Roman Novak, Lechao Xiao, Yasaman Bahri, Jaehoon Lee, Greg Yang, Daniel A. Abolafia, Jeffrey Pennington, and Jascha Sohl-dickstein. Bayesian deep convolutional networks with many channels are gaussian processes. In International Conference on Learning Representations, 2019. URL https:t/pp// / /pwnnevieeeee..c//////e/e net /tt.//ormmmddd11300000 (Cited on 3)
[21] Max A Woodbury. Inverting modified matrices. Statistical Research Group, 1950. (Cited on 4)
[22] Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma. Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211-22 1-222, 2015. (Cited on 6)
[23] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770-778, 2016. (Cited on 6, 7)
11

---

[24] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. International Conference on Machine Learning, pages 1321-1330. PMLR, 2017. (Cited on 6, 7)
[25] Ranganath Krishnan and Omesh Tickoo. Improving model calibration with accuracy versus uncertainty optimization. Advances in Neural Information Processing Systems, volume 33, pages 18237-18248, 2020. (Cited on 6,
[26] Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. In International Conference on Learning Representations, 2019. URL
(Cited on 6)
[27] Weitang Liu, Xiaoyun Wang, John Owens, and Yixuan Li. Energy-based out-of-distribtioon detection. Advances in Neural Information Processing Systems, 33:21464-21475, 2020. (Cited on 6)
[28] Bhavya Kailkhura, Jayaraman Thiagarajan, Charvi Rastogi. Pramod K Varshney, and Peer- Timo Bremer. A spectral approach for the design of experiments: Design, analysis and algorithms. The Journal of Machine Learning Research, 19(1):111-12259 2018. (Cited on 8)
[29] Bobak Shahriari, Kevin Swersky, Ziyu Wang, Ryan Adams, and Nando De Freitas. Taking the human out of the loop: A review of bayesian optimization. Proceedings of the IEEE, 104 (1):148-175, 2015. (Cited on 8)
[30] Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine learning algorithms. Advances in neural information processing systems, 25, 2012. (Cited on 8)
Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes]
(c) Did you discuss any potential negative societal impacts of your work? [Yes]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes] Main claims presented in the paper, further derivations are given in the supplement
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as URL)? [Yes]
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A]
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes]
(b) Did you mention the license of the assets? [N/A] We are using standard open sourced datasets and models
(c) Did you include any new assets either the supplemental material or as URL? [No]
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A]
12

---

(c) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsoureing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "qhiIQIjv2Kw", "reviewer": "Reviewer_DQAh", "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.", "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.\n\n* As noted earlier, can we look at the raw performance numbers for the experiments in Table 1?", "limitations": "Yes.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n* 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.\n\nWeaknesses:\n\nThe 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.\n\nAlongside 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.\n\nOverall, I think this is technically sound and a promising direction to investigate further.\n\nMinor:\n\n* The GAN experiments could use a more elaborate description", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "q0UKGFGS1wU", "reviewer": "Reviewer_KbkE", "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.", "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:\n1) Derive the NTK for the shifted inputs (eq (2)) and linearized the infinite width limit of the prediction (eq (5)) in the original space ($(x-c)$).\n2) 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).\nSo 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.\n\n- 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? \n\n- 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?\n\n- The main argument (NTK) assumes infinitely wide neural networks; does the proposed method work well for narrow neural networks?\n\n- 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.\n- 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.\n- 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.\n- 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?\n\nP. S. \nThere 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.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength\n- 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. \n- The experimental results are promising and convincing.\n\nWeakness\n- The main argument based on NTK does not correspond to the actual settings on which the neural networks are trained.\n- It is rather unclear how the single neural network argument is justified (see below questions).\n- No experimental results on the most standard task - image classification & uncertainty quantification for in distribution data.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Wwhbm5I8qzI", "reviewer": "Reviewer_Gnzc", "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.\n\nThe authors provide experiments on ImageNet dataset using ResNet50, experiments with sequential optimisation and GAN.", "questions": "\nFig 1: It appears that the $\\Delta - UQ$ is more overconfident on left hand side than the second fit?\n\nDoes 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.)\n\nIt 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?)\n", "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.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "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.\n\nThe authors provide experiments on ImageNet dataset using ResNet50, experiments with sequential optimisation and GAN.\n\n\nLike 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.\n\nAs the proposed method is simple, this paper attempts to put emphasis on the experimental side.\nWhile the experiments are interesting, I think the results are not clearly demonstrating that the proposed method is superior. \n\nFirst, 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.\n\nSecond, 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).\n\nThe 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).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Ev5FiQnYTSn", "reviewer": "Reviewer_zA8F", "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.\n\nAuthors 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.\n\n", "questions": " * Effectiveness of training-time augmentation.\n   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.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**\n\n * A novel discovery that stochastic data centering is sufficient in producing effective predictive uncertainty.\n * A clear NTK analysis that strong motivates the approach, and elucidates it's impact on model's output uncertainty.\n * The method is simple to implement, and performs strongly on realistic benchmarks.\n\n**Weakness**\n\nI 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**\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept