A Geometric Perspective towards Neural Calibration via Sensitivity Decomposition
Authors: Junjiao Tian
Dylan Yung
Yen-Chang Hsu
Zsolt Kira
Venue: NeurIPS 2021 Spotlight (2021)
Keywords: uncertainty estimation
calibration
sensitivity

Abstract: It is well known that vision classification models suffer from poor calibration in the face of data distribution shifts. In this paper, we take a geometric approach to this problem. We propose Geometric Sensitivity Decomposition (GSD) which decomposes the norm of a sample feature embedding and the angular similarity to a target classifier into an instance-dependent and an instance-independent com-ponent. The instance-dependent component captures the sensitive information about changes in the input while the instance-independent component represents the insensitive information serving solely to minimize the loss on the training dataset. Inspired by the decomposition, we analytically derive a simple extension to current softmax-linear models, which learns to disentangle the two components during training. On several common vision models, the disentangled model out-performs other calibration methods on standard calibration metrics in the face of out-of-distribution (OOD) data and corruption with significantly less complexity. Specifically, we surpass the current state of the art by 30.8% relative improvement on corrupted CIFAR100 in Expected Calibration Error.


A Geometric Perspective towards Neural Calibration via Sensitivity Decomposition
Junjiao Tian
Dylan Yung Georgia Institute of Technology dyung60gatech. edu
Georgia Institute of Technology j jtian30gatech. edu
Yen-Chang Hsu Samsung Research America yenchang hww.samsung...... com
Zsolt Kira
Georgia Institute of Technology zkiraβgatech. edu
Abstract
is well known that vision classification models suffer from poor calibration in the face of data distribution shifts. In this paper, we take a geometric approach this problem. We propose Geometric Sensitivity Decomposition (GSD) which decomposes the norm of a sample feature embedding and the angular similarity to a target classifier into an instance-dependent and an instance-indepe inneeeedende com- ponent. The instance-des deeendee component captures the sensitive information about changes the input while the instance- indeeeedende component represents the insensitive information serving solely to minimize the loss the training dataset. Inspired by the decomposition, we analytically derive simple extension to current softmax-linear models, which learns to disentangle the two components during training. several common vision models, the disentangled model out- performs other calibration methods standard calibration metrics in the face of out-tf-distribution (OOD) data and corruption with significantly less complexity. Specifically, surpass the current state of the art by 30.8% relative improvement on corrupted CIFAR 100 Expected Calibration Error. Code available at ht ttsps: //github com/ RRI.//e.I.eeeeeeeee werrt-ieiiiiiiiiiiie wwwriity-DDcmmppee tion. git.
Introduction
During development, deep learning models trained and validated data from the same distribu- tion. However, in the real world sensors degrade and weather conditions change. Similarly, subtle changes in image acquisition and processing can also lead to distribution shift of the input data. This often known as covariate shift, and will typically decrease the performance (e.g. classification ac- curacy). However, has been empirically found that the model's confidence remains high even when accuracy has degraded [1]. The process of aligning confidence to empirical accuracy is called model calibration. Calibrated probability provides valuable uncertainty information for decision making. For example, knowing when a decision cannot be trusted and more data is needed is important for safety and efficiency in real world applications such self-driving [2] and active learning [3].
comprehensive comparison of calibration methods has been studied for in-distribution (IND) data [4], However, these methods lead unsatisfactory performance under distribution shift [5]. To resolve the problem, high-quality uncertainty estimation [6. 5] is required. Principled Bayesian meth- ods [7] model uncertainty directly but computationally heavy. Recent deterministic methods [8, 9] propose to improve a model's 's sensitivity to input changes by regularizing the model's intermediate layers. In this context, sensitivity is defined as preserving distance between two different input samples through layers of the model. We would like to utlize the improved sensitivity to better detect
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Lioof-Distributionn (OOD) data. However, these methods introduce added architecture changes and large combinatorics of hyperparameters.
Unlike existing works, we propose to study sensitivity from a geometric perspective. The last linear layer in softmax-linear model can be decomposed into the multiplication of a norm and cosine similarity term [10, 11, 12, 13]. Geometrically, the angular similarity dictates the membership of an input and the norm only affects the confidence in softmax-linear model. Counter-intuitivelly the norm of a sample's s feature embedding exhibits little correlation to the hardness of the input [1 I]. Based on this observation, we explore two questions: 1) why is a model's 's confidence insensitive to distribution shift? 2) how do we improve model sensitivity and calibration?
We hypothesize that in part an insensitive norm is responsible for bad calibration especially shifted data. We observe that the sensitivity of the angular similarity increases with training whereas the sensitivity of the norm remains low. More importantly, calibration worsens during the period when the norm increases while the angular similarity changes slowly. This shows a concrete example of the inability of the norm adapt when accuracy has dropped. Intuitively, training on clean datasets encourages neural networks to always output increasingly large feature norm to continuously minimize the training loss. Because the probability of the prevalent class of an input proportional to its norm, larger norms lead to smaller training loss when most training data have been classified correctly (See Sec. 3.1). This renders the norm insensitive to input differences because the model is trained to always output features with large norm on clean data. While we have put forth that the norm is poorly calibrated, we must emphasize that it can still play an important role in model calibration (See Sec. 4.
To encourage sensitivity, propose decompose the norm of sample's s feature embedding and the angular similarity into two components: instance-dependent and mnstance-independer instance-dependent component captures the sensitive information about the input while the instance- independent component represents the insensitive information serving solely to minimize the loss on the training dataset. Inspired by the decomposition, we analytically derive a simple extension the current softmax- linear model, which learns to disentangle the two components during training. We show that our model outperforms other deterministic methods (despite their significant complexity) and is comparable to multi-pass methods with fewer training hyprrparameters in Sec. 4.1.
The
In summary, our contributions are four fold:
We study the problem of calibration geometrically and identify that the insensitive norm is responsible for bad calibration under distribution shift.
We derive principled but simple geometric decomposition that decomposes the norm into an instance- -eeeeneed and
component.
Based on the decomposition, we propose a simple training and inference scheme to encour- age the norm to reflect distribution changes.
We achieve state of the art results in calibration metries in the face of corruptions while having arguably the simplest calibration method to implement.
2
Related Work
Methods dedicated to strengthening calibration can be divided into two camps: multi-pass models and single-pass deterministic models. The current state-of-the-art multi-pass models are: Bayesian Monte Carlo Drop Out (MCDO) [7] and Deep Ensembles [14]. Bayesian methods are the most principled way to model uncertainty. Instead of optimizing max likelihood for a single set of parameters, Bayesian methods obtain a posterior distribution over possible parameters given prior distribution over parameters and calculated data likelihood assuming some process noise. The posterior distribution over parameters captures epistemic uncertainty uncertainty due to the limits of what the model knows. The final predictive distribution is obtained by marginalizing out model parameters. While Bayesian methods theoretically sound, they are intractable in practice. Deep Ensembles averages multiple models trained using different random initialization so they learn different classification functions.
recent trend is to use a single-pass deterministic non-Bayesian model to improve uncertainty estimation. Two recent works DUQ [8] and SNGP [9] propose to improve mncettainty-wwarenesss of

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deterministic networks by improving the networks' sensitivity to input changes. Intuitively, a sensitive model should map samples further from the training data as they become more out-of-distribution. This can be achieved at two levels: feature level and output level. At the feature level, both methods require the feature extractors (CNNs) to be regularized to prevent feature collapse, which is the mapping of two different data points to the same embedded vector. This is ensured by having input distance awareness, which is equivalent to ensuring bi- Di-Lipschitz continuity [15].
In order to achieve this, DUQ [8] uses a two-sided gradient penalty [16] and SNGP [9] uses bounded spectral normalization [15]. The output level needs reflect the changes feature space. This can be done by adopting distance-awart classifiers. DUQ [8] uses a RBF networks with learned centroids for each class and SNGP [9] uses an approximate Gaussian Process layer. We were inspired by temperature scaling [4], which is another method for bettering calibration, but fails under distribution shift [5]. Our method does not require input distance awareness and instead leverages the geometric intuitions about the output layer, specifically properties of the norm of the input embedding.
3 Method
Following our hypothesise that the insensitivity of the norm is responsible for bad calibration on distribution shifted data, we propose geometric sensitivity decomposition (GSD) for the norm. We first introduce the geometric perspective of the last linear layer in Sec. 3.1 and then derive GSD in Sec. 3.2. To improve sensitivity of the norm and model calibration on shifted data, we propose a GSD-inspired training and inference procedure in Sec. 3.3 and Sec. 3.4.
3.1 Norm and Similarity
The output layer of a neural network can be written as a dot-product < x, Wy >, where x is the embedded input and wy is the weight vector associated with class y. Though seemingly simple there are strong geometric and calibration related intuitions drawn from this. Several prior works [10, 12, 11] have studied the effects decomposition of the last linear layer in softmax model can have on classification. The output layer can be decomposed into angular similarity cos by and norm ||x 122
exply
cxp (l/wy ||||| cos Φy) ΣS= exp w
Plylx) Σラ= exply
(1)
la |x||2 cos Φz)
where ||wylla ll2 is the norm of a specific classifier in the linear layer. We'll use this geometric view of the linear layer instead of the dot-product representation.
Based on this perspective, we base the foundation of our work on the following observations from prior works [10, 12, 11]: 1) The probability/confidnnce of the prevalent class of an input is proportional to its norm [12]. 2) While the norm of a feature strongly scales the predictive probability, due to it's unregularized nature the norm is not sensitive to the hardness of the input [11]. In other words, the norm could be the reason for bad sensitivity of the confidence to input distribution shift. Consequently, the insensitive norm can be causally related to bad calibration. We will examine a strong correlation between the quality of calibration and the magnitude of norm in Sec. 4.2.
3.2 Geometric Sensitivity Decomposition of Norm and Angular Similarity
To motivate the subsequent geometric decomposition, we can revisit the softmax model, P(y]z) o exp (|]wyl|a||z |||| 22 cos dyy) There are three terms contributing to the magnitude of the exponential function, |wy 1l, |x||2 and cos ey. Due to weight regularizations, ||wylle l22 is most likely very small, while cos Φ, [-1, 1]. Therefore, the only way to obtain a high probabiliy/cconfidenee on training data and minimize cross-entropy loss is to 1) push the norm ||x||2 la to a large value and 2) keep cos Joyl of the ground truth class close to one, i.e., 1 close to zero This is further supported by [17], where it was shown that logits of the ground truth class must diverge to infinity in order to minimize cross-entropy loss under gradient descent. In this process, models tend towards large norms and small angles for all training samples.
Therefore, we propose to decompose the norms of features into two components: an instance- independent scalar offset and an instance-dependen variance factor, which we define in Eq. 2. The role of the instance-independee independent offset Cp is to minimize the loss on the entire training set and
3

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the Instance-depentee component Ar accounts for differences in samples. Therefore, if we can disentangle the getance-independeee component from the instance-depea -eaennddd component, we can obtain a norm that is sensitive to the hardness of data. Following this logic, we decompose the norm into two components.
| = | + C.
(2)
Similarly, we relax the angles such that the predicted angular similarity does not need to be close to the training data, i.e., making the angles larger. To achieve this, we introduce instance- independent relaxation angle Co and an instance-dependent angle Aøy. Analogous to the norm decomposition, the scalar C. serves solely to minimize the training loss while the instance-dependent Aø, accounts for differences in samples. Because we need to account for the sign of the angle, we put absolute value on it.
bdyl و [Cal
The || Aoyl are the instance-dependeen components and Cr. |Cal are the iistace-indepenni components. We can rewrite the pre-softmax logits in Eq. 1 with the decomposed norm and angular similarity. (Detailed derivation in Sec. A. in the Appendix.)
||||| cos By = ||x||2 cos (oy, (||Ax|12 + C.) cos (1Adyl (C
(4)
1
Adyl (1 (1- sin |Col (1- cos [Co sin ا
= -1|Ax|12 + Cz)
cos
cos Capl
sin Col cos A
We can simplify the equation by assuming cos Joyl is close to one, which means 1 is small. This due to the fact that l is the angle between the correct class weight and æ, which means as training ensues, the angle converges to 0 and thus the cosine similarity converges to 1. (Please Sec. A.2
for empirical support.)
cos |Cp sin Adyl
sin (,اØا) + CCll + sin b = 1
(5)
sin Col cos Aøyl sin (, (C,l) sin loyl
Therefore, Eq. 4, omitting the absolute value angles because cos is even function, simplifies:
||x||2 cos dy (|/x||2 Cz)
cos Ady
cos cos Co
1
||x||2 oos Co
Cr cos Ady
cos Cab
B
|x|l2
cos Ady
Because cos C. and C₂ are iustance-indepencee we denote them as and respectively. This geometric decomposition of norm and cosine similarity inspires us to include and | as free trainable parameters in a new network and the network can learn to predict the more input- sensitive ||x||| and Ady instead of the original ||x||2 and oy. While both the angle and norm can be decomposed we direct the focus the norm as the angle already calibrated to accuracy [11]. In other words, angles have been shown to be sensitive to input changes in [11].
3.3 Disentangled Training
Following the derivation in Eq 6, we replace the norm, ||||2, Eq. by (11Ax112+ 2) and dy by Ady |Ax||2 and Ad, now learned outputs from new network instead shown in Eq. 6:
exp (1 wyyl2 (11l _) cos Arby)
exply
P(y|x) = ΣS=1 exply Σラ= exp (l/w)l22 |x|12
| + 8) cos Arb;)

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a [|xll] +
(1-
[ I
|+lll
|
|1xll
1
[14.1|
Indioii.utoond data
Busten
Ousefderrbutteddrr In-Simiiuution 5316
(a) Temperature Scaling
(b) Ours: calibration Step
(c) Ours: Calibration Step 2
Figure Calibration Procedure (a): Temperature Scaling [4] changes the slope of the effective
norm based on in-distribution (IND) data (See A.9 in Appendix)
The new model can be trained using the same training procedures the vanilla network without ad- ditional hyperparameter tuning, changing the architecture or extended training time. Even though the outputs of the new network, |Ax||2 and Δøy, only approximate the original geometric relationships with Eq. 6, the effect of a and / reflects the decomposition in Eq. and Eq. 2.
3 encodes an iustance-indepenne smaller instance-dependen component || |
scalar C₂ of the norm. larger corresponds
a encodes the cosine of a relaxation angle Cg. A larger arccos a corresponds to a larger Co and therefore larger Adj.
Because 3 encodes the independent component, the new feature norm | becomes sensitive to input changes and maps OOD data to lower norms than IND data regularize a such that the instance-indee ieneeendenn component Co is small. Specifically, we penalize ||o - 1 because a = cos i.e., if a 1, Co = O. We empirically found that larger relaxation angle C. deteriorates performance because the angular similarity already correlates well with difficulty of data [11] and we do not need to encourage a large relaxation. Sec. 4.3 will empirically verify this.
can see in Fig. 3a, 3b. We
3.4 Disentangled Inference
The decomposition theory in Sec. 3.2 provides a geometric perspective on the sensitivity of the norm and the angular similarity to input changes and inspires a disentangled model in Sec. 3.3. The new model uses a learnable affine transformation on the norm |||x||2. Let's denote the affine transformed norm as the effective norm ((Ax) = A|lxx||a + & However, the training only separutes the sensitive components of the norm and angular similarity, the model can still overconfident due to the existence of insensitive components. Therefore, we can improve calibration by modifying insensitive components, c.g., ,3 in our case. We propose two-step calibration procedure that combines in-distribution calibration (Fig. 1b) and out-of-distribution detection (Fig. Ic) based on two observations: 1) overconfident IND data can be easily calibrated on validation set, similar to temperature scaling [4]. 2) for OOD data, without access to calibration set for OOD data, the best strategy is to map them far away from the IND data given that the model clearly distinguishes them.
The first step is calibrating the model on IND validation set (note our method does not rely on OOD validation data), similar to temperature calibration [4]. However, instead of tuning a temperature parameter as shown in Fig. Ia, we simply tune the offset parameter 3 on the validation set in one of two ways: 1) grid-search based on minimizing Expected Calibration Error (see Sec. 4) 2) SGD optimization based on Negative Log Likelihood [4]. Because these post-training procedure, both methods are very efficient. We denote the new parameter as 3'. As shown in Fig. Ib, by changing the offset, we decrease the magnitude of the norms after the affine transformation. Formally,
N(Ax) = || B N(Ax) = = Ax +
8'
(8)
a
The second step approximates the calibrated affine mapping in Eq. 8 by non-linear function which covers a wider range of the effective norm as shown in Eq. 9 and maps OOD data further away from IND data. Intuitively, when sample is more likely IND, the loneirnear function maps it closer the
5

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calibrated transformation. When a sample is OOD, the non-linear function maps it more aggressively smaller magnitude, exponentially away from the IND samples.
- -1 -1-e a
((Ax
where e is a hyperparameter which can be calculated as in Eq. 10. The non-linear function grows exponentially close to the calibrated affine mapping in Eq. 8 dictated by 1 - exllAxii|z as shown in lc. Therefore, e- exllAxil2 can be viewed as an enror term that quantifies how close the non- non-linear function is to the calibrated affine function in Eq. 8. Let Mær and 0₂ denote the mean and standard deviation of the distribution of the norm of IND sample embedding calculated on the validation set. We use the heuristic that when evaluated one standard deviation below the mean, |xx||2 = Pa oz. the approximation error e-clu= 0.1. Even though the error threshold is hyperparameter, using an error of 0.1 lead to state-of-the-att art results across all models applied.
-In(1 error)
-0n9).9
(10)
Mx Oz
Pe Ox
In summary, the sensitive norm || is used both as a soft threshold for OOD) detection and as a criterion for calibration. While similar post-processing calibration procedure exists, such as temperature scaling [4] (illustrated in Fig. la and further introduced in A.9) it only provides good calibration on IND data and does not provide any mechanism to improve calibration on shifted data [5]. Our calibration procedure can improve calibration on both IND and OOD data, without access to OOD data, because the training method extracts the sensitive component in principled manner. Just as temperature scaling, the non-linear mapping needs only to be calculated once and adds no computation at inference.
4 Experiments
4.1 Experiments on Calibration
Table I: ResNeet-28-10 on CIFAR10 averaged over 10 seed. denotes results from [9]. Our method outperforms other single-pass methods and is comparable Deep Ensemble [14] on corrupted data. While the ensembled version of our model beats all multi-pass models.
Method
Accuracy
ECE
NLL
Clean
Corrupted
Clean
Corupted
Clean
Cormpted
Vanillat
96.0=0.01
72.9:0.01
0.02310.002
0.15310.011
0.15810.01
1.059:1.022
DUQi
94.7=0.02
71.6:0.02
0.034440002
0.183=0.011
0.239:10.02
1.348:00.01
Single-Pass
SNGP:
95.920.01
74.610.01
0.018:10001
0.090=0.012
0.138:10.01
0.93510001
Ours 3' Grid-Searched
95.9=0.01
74.900.05
0.018+40.003
0.067=0.010
0.148:10.003
0.826=00.03
Ours 38' Optimized
95.9==0.11
7..9:0.05
0.008:10..02
0.085=00012
0.140::0.004
0.853::0.04
Deep Ensemblest
96.6=2001
77.9=0001
0.010-10.001
0.087=00.004
0.114:00.01
0.815::0.01
Multi- Pass
MC Dropout
96.0+0.01
70.0+0.02
0.021+10.02
0.116:0.009
0.173:10.001
1.152+0.01
Ours 3 Sraic-eereeeed
96.62
77.9
0.007
0.069
0.108
0.773
Table 2: ResNet-28-10 on CIFARI00 averaged over 10 seeds. t denotes results from [9]. Our
method outperforms other single-pass -pass methods and Deep Ensemble [14] on corrupted data. While the ensembled version of our model beats all multi-pass models
Method
Accuracy?
ECE
NLL 1
Clean
Corrupted
Clean
Comuptee
Clean
Corrupted
Vanilla DUQI
79.8:20.02
50.510.04
0.085=0.004
0.239:10.020
0.872=0.01
2.756:20.03
78.5.10.02
50.410.02
0.11910.001
10.011
0.980.1..02
2.801
Single-Pase Pass
SNGPt
79.9--0.03
49.0+-0.02
0.025-0.011
0.117440.01
0.847-0.01
2.626++0.01
Ours 8 Grid-Searched
79.8=00.33
49.8 0 0.003
0.027=0.003
0.081 0.007
0.787=0.009
2.23-0.02
Ours /3' Optimized
79.8:20.03
49.820.03
0.027=0.003
0.088+1.007
0.784=0.011
2.23620.021
Deep Ensemblest
80.2+0.01
54.11:0.04
0.021=0.004
0.138:00003
0.666=20.02
2.281++0.33
Multi-Pass
MC Dropout]
79.6:10.22
42.610008
0.050=0.003
0.202-10.010
0.825=0.01
2.88110.011
Ours !!' Grid Searched
83,09
54.1
0.018
0.086
0.614
2.042
The ultimate goal of the paper is to improve model calibration under distribution shift by improving sensitivity. Popular metrics for measuring calibration include: Negative Log-Likelihood (NLL [18]),

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Table 3: Generalizability Experiments Our method is effective with different feature backbones.
Clean
CorrupURotate
model
dataset
accuracyt
ECE|
NLL
Brier.
accuracyt
ECE|
NLL
Brier,
ResNet34
CIFARIO CIFAR10
95.63%
0,026
0.186
0.007
81.96%
0,164
1.114
0.039
GSD ResNet34
95.9%
0.005
0.148
0.006
76.54%
0.088
0.882
0.037
ResNet50
CIFARIO
95.32%
0.03
0.203
0.008
76.32%
0.17
1.23
0.039
GSD ResNet50
CIFARIO
95.82%
0.008
0.147
0.007
76.23%
0.057
0.766
0.033
ResNet101
CIFARIO
95.61% %
0.028
0.197
0.007
77.59%
0,154
1.118
0.037
GSD ResNet101
CIFARIO
95.62%
0.007
0.158
0.007
77.21%
0.075
0.852
0.036
ResNet152
CIFARIO
95.7%
0,028
0.196
0.007
75.2%
0.179
1.337
0.041
GSD ResNet152
CIFAR1O
95.63%
0.007
0.151
0.007
76.58%
0.058
0.765
0.033
ResNet34
CIFAR 100
78.81% %
0,071
0.868
0.003
51.16%
0.19
2.387
0.007
GSD ResNet34
100
78.02%
0.037
0.938
0.003
49.27%
0.098
2.361
0.007
ResNet50)
CIFARTO0
79.28%
0.075
0.861
0.003
49.71%
0.213
2.477
0.007
GSD ResNet50
CIFAR100
78.97%
0.033
0.879
0.003
50.12%
0.08
2.264
0.006
ResNet101
CIFAR 100
80.17%
0.092
0.846
0.003
58.19%
0.253
2.575
0.007
GSD ResNet101
CIFARIO0
79.82%
0.034
0.834
0.003
53.14%
0.082
2.11
0.006
R5sNet152
CIFAR 100
80.71% 79.85%
0,090
0.815
0.003
54.2% 53%
0.233 0.078
2.45
0.007
GSD ResNet152
100
0.036
0.827
0.003
2.12
0.006
Table 4: Importance of Norm While norm is poorly calibrated, it is important for calibration.
ECE
NLL
Brier
Entropy
Accuracy
Vanilla (||w, |||| cos 6y)
0,025 001
0.186=0,006
0.001
0.082:00.002
95,4=0.19
No Weight Norm (w/o [|w,,l)
0.061+0.0003
0.206-0.006
0.001+0.0
0.527-0.014
95.4-0.1%
No Norm (w/o [|x]]
0.893-10.002
2.837=0.005
0.00910.0
4.537-0.001
95.4=0.1%
Only Cosine (w/o [xll
0.914-t0.001
3.235=0.001
0.009+0.00
4.546=0.000
95.3=0.199
Brier [19] and Expected Calibration Error (ECE [20]). Our goal is for our model is to produce values
close 0 in these metries, which maximizes calibration. Please refer to Sec. A.3 (Appendix) for more detailed discussion on these metrics. Following prior works [9, 8, 5], we will use CIFARIO and CIFAR 100 the in-distribution training and testing dataset, and apply the image corruption library provided by [1] to benchmark calibration performance under distribution shift. The library provides 16 types of noises with severity scales. In this section, we show that our model outperforms other deterministic methods (despite their significant complexity
Compared Methods We compare to several popular state-of-the-ar models including stochastic Bayesian methods (multi-pass): Deep Ensemble [14] and MC dropout [7], and recent deterministic methods (single pass): SNGP [9] and DUQ [8].
Results Tab. and 2, we compare our model the most recent state of art deterministic methods SNGP and DUQ using Wide ResNet 28-10 [21] as the model backbone and each model evaluated us- ing the average of 10 seeds. We report accuracy, ECE and NLL on clean and corrupted CIFAR10/100 datasets [1]. Our method outperforms all single-pass methods on calibration when data is corrupted, and even surpass ensembles on error metrics for corrupted data. We had versions of our model: Grid Searched: grid search 3' on the validation set to minimize ECE and Optimized: optimize 3' on the validation set via gradient decent to minimize NLL for 10 epochs, similar to temperature scaling. We report additional results with ResNet18 in Sec. A.4 and Sec. A.5 (Appendix) with image noise and rotation respectively.
Generalizability We explored how generalizable our method (Grid Searched) is by applying it to 12 different models and 4 different datasets in Tab. 3. We can see consistently that our model had stronger calibration across all models and metrics, including models known to be well calibrated like LeNet [22]. All models were tested on CIFARIOC and CIFAR 100C datasets offered by [1] where the original 10 and CIFARION 100 were pre-corrupted; these were used for consistent corruption benchmarking across all models. All non-CIFAR datasets were corrupted via rotation from angles [0,350] with 10 step angles in between and the average calibration and accuracy taken across all
7

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Cean goussian
geussian
clean
Raussian
06
dIf
3.8
Coure larts Officncc
0.2
0.15
02
435
(a) CIFARIO0 Accuracy
(b) CIFARIOO ECE
(c) CIFARIOO Norm
(d) CIFARIO0 Cosine
Figure 2: Accuracy ECE, norm and cosine similarity CIFAR100 validation set with clean and Gaussian noise trained on vanilla ResNet. In the shaded region, increase in norm is responsible for increase in ECE because cosine similarity is relatively flat. Throughout training, sensitivity of the cosine similarity improves while that of the norm remains insensitive.
Table 5: Pearson Correlation of Cosine Similarity and Norm vs. ECE during training on CIFARI00. Norm is consistently positively correlated with ECE whereas the similarity is either negatively or not correlated with ECE.
ResNet18 Gaussian
RecNet34
ResNet101
ResNet152
shot
Defocus
shot
Gaussian
Defocus
shot
Gaussian
Defocus
shot
Gaussian
Defocus
Sim
0.09
0.73
0.09
0.03
-0.04
-0.97
Norm
0.82
0.82
0.78
0.82
0.81
0.78
0.87
0.87
0.85
0.86
0.85
0.81
degrees of rotation. Our models included: DenseNet [23], LeNet [22] and 6 varying sizes of ResNet, which are described in [24]. The datasets we experimented on CIFAR10 [25], CIFARI00 [25], MNIST [26] and SVHN [27], CIFARIOC [1], CIFARI00C [1]. We report Optimized results in Tab. 14 in A.7 (Appendix). Both tuning methods yield similar performance.
Qualitative Comparison The current state-of-the-art single pass models for inference OOD data, without training on OOD data, are SNGP [9] and DUQ [8]. The primary disadvantages of these models are: 1) Hyperparameter Combinatorics: Both DUQ and SNGP require many hyperparameters as shown in Tab. 13 in A.6 (Appendix). Our model only has one hyperparameter that tuned post-training, which is quicker and less costly than the other methods that require pre-training tuning. 2) Extended Training Time: DUQ requires a centroid embedding update every epoch, while SNGP requires sampling potentially high dimensional embeddings of training points, thus increasing training time while our model trains in the same amount of time as the model it is applied to. Bayesian MCDO [7] and Deep Ensemble [14] are considered the current state-of-the-as methods for multi-pass pass calibration. Bayesian MCDO requires multiple passes with dropout during inference. Deep Ensembles requires N times the number of parameters as the single model is ensembling where N is the number of models ensembled. The main disadvantage of multi-pas, pass models is high inference complexity while our model adds no overhead computation at inference.
Importance of the Norm While we have shown and conjectured that the norm of x is uncalibrated to OOD data and not always well calibrated to IND data, one might suggest to simply remove the norm. We show in Tab, 4 though the norm is uncalibrated it is still important for inference. We trained ResNet18 on CIFARIO and then ran inference with ResNet18 modified in the following: dividing out the norms of the weights for each class, dividing out the norm of the input and then dividing out both. As we can see the weight norm contributes minimally to inference as accuracy decreased by 0.03% without it and as previous work has shown the angle dominates classification. We can see with ||x|] removed the entropy is at it's highest while calibration is very poor, implying the distribution much more uniform when it should be peaked, as larger entropy implies a more uniform distribution. Thus the root of the issue does not lie in the existence of the norm, but it's lack of sensitivity.
4.2 Reasons for Bad Calibration under Distribution Shift
To identify the cause of bad calibration, we record the accuracy, ECE, norm and cosine similarity of a model during training of a vanilla ResNet model. Specifically, we record the evaluation statistics on clean data and also on data corrupted with Gaussian noise on CIFARIO0 100. Fig. 2a and 2b show the accuracy and ECE respectively. We observe that evaluation on Gaussian noise corrupted data yields lower accuracy and higher ECE compared to evaluation on clean data. This demonstrates that
8

---

cifar10
dfar10 syhn
In svhn
svhn
svhn
90.48
AUROC:
AUROC:
AUROC 99.98
AUROCI
ResNetta
ResNett1
GSD
(a) CIFARIO vs. SVHN
(b) CIFARIOO vs. SVHN
Figure 3: Histogram of Norm Distribution Our model (a-regularized) improves separation of norm
between IND and OOD data.
the model's confidence fails to adapt to the decreusing accuracy. Fig. 2c and 2d show the change of average norm and average cosine similarity throughout training. The difference between Gaussian noised data and clean data is also reported. We observe that the norm of clean data and the norm of Gaussian noised data close and the difference remains constantly low whereas the cosine similarity of the two diverges with training. This indicates that sensitivity of cosine similarity increases whereas sensitivity of the increases the most, we observe that the norm also increases but the cosine similarity only increases slowly. The observation also holds for other noises and architectures. We further present Pearson correlation between ECE and cosine similarity or norm on 4 models and 3 noises in Tab. 5. A large correlation coefficient indicates higher positive correlation. Norm consistently positively correlated with ECE whereas the similarity is either negatively or not correlated with ECE. This shows that the worsening of ECE (large ECE) is correlated with the increasing norm. Based on supporting literature [12], [11] and this correlation, the observation supports the conjecture that the insensitivity of the norm is responsible for bad calibration.
remains low with training. In the shaded region of Fig. 2b-2d where ECE
4.3 Empirical Support for the Disentangled Training
Table OOD AUROCA using Norm and Similarity We show OOD detection results using norm and cosine similarity. SVHN [27] is used as the OOD dataset. Our method (a-regularized) signifi- cantly increases the sensitivity of feature norm.
ResNot18
CIFARIO
CIFARIO (Incorrect)
ResNer18
Criterion
CIPAR100
CIPAR100 (Incorrect)
Vanilla
Similarity
Norm
93.87
90.48
56.98
67.23
vanilla
Similarity
Norn
82.26
79.38
$5.54
62.66
o- regularized
Similarity
Norm
99.05
93,16
regolueized
Nama Similarity
94.46
$6.67
untogulanized
Similarity
Norm
98.20
94.72
88.29
60.63
or unregularized
Similarity
Narm
72.61
84.78
73.11
42.90
(a) CIFARIO vs. SVHN AUROC
(b) CIFARI00 SVHN AUROC
In the first set experiments, we show that a and 3 reflect the effects of the geometric decomposition as claimed in Sec. 3.2 with different
3 configurations. From Fig. 4a -d, we observe that the norm decreases linearly with 3 for fixed a. From Fig. 4e - 4h, we observe that the angle increases linearly with cr)) The observations are consistent with the original geometric motivation. B encodes an instance-indes -nneneeddddeeedentt portion, C,, of the norm. As 3 increases, C2 increases and therefore the magnitude of the dependent component, ||Ar||| decreases linearly. encodes the inverse of the cosine of a relaxation angle, Cg. As arccos( (a: increases, the resulting angle, Aó increases linearly due to the increased relaxation angle encoded by a.
In the second set of experiments, we show that the new model effectively increases the sensitivity of both the norm and the angle to input distribution shift claimed in Sec. 3.3. Specifically, we measure OOD detection performance of the models using both the norm and the cosine similarity with the Area Under the Receiver Operating Characteristic (AUROC) curve metric. We use CIFAR10/ as
Table 7: Average norm and accuracy across different corruptions on GSD ResNet18. The table organized in decreasing accuracy order.
ReiNat GSD
dean
Inidhacss
Eoe
clasiic
defocus
frost
motion blar
jpsr
zoom blur
piselat
contrast
shot
plas biu
impulac
Gasssias
ascunar
93.83
$3.87
79.71
70.31
76.4
59 46
47.33
Bonm
0442
0.46
0.28

---

R*-0.938 0.9386
0..9109
||ad|]
R-- 0.929
0.9405
0
0
3
6
8
8
(8) a
(b) a = 1.5
(c) C = 2.0
(d) or = 2.5
44
# 5943
42
40
39
5788
40
30
41 1.25
40
55
65
55
65
75
45
55
55
65
75
arocos(ot)
arccos(a)
arccos(e)
arccos(a)
(c) B =
(1) B 2
(g) 9 =
(h) B =
Figure 4: Properties of || and Aó. (a) (b): || decreases linearly with 3 for fixed a reflecting Eq. 2 and 6. (e) (h) Aø increases linearly with arccos(a) for fixed 3 reflecting Eq. 3 and 6. All plots include R-squared values to indicate goodness-of-fi of the linear relationship.
the IND data and SVHN [27] the OOD data. In Tab. 6a and 6b we show two configurations of models in addition to vanilla ResNet18: (a-regularized) we regularize a such that it stays close to one as described in Sec. 3.3; (o-unregularzed) we optimize both a and freely without constraints. Compared to vanilla ResNet, the norms predicted by our models achieve significant improvement in separating IND data from OOD data. Additionally, we visualize the distribution of norms in Fig. 3a and 3b. The separation between IND and OOD data increases significantly compared to vanilla ResNet18. However, a large a (see c-unregularzed in Tab, 6a and 6b) leads marginal cosine similarity sensitivity improvement on CIFAR 10 and CIFARI00. This indirectly confirms our observations in Sec. 4.2 and in prior works [11] that cosine similarity correlates well with distribution shift. Introducing further angle relaxation might not be always beneficial. While we mainly focus on calibration, our method also strengthens its base model's ability for OOD detection.
The assumption that OOD data have smaller norms is based on the expectation that a model should be less confident on OOD data. Practically, the norm acts as temperature in softmax as shown in Eq. 1. Intuitively, larger always yields more peaked/confident predictions, and smaller always yields flatter predictive distributions. Therefore, we expect less confident data such as OOD data to have smaller because we expect the output distribution to be flatter. The assumption is supported by the following empirical evidence. In Tab. 7 we show the norm of in-distribution and out-of-distribution data on CIFARI0 using ResNet50-GGS paper. OOD data have consistently smaller norms and the accuracy decreases with decreasing norm with Pearson correlation of 0.9 as an indicator of more out-of-din titto---istibutton..
(ours). The OOD data is produced by the 15 corruptions used in the
5 Conclusion
In this paper, we studied the geometry of the last linear decision layer and identified the insensitivity of the norm as the culprit of bad calibration under distribution shift. To encourage sensitivity, we derived a general theory to decompose the norm and angular similarity. Inspired by the theory, we proposed simple yet very effective training and inference scheme that encourages the norm reflect distribution changes. The model outperforms other deterministic single pass- mesthds in calibration metrics with much fewer hyperparameters. We also demonstrated its superior generalizability variety of popular neural networks. Note that our problem and method have positive societal impact, as calibration under shift improves overall confidence and robustness of these models.
10

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6 Acknowledgements
This work was partially supported by ONR grant
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12


Reviews:
[{"review_id": "ZtFWzoehBA8", "reviewer": "Reviewer_7rY8", "summary": "This paper studies the problem of model calibration in the presence of distribution shifts for vision classification tasks. The authors propose a sensitivity decomposition on the last softmax layer of the model. The resulting calibration approach demonstrates improved performance compared to existing calibration methods under distribution shifts.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Strengths:\n1. The paper takes a geometric perspective to study model calibration. The finding that the insensitivity of the norm accounts for the poor calibration under distribution shifts is interesting.\n2. The approximate decomposition is elegant and easy to implement. It does not impose additional computational burden.\n3. The proposed calibration approach is simple. It achieves good performance without increasing training and inference complexity significantly.\n\n\nMajor comments:\n1. Do you have intuition on why the decomposition should be additive besides its simplicity? What happens if a product term is included in the decomposition?\n2. I found the interpretation of $\\lVert \\Delta x \\rVert_2$ confusing. For instance, in the first bullet on the top of page 5, when $\\alpha$ is fixed, it is clear that a larger $\\beta$ corresponds to a larger $\\mathcal{C}_x$, but why a smaller $\\lVert \\Delta x \\rVert_2$? Moreover, in Figure 1c, all OOD data have small values of $\\lVert \\Delta x \\rVert_2$ (this is also implicitly assumed in Eq (9)), why can’t they have large values? If a model outputs moderate values on the training set but large values on the test set, this is also a sign of distribution shifts, right?\n3. In the proposed calibration method in Sec 3.4, only $\\beta$ is tuned on the validation set. Why don't you also tune $\\alpha$?\n4. I would like to see standard errors in Table 3-5.\n5. In Sec 4.2, the findings are for corruptions with Gaussian noise. Do the findings hold under other types of corruptions?\n6. It is unclear to me what the metric AUROC is measuring and how it is related to the OOD detection performance. I would like to see more explanation.\n\nMinor comments:\n1. The sentence “While both … calibrated to accuracy” on line 156-158 is unclear to me.\n2. On line 161, “\\Delta \\phi_j” —> “\\Delta \\phi_y”.\n\n\n-----\nUpdates: they answered most of my questions, so I decided to raise my score to 7.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "See comments above for suggestions.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6"}, "extra_text": {}}, {"review_id": "SEWOxrnLGMS", "reviewer": "Reviewer_vKqM", "summary": "This paper studies the problem of neural calibration from a novel geometric perspective and finds that the insensitive norm is responsible for bad calibration under distribution shift. Based on this insight, this paper proposes Geometric Sensitivity Decomposition (GSD), which decomposes the norm and angular similarity into an instance-dependent and instance-independent component. Subsequently, this paper proposes a training and inference scheme, to encourage the norm to reflect distribution changes. The proposed method is easy to implement and achieves SOTA results in calibration metrics in the face of corruptions.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Originality: This work studies the problem of neural calibration under distribution shift from a novel geometric perspective. This work leverages properties of the norm of the input embedding to strengthen calibration, which clearly differs from previous works. However, this work refers to [10][11][12] (references of the paper), the difference between them is not made clear.\nQuality: Overall, this work is technically sound and the claims are well supported by experimental results. The finding that the insensitive norm is responsible for bad calibration under distribution shift is key to this work, we hope the authors can give more theoretical analysis or experimental results (e.g., different datasets and base models) to support and strength this point. Meanwhile, some definitions are confusing and should be further explained: why Eq.2 uses “+” while Eq.3 uses “-“? why use such a kind of non-linear function in Eq.9?\nClarity: This paper is well organized. Some writing details need to be worked on:\n1.hard-to-understand sentences: line140, “…making the angles larger”; line206, ”…=µx – σx”\n2. misquotation:Line255, “Tab9”\n3. a table like Tab. 10 may be better to compare “Extended Training Time”.\nSignificance: Studying the problem of calibration under distribution shift from a geometric perspective has some implications for other researchers or practitioners. This paper achieves promising performance on standard calibration metrics in the face of out-of-distribution (OOD) data and corruption. This problem and method have positive impact. Since I am not very familiar with the neural calibration literature, I am not sure the novelty of this paper reaches the bar of \"accept\".\nThe possible limitations and future works of this paper needs to be discussed.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Nil", "review_text": "", "extra_scores": {"ethical_concerns": "Nil", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "24 hours"}, "extra_text": {}}, {"review_id": "Nzf_7m6e5E2", "reviewer": "Reviewer_3KLr", "summary": "The paper proposes Geometric Sensitivity Decomposition (GSD) as the new calibration method in the face of out-of-distribution (OOD) data. The paper identifies that the insensitive norm is responsible for bad calibration under distribution shift. Based on GSD, the paper proposes a simple and quick training and inference scheme. The paper achieves SOTA in calibration metrics in the face of corruptions while having arguably the simplest calibration method to implement.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "**[Update after the rebuttal] I am satisfied with the extra mile that the authors went to show the proposed method's effectiveness against the ensemble models. The provided reason behind why it was not a fair comparison before and how it is now fair comparison is sound. Updated my rating accordingly.**\n\nOriginality: Are the tasks or methods new? Is the work a novel combination of well-known techniques? (This can be valuable!) Is it clear how this work differs from previous contributions? Is related work adequately cited?\n\n**The method is new. It is clear how this work differs form previous contributions. Related works are adequately cited.**\n\nQuality: Is the submission technically sound? Are claims well supported (e.g., by theoretical analysis or experimental results)? Are the methods used appropriate? Is this a complete piece of work or work in progress? Are the authors careful and honest about evaluating both the strengths and weaknesses of their work?\n\n**Theoretical support is good. Experimental results seem weak given that many metrics are not SOTA compared to prior arts. It is difficult to figure out the pattern when this method yields better results (i.e. Table 3, Table 2); I think it is a work in progress. Yes, honest in reporting all the numbers.**\n\nClarity: Is the submission clearly written? Is it well organized? (If not, please make constructive suggestions for improving its clarity.) Does it adequately inform the reader? (Note that a superbly written paper provides enough information for an expert reader to reproduce its results.)\n\n**It is clearly written. Well organized.**\n\nSignificance: Are the results important? Are others (researchers or practitioners) likely to use the ideas or build on them? Does the submission address a difficult task in a better way than previous work? Does it advance the state of the art in a demonstrable way? Does it provide unique data, unique conclusions about existing data, or a unique theoretical or experimental approach?\n\n**The theoretical idea and its simplicity are promising. The empirical results are not so much as it is presented today.**", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Good.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "9"}, "extra_text": {}}, {"review_id": "NKD1BzOvaPm", "reviewer": "Reviewer_ssfF", "summary": "The paper presents an approach for calibration of neural networks. The proposed approach is based on the analysis of the softmax logits and decomposing it into two components that depends on the norm of the feature vector (norm component), and the other that depends on the similarity (similarity component) with the weight vector corresponding to a class. The paper then derives learnable parameters that could control each of the components and thus can be used for calibration of the predicted probabilities. The paper claims that for out-of-distribution (OOD) scenarios, this calibration scheme outperforms previously proposed approaches. The empirical results are reported for in-distribution (IND) on CIFAR-10 and CIFAR-100 and their corrupted versions of these for OOD, for various different models. ", "questions": "", "limitations": "", "rating": 9, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Originality: \nThe idea of geometric decomposition is both interesting and insightful and as far as I know, original. The derivation that leads to trainable parameters that can control the norm and the similarity components is original and seems to be quite generally applicable. This is also validated from the experiments. \n\nQuality: \n+ The paper brings new insights over previous approaches [10,11,12] by the proposed geometric decomposition. \n+ The experimental results are thoroughly done and presented, validating the claims made. \n\nClarity: \nThe paper is very well written and easy to follow. The approach is well motivated and the related work description seems to be complete (AFAIK). \n\nSignificance: \nI believe the geometric decomposition of the norm component and the similarity component, and its parameterization is an important step towards making predicted probabilities more meaningful. The ideas in this paper could potentially inspire more research as well. \n\nHowever, I hope the authors can address the following comment in the rebuttal. \n- Line 148 states an assumption that \\cos|\\phi_y| is close to zero. This may be true after sufficient training of a high-capacity model for datasets like CIFAR and other similar ones. However, what happens when this assumption is violated? This may happen for certain classes when there is a high amount of imbalance. Kindly discuss the impact on the performance of the proposed approach. \n\n\nUpdate after rebuttal. \nMy concern has been addressed. I retain the rating. I think it is a pretty good paper. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The limitations and societal impact are pointed to. Perhaps potential failure scenarios of the said method could be pointed to, so that follow up works may explore that as well. \n", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5 hours"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors study the problem of model calibration in the presence of distribution shifts for computer vision. The authors propose a sensitivity decomposition on the last softmax layer of the model. This calibration approach demonstrates improved performance compared to existing ones.

All reviewers recommended to accept. 

Accept.
Decision: Accept (Spotlight)