Mix-MaxEnt: Creating High Entropy Barriers To Improve Accuracy and Uncertainty Estimates of Deterministic Neural Networks
Authors: Francesco Pinto
Harry Yang
Ser-Nam Lim
Philip Torr
Puneet K. Dokania
Venue: ICLR 2022 Submitted (2022)
Keywords: regularizer
maximum entropy
uncertainty estimation
data-shift robustness
calibration
out-of-distribution detection

Abstract: We propose an extremely simple approach to regularize a single deterministic neural network to obtain improved accuracy and reliable uncertainty estimates. Our approach, on top of the cross-entropy loss, simply puts an entropy maximization regularizer corresponding to the predictive distribution in the regions of the embedding space between the class clusters. This is achieved by synthetically generating between-cluster samples via the convex combination of two images from different classes and maximizing the entropy on these samples. Such a data-dependent regularization guides the maximum likelihood estimation to prefer a solution that (1) maps out-of-distribution samples to high entropy regions (creating an entropy barrier); and (2) is more robust superficial input perturbations.
Via extensive experiments on real-world datasets (CIFAR-10 and CIFAR-100) using ResNet and Wide-ResNet architectures, we demonstrate that Mix-MaxEnt consistently provides much improved classification accuracy, better calibrated probabilities for in-distribution data, and reliable uncertainty estimates when exposed to situations involving domain-shift and out-of-distribution samples.


Under review as a conference paper at ICLR 2022
MIX-MAXENT CREATING HIGH ENTROPY BARRIERS TO IMPROVE ACCURACY AND UNCERTAINTY ESTI- MATES OF DETERMINISTIC NEURAL NETWORKS
Anonymous authors Paper under double-blind review
ABSTRACT
We propose an extremely simple approach regularize a single deterministic neural network to obtain improved accuracy and reliable uncertainty estimates. Our approach, on top of the cross-entropy loss, simply puts an entropy maximization regularizer corresponding ding space between the class clusters. This is achieved by synthetically generating between-cluster samples via the convex combination of two images from different classes and maximizing the entropy on these samples. Such a data-dependent regularization guides the maximum likelihood estimation to prefer solution that (1) maps out-of-disto -i-diiiirituionn samples to high entropy regions (creating an entropy barrier); and (2) is more robust the superficial input perturbations. Via extensive experiments on real-world datasets (CIFAR-10 10 and CIFAR- 100) using ResNet and WiseeRReNeet architectures, we demonstrate that MixxMt consistently provides much improved classification accuracy, better calibrated probabilities for in-distribution data, and reliable uncertainty estimates when exposed situations involving domain-shift and nut-of-distributen samples.
the predictive distribution in the regions of the embed-
INTRODUCTION
particularly thriving sub-field of research in Deep Neural Networks (DNNs) concerns devising efficient approaches towards obtaining reliable predictive uncertainty. NNs are known to be overcon- fident for both, in- and out-distribution samples (i.e. for samples coming from the same distribution from which the training distribution has been sampled (IND) samples) and for samples not coming from such distribution). leading to highly unreliable uncertainty estimates. They can be wrong with very high confidence (Guo al., 2017) not only on the test data similar to the one they have been trained on, but also when facing previously-une dence problem becomes even more concerning when just slight changes in illumination, atmospheric conditions or in the image capturing process (domain-shift) can severely damage the actual accuracy of the model (Taori et al., 2016). A desirable property of any model is to be robust such superficial changes that do not affect the label of the classified image, and to become uncertain (or indecisive) when exposed samples from distribution different from the training distribution.
conditions (Taori et al., 2016). The overconfi-
While the literature suggests that classifiers whose predictive distributions increase their entropy the further away the test input gets from the training data are desirable, the implementation of models satisfying such a property is not scalable, and can only occur at the cost of performing crude approximations and by nriviaivial modifications the architecture of the neural network (Liu et al., 2020a; Kristiadi et al., 2020). Such modifications sometimes lead to degraded accuracy.
Our motivation behind this work is based on the following observations. (1) We find that a trained network (DNN) do not project out-of-distribution (OOD) and ily away from the training data. In fact, all the OOD and domain-shi inputs that we considered (detailed discussion in Section 3) are projected within the smallest hypersphere that contains all of the IND test data. Therefore, it might not be necessary to enforce the network to be uncertain every- where away from the data, and perhaps, in-distribution data can be used mimic the regions where OOD and DS samples are being projected. (2) DNNs tend to embed OOD and DS samples in high confidence (low-entropy) regions (see Figure 1). This contradicts the desired ideal behaviour (Liu
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et al., 2020a; Pereyra et al., 2017), that would require model to map such samples in high-entropy regions.
Given the observations above, we propose simple entropy maximization regularizer (called Mix-MaxEnt), that induces a high entropy barrier between class clusters, while the cross-entropy optimization objective keeps the entropy low close to the class clusters. The entropy regular- izer is enforced for samples synthesized using the convex combination of pair of samples from two different classes of the in-distribution training data. When combined with the cross- entropy loss, this regularizer prefers a maximum likelihood solution such that the uncertainty of the network increases while moving away from the embeddings of one class in the direction of another, and learns features that are robust to local input perturbations.
Figure 1: DNN (Left), Ours (Right). Interpola- tion experiment on CIFARIO show embeddings of the linear interpolation of two randomly picked input samples from class (purple) and class (yellow). Red and green samples are classified as class I, and orange and blue samples as class 2. As the color changes from red to green, the predictive entropy increases. Same for the color change from blue to orange. Note, DNN classifies interpolated points with very high confidence (low entropy) even if the samples shift drastically from the data. However, Mix-MaxEnt maps these samples into a wide high-entropy region. Details of this visualiza- tion are provided in Appendix F. More exhaustive visualizations of this phenomenon are provided in Section 5.2.2.
Through extensive experiments using WideRessNet2-10 and ResNet50 architer- ctures on CIFARIO 10 and CIFARI00 datasets, we demonstrate that our method outperforms all existing single model baselines in provid- ing clean data accuracy. On Domain- Shift experiments, it provides remarkably improved accuracy compared to all the baselines including Deep Ensembles (DE) (lakshminarayanan et al., 2017). For instance, it provides 4.8% and 4.7% improvements over the highly competitive DE and SNGP (Liu et al., 2020a), respectively, on CIFAR-I0 using WideResNet. In terms of reliable uncertainty estimates, it is either the best in few cases, extremely competitive with respect to the best performing one, with only slightly inferior performance. Overall, our experiments show that Mix-MaxEnt is by far the best performing one compared to the existing single model approaches.
We would like to highlight that, along with its effectiveness, one of the core strengths of our approach its simplicity. As opposed the recently proposed SNGP and DUQ (van Amersfoort et al., 2020), does not require any modifications to the architectures and does not trade accuracy in order to improve uncertainty estimates. And, as opposed to the extremely competitive DE, it is single deterministic model, hence, extremely efficient.
2 RELATED WORKS
Modern NNs have been shown to be miscalibrated (Guo et al., 2017), implying, a mismatch between model's confidence and its accuracy. While recent approaches have been proposed to fix this issue on in-distribution data (Guo et al., 2017; Mukhoti et al., 2020; Lakshninarayanan et al., 2017), such calibrated models do not necessarily behave as well under domain- shift (Ovadia et al., 2019).
Uncertainty estimation methods The current state-of-the-art in producing reliable uncertainties is the extremely expensive Deep Ensembles (Lakshminarayanan et al., 2017), whose training and inference costs scale linearly with the number of members in the ensemble. This significantly limits its applicability in real-world applications, where time and compute are of the essence. But even in this case, the accuracy and calibration dramatically drop when facing data-shift, current research trend is to either emulate deep ensemble behaviour while avoiding its disadvantages (Havasi et al., 2021; Wen et al., 2020; Huang et al., 2017) ) or combining it with other methods (Ashukha et al., 2020; Rahaman & Thiery, 2020; Wen et al., 2021).
An alternative approach to obtain a reliable predictive distribution is to rely on Bayesian NNs (Chen et al., 2014; Zhang et al., 2020; Durmus et al., 2016). However, obtaining the exact posterior for NNs is computationally infeasible, for this reason, it is necessary to rely on approximate inference schemes whose effectiveness is in question (Hron et al., 2018; Foong et al., 2020). A recent trend in
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literature has tried reducing the application of Bayesian inference only to the last layers get the benefit of the effectiveness of training deterministic networks with standard training procedures, and the improved uncertainty estimation provided by Bayesian approaches (e.g. SNGP (Liu et al., 2020a), KFAC-LLLA (Kristiadi et al., 2020), BM (Joo et al., 2020), DUQ (van Amersfoort et al., 2020))-
Unfortunately, many of these models often trade aceuracy (due to changes in architecture training procedures) in exchange of marginal improvements in uncertainty estimation, and are hard to use in a plug-and-play fashion since they are extremely sensitive to hyperparameter tuning.
Learning better features via augmentation or normalisation It has been shown (Shah et al., 2020) that if NNs can identify simple (spurious) features to separate training data, they will prefer those to semantically meaningful features. NNs also heavily rely on textures (Geirhos et al., 201 8). This agrees with recent literature, that has shown self-supervision (Winkens et al., 2020; Hendrycks et al., 2019) or simple augmentation techniques like mixup (Thulasidasan et al., 2019) can produce representations that yield better uncertainty estimation. A way to make NNs insensitive to noise is imposing constraints on their sensitivity. An extensive literature studies how controlling Lipschitzness impacts on the robustness and generalization performance (Bartlett et al., 2017; Neyshabur et al., 2018; Yoshida & Miyato, 2017; Gouk et al., 2020; Arora al., 2018).
ANALYSIS AND OBSERVATIONS
Consider the supervised task of learning
11
predictive distribution p[y|x; 0), where y E
14
(1....,K K} is the set of K-classes, x X' C Iº is the space of input data, and 0 respresents the parameters of the underlying model (e.g. neural network). To obtain an optimal 8, the standard approach is to collect a training data in i== and maximize the log- set D (x;, yi
Figure 2: Histograms of the Euclidean distance of the embeddings (model trained on CIFAR-10). (Left) CIFAR-10 vs -IFAR-10-C. (Right) CIFAR- 10 (IND) vs CIFAR- 100 (OOD) and SVHN (OOD). Note, both corrupted and OOD embeddings lie within the hypersphere as their histograms are left shifted relative to the histogram of CIFAR-10. Also, the SVHN histogram is more shifted towards the center than that of CIFAR-100, showing an implicit ordering.
likelihood p(y) x E X) under D. Therefore, the nature of the predictive distribution heavily relies on the training data and the model under consideration. In practice, however, because of the high dimensionality of the input space X. the training data D is normally collected from a sub- set X2 C X, where X2 denotes the in-domain data manifold. Let us denote the out-domain data manifold as Xo = X \ Xz.
Where do neural networks map Xo? Modern softmax- based neural classifiers operate under two essential assumptions: (1) world assumption (Bendale & Boult, 2015); (2) an implicit clutering assumption (Hess et al., 2020; Lee et al., 2018). The closed-world assumption implics that the classifier must select one out of K classes fixed at training time, even if the input belongs to none of them. The clustering assumption depends on the presence of the softmax layer: the authors of (Hess et al., 2020) have shown that softmax neural classifiers perform K-means clustering (which, we recall, can be interpreted as a limit subcase of poppectation-Maxiissaae Model (GMM) assumptions (Hastie et al., 2001: Bishop, 2006)) with K centroids at equal distance from the center of the embedding space. The authors of (Lee et al., 2018) observed the embeddings of modern neural network do empirically adapt well to GMM assumptions' Following the above insights we consider WideResNet (WRN) trained on CIFAR-10 (X₂) and obtain the smallest hypersphere in the feature space that contains all the in-domain test samples. Note, as already mentioned in Hess et al. (2020), we also found the mean of the embeddings of X2 to be very close to a zero vector, therefore, S is practically centred at zero. Given S, we would like to understand where do Xo (CIFAR-100 and SVHN) and data-shifted samples (corrupted CIFAR-10 test samples denoted as CIFAR-10-C) lie with respect to S. These are our observations:
under Gaussian Mixture
'It is important to observe that, given this empirical evidence, one would think that using the log- probbbbbyyylity of GMM could be an optimal uncertainty measure for this task. However, in Appendix E. we show that the high-dimensional nature of the latent space makes the computation numerically unstable.
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The NN maps all (except one) distrf--itttibutionn (Xo) and data-shifted samples (corrupted) inside the hypersphere S that was obtained using the test data belonging to Xz.
There exist an implicit ordering in the sense that CIFAR-100 is mapped closer to CIFAR- 10 than SVHN. This is clearly shown in the histogram (refer Figure 2) of the Fuclidean distances of the OOD and data-shifted embeddings from the center of the hypersphere S. This behaviour can be explained by the fact that convolution kernels are learned using Xy, therefore, will produce stronger activations if the similarity between the kernels learned on X and the features present in the input is high. On the other hand, dissimilarity between kernels and features present in the input will produce weaker activations, thus driving the embeddings closer to the origin.
Forcing classifiers to be uncertain where truly matters Recall that an ideal classifier's predic- tive distribution p(y)x x E X; would assume the following form
ply|x Xy; f))(px E Xg) + p(y|x E 0o:x) E Xo).
maximum likelihood estimate
unknown diaring training
Since Xo is unknown during the maximum likelihood estimation (MLE) of 6, the model is com- pletely unaware of the second part of the predictive distribution that depends on the presence of No. Therefore, in order for the model to be indecisive for out-of-distribution and domain-shifted samples (assuming they are away from the in-distribution data), it is desirable that the model's s uncertainty increases proportionally to the test sample distance? from the training data X,. This is the prime motivation behind the recent work called SNGP (Liu et al., 2020a) where the uncertainty estimates of a NN is being improved by taking inspirations from the classic Gaussian Processes literature (Rasmussen & Williams, 2005). While techniques like SNGP (Liu et al., 2020a) and KFAC-LLLA (Kristiadi et al., 2020) try to increase the entropy everywhere away from the training data, they need resort to crude approximations of the theoretical models that provide such property (both) and also have to modify the network's 's architecture (SNGP). Most approaches also compromise with the training accuracy because of such approximations.
Given the observations from our analysis, we find that the OOD and domain-shifter samples are mapped to specific embedding space 8 where the network tends to concentrate all its embed- dings.Thi This indicates that correcting its uncertainty estimates within this space S could be sufficient to achieve better performance, rather than trying to increase entropy everywhere away from the training data (i.e. mostly in regions where the network never projects its inputs). This is precisely the motivation behind our work. In what follows, we present an extremely simple approach to regularize the embedding space S such that entropy barriers are being created between different classes.
4 METHODOLOGY
Mix-MaxEnt Motivated by the evidence we collected, we define simple regularization approach to make the maximum likelihood estimate aware (increasing the uncertainty and becoming less confident) of what is unknown to it during the training by simply putting a maximum entropy regularizer (Pereyra et al., 2017) on synthetically generated samples that leverage the knowledge of X, to drift towards Xo. Specifically, since Xo is unknown during the training, we take simple approach where we synthetically create samples & E as follows:
X = tox; + + (1- Ao)X,, if y: + yj--
(1)
Here, X,} E X2, and Ao ~ Betaaa, 3). where a and | are the parameters of the beta distribu- tion. Note, we only choose pair of samples belonging to two different classes (y # yj), so that x will mix features mimicking an off-data manifold image that has intermediate properties between those of each class. We further ensure this by picking a = 3 >> 1 as, in this case, the beta distribution will have the peak in the middle and the sharpness of the peak increases collection of such synthetic intermediate samples Xo. Then, our final objective function adds to the usual cross-entropy optimisation term, an entropy maximisation term on such samples:
grows. Let us call the
min 8
log p(y|x Xy; Hg(p(-Ix E Xo:8)):
2 Note, although the theory assumes Fuclidean distance, it is arguable this distance notion reflects perceptual dissimilarity in the image space.
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where, (.) is the entropy defined over the label support set f {vt: V3 }- It is important to observe that since the imput image will contain features coming from the inputs of the two classes, it is reasonable maximise the entropy only over such classes while keeping the probability assigned to the other classes to zero (since no input features endorses the presence of these classes). Indeed, in our experiments show that maximising the entropy over all the classes severely degrades the performance while making the network underconfident. Since we increase the entropy only for mixed (interpolated) samples from different classes, we call our approach Mix-MaxEnt.
Mix-MaxEnt vs mixup (Zhang et al., 2018) Though the sample interpolation in our approach is similar to the one presented mixup, there are fundamental differences between the two. The mixup training goal is to reduce memorization by training around the vicinity of the samples (vicinal risk minimization). Effectively, it slightly perturbs the data and uses its perturbed version for the log- likelihood estimation. It optimises log p( Xx) where and y/ and yj might be the same. Practically speaking, a small value of Beta distribution parameters (a B < 1) is typically chosen for mixup (Zhang al., 2018; Thulasidasan al., 2019; Wen et al., 2021), thus, Ao in this case is either R 0 or A 1. Because of this, one of the two interpolating samples heavily dominates the interpolated sample x. Note, this is necessary for mixup as otherwise the interpolated sample distribution would be different from the original data ditt-distribution and, because of the absence of the tog-likelihook term over the clean samples (as in Mix- Mix-MaxEnt), there will be drop in the performance at test time.
(1-Xo)y, X = λgx; + (1-)))x
(refer Eq (2)) uses clean unperturbed samples for the log- lideeeiddd estimation while interpolated samples are being used to regularize the network so that it prefers a solution having high-entropy at regions in in-between samples of different classes. Since the purpose of the interpolated samples here is only to guide the log-likelihood solution, thus, as opposed to mixup, high interpolation factor can be used in this case. Hence, we use Ao As 0.5 to interpolate pair samples from different classes so that the interpolated sample mimic points from outside the data manifold for entropy maximization.
However,
5 EXPERIMENTS
Training Datasets and Network Architectures We employ WideResNet28-11 10 (WRN) (Zagoruyko & Komodakis, 2016) and ResNer50 (RN50) (He et al., 2016) architectures that have been shown produce state-of-the-arg art classification accuracies on real-world datasets. We train them on CIFAR-I0 (C10) and CIFAR-100 (C100). For Domain-Shift experiments, we resort to the widely used CIFARIO-C and CIFAR100-C, corrupted versions of C10 and C100 (Hendrycks & Dietterich, 2019). For Out-of-Distribution detection experiments, following SNGP (Liu et al., 2020a), we use C100 and SVHN as OOD for models trained on C1O. Similarly, for models trained on C100, we use C10 and SVHN as OOD.
Methods considered for comparisons We consider both deterministic and Bayesian approaches for comparison. Following (Liu et al., 2020a), we also create two additional strong and simple baselines where a ResNet is enforced to be bi-Lipschitz using Spectral Normalization (SN) (Miyato al., 2018a) and Stable Rank Normalization (SRN) (Sanyal al., 2020). Note, we are the first to consider SRN for these experiments as it induces more compact clusters in the feature space than SN (we provide a simple mathematical proof of this in the Appendix A). Therefore, we compare our approach with the following baselines:
DNN: Standard deterministic neural network trained using cross-entropy loss.
DNN-SN:DNN with SN (Miyato et al., 20)
DNN-SRN DNN with SRN (Sanyal et al., 2020).
SNGP: Spectrally Normalized Gaussian Process (Liu et al., 2020a).
DUQ: Deterministic Uncertainty Quantification (van Amersfoort et al., 2020).
Mixup: Standard Mixup training (Zhang et al., 2018).
KFAC-LLLA: KFAC-Laplace Last Layer Approximation (Kristiadi et al., 2020). A method that makes model Bayesian at test time by taking Laplace approximation of the last layer using a Kronecker-F Factored approximation (Ritter et al., 2018). For the sake of completeness, we provide simple outline of this approach in Appendix B.
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Clean Dula
Dommii-Shiit
Out-ot Distibbici
CIEARI00 (Tos))
CIFARIOBCC (CE(4)
CIFARIO
SVHN
Methods
Accuracy (1)
ECE
AdaECE ())
Aceurncy (1)
AdaECE ())
(1ROC
AUPR. (7)
ALEOCC (*)
AUPR (1)
DNN
81.58 + 0.13
3.88 = 0.25
3.54 0.24
52.54 0.3.1
9.06 0.221
9.94 022
81.06 = 0.29
0..35
79.68
= 4.81
88.46 2.58
DNNUSN
81.60 2: 0.15
3.94 = 0.23
3,81 0:2
52.61 = 0.23
11,62 a 0.4
11.39 0004
81.10 2 0.35
17.34
1..19
83.43
2 3.63
91.01 + 2.05
DNN-SKN
81.38 x 0:25
3,82 0.27
3711 0.26
52.54 0.17
11:04 0.77
11:00:0000
81:26
x 0.18
77.36
0..90
85.51
8 1.18
91.84
Doep Brecuibles
83.45
3.31
3.30
0.13
12.36 0:15
83.36
7982 6.27
85.07
91.65
SNOP
79.20 4:
1,05
1 0.28
57.23
1.56
79:05
0.34
86.78
93.30 1.08
KINGILLA
81.56
7.00 0.51
0.27
81:04
77.36
89.05 3:300
Mixu
82.60
1,77 0:49
1,58 0.43
$6.99
10:32 0.64
10,45
76.37
73.05
0.96
58.37
1.59
Mix-MaasFal
83.23 0.22
1.67
0.89
1.76 0.62
59.39
0.72
7,53 + 0.84
7,98 0.84
81 0.48
77.38
+ 0.35
89,32 1.11
94.45 0.90
Table I: WideResNer28- 10 trained on C100. See Appendix for the cross validtta hyperparameters.
Clean Date
Dumain-Shift
Ontee--ietrilntien
CIFARIO (Test)
CIFARIO-C FCE()
CIFARIOO
SYIIN
Methods
Accurayy (*)
(1)
AdsRCR( (1)
Accuracy (1)
AddRCE
AUBOC
(TRROC())
(1)
DNN
0..04:00.06
1.26 0.05
1.34
0.25
12.64
0.77
58,51
85.91
0.21
96.00
98.05
DNN-S
96.22--0.11
0.91 0.14
1 0.16
16.56 10:6-000 0.34
11.13 0.33
11:15
031
58.56 + 0.56
8901
0.34
95.99-0044 0.49
97.85 +:32
DNN-SRN
96.12+00.10
1:34 0.08
1.36 0.15
72:31 0.99
11 0:40
11:56
0.4
68.46 + 0.36
88.84
0.37
96.12.+ 1.61
98.10
1:10++0..5
Deep Ensembles
0.00
0,81 0.16
1
78.32 0.05
10.11 4-017
10.31
0.10
91.25 2: 0.14
91.13 0.15
SNGP
95.08 0.11
0.84 9.11
087 0.116
78.30 (.77--223
11:33-1 + 0.38
11:34
0.99
90A1 1 0.00
00.30+00.12
0.55 95.25
97.9K 0.1.8
DUQ
96.7 0.02
3.4+00.3
71 0.02
18.3 + 1.1
K34
105
97.3+ 1.0
KFAC LLLA
96.11
2:01
1.00 0.00
1.124 0.07
0.48 76.56
11.69
1 0.76
11.07 0.76
89.33 4 0.23
85.52
2: 0.20
94.17
96.99
0.94
Mizup
97.01 0:11
0.96++0.30
1.16+ 0.19
81 0.67
7.54 0.83
7.83 1.2
K3.17 +
85:47
0:45
67.53 6:07
95
Mix-MaxEas
07044:00006
0.63 0.08
0.08
1.33
1.59
59.13
0.31
96.22
98.1
Table 2: WideResNNe12-100 trained on C10. See Appendix C for the cross-validated hyperparameters.
DE: Deep Ensembles (Lakshminarayanan al., 2017) with 5 members. Note, it is almost 5x slower than all other approaches mentioned above.
Code base For fair comparisons, we developed our own code base for all the approaches mentioned above (except SNGP and DUQ) and performed an extensive hyperparameter search to obtain the strongest possible baselines. For SNGP, we used the available code and made sure that we follow exactly the same procedure as mentioned in their original paper. For DUQ, the original paper did not perform large scale experiments similar to ours. Unfortunately, we could not manage to make their code work C100 it exhibited unstable behaviour. For this reason, we borrowed numbers for DUQ from the SNGP paper. Please note that the authors of SNGP performed non-trivial modifications to the original DUQ methodology to make it work on C100. Further details provided in Appendix C.
Optimization and Hyperparameters We use SGD with Nesterov momentum 0.9 and a weight decay of 10-4, For WRN, we apply dropout p 0.1 at train time. We perform extensive cross-validation of all the hyperparameters for all the baselines. Details provided in Appendix C.
Evaluation Metries For calibration, we employ: (1) the widely used Expected Calibration Error (ECH) (Guo et al., 2017), and (2) the recently proposed Adaptive ECE (AdaECE) (Mukhoti et al., 2020). For all the methods, the ECE and AdaECE are computed after performing temperature scaling (Guo et al., 2017) with a cross-validated temperature parameter. Metrics and uncertainty measures used for -ut-of--istribution detection are discussed in detail in Section 5.1.3.
5.1 EXPERIMENTAL RESULTS
5.1.1 ACCURACY AND CALIBRATION ON CLEAN DATA
Our first set of experiments evaluates the accuracy of the proposed method on the test sets of C100 and C10. As it can be observed from the 'Clean Data" column of Tables I and 2, our method provides a remarkable increase in accuracy compared to recently proposed approaches for improved uncertainty estimation. For example, in the case of C100, it is 4.03% and 1.67% better than SNGP and KFAC, respectively. Most of the time, while improving uncertainty estimates, such approaches degrade the accuracy. However, Mix-MaxEnt not only provides much improved accuracy compared to these approaches, it also shows either comparable or improved accuracy compared to DE as well. Note, DE is computationally much more expensive than other approaches both at training and test time.
In terms of calibration, our method remarkably improves the ECE and AdaECE outperforming all the baselines. Mixup turns out be an extremely strong baseline in this regard. We provide ECE

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and AdaECE without temperature scaling in Appendix D. Due to lack of space, results using the ResNet50 architecture are presented in Appendix D.
5.1.2 ACCURACY AND CALIBRATION ON CORRUPTED DATA (DOMAIN-SHIFT
To evaluate the behaviour of various models under domain-shift, we resort to the widely used CIFAR- 100-C and CIFAR-IO-C datsets, corrupted versions of the C10 and C100 datasets (Hendrycks & Dietterich, 2019). The dataset is made by applying 15 synthetically generated but realistic corruptions degrees of intensity on the test sets of C100 and C10, respectively. The desired behaviour would be to preserve the classification accuracy as much as possible as these corruptions do not impact the underlying label, and to have an appropriate reduction in the confidence when the accuracy of the model degrades so that it is not incorrect with very high confidence. We report the expected accuracy, ECE and AdaFCE, averaged over all the corruptions and degrees of intensities in the column 'Domain-Shif" Shift' of Tables 1 and 2. It is evident that our approach produces a remarkable improvement in the average accuracy compared to all the baselines. For instance, as shown in Table (for C100), our method achieves an accuracy improvement of 6.85% over DNN, of 3.8% over DE, and of 2.4% % over Mixup. Similarly, in Table (for C10), our method achieves an improvement of almost 6.5% over DNN, of 4.78% over DE, of 1.42% over Mixup. Thus, setting new tfat-oo-tthe-ne with significant improvements. In terms of calibration, Mixup outperforms all the approaches for C10 experiments. Mix-MaxEnt turns out to be strong runner-up in this case performing at least as good as the expensive DE and outperforming SNGP and KFAC. However, for C100 experiments, our approach again outperforms all the baselines. For instance, in terms of ECE (refer Table 1). Mix-MaxEnt obtains a 4.5% % improvement over DE, 2.4% % over Mixup, and 2.5 % over SNGP.
5.1.3 PERFORMANCE WHEN EXPOSED TO OUT-OF DI-----DDSTIIUTIOON SAMPLES
Following the standard evaluation methodology (Liu et al., 2020a), we report the performance in terms of Area Under Receiver Operating Characteristic (AUROC) and Area Under Precision- Recall (AUPR) curves for the binary classification problem between in- and out ditt-iittbbttin samples. The uncertainty of the prediction of the model is normally used to obtain these curves. Given an uncertainty measure, it is important for models to be able to provide reliable uncertainty estimates to obtain good AUROC and AUPR. However, there is no consensus in the literature regarding which uncertainty measure to use. Choice of the measure itself can have significant impact the quality of the uncertainty estimation. In fact, there are various such metrics and they are not yet compared and evaluated properly on large scale experiments. To fill this gap, we performed extensive experiments with the following metrics to understand their behaviour and choose the one that is stable and provides the strongest possible baseline:
Entropy: (p(x))) = - P; log P;.
Dempster-Shafer (Sensoy et al., 2018): DS(x) = K/K-E exp(x:).
log >K 1 exp(s;) (ignoring the temperature parameter). This metric was used in (Liu et al., 2020b) for OOD.
Energy: Fx()
Maximum Probability Score: MP) max; Pt-
Feature Space Density Estimation (FSDE): Assuming that the features of each class follow Gaussian distribution, there are several ways one can estimate the belief of test sample belonging to in-distribution data and treat it as a measure of uncertainty. One such approach is to compute the Mahalanobis score arg min - ",)"E, (o(x) - p4), where μa and Σ, are class-wise mean and the covariance matrices of the train data. We define and use other variants as well: a detailed discussion is provided in Appendix E.
Remarks regarding various metries: We would like to highlight a few important observations that we made regarding these metrics. (1) DS and E are equivalent as they are both decreasing
functions of Σ Zist exp(s;), and since log does not modify the monotonicity, both will provide the same ordering of set of samples. Hence, will give the same AUPR and AUROC values. (2) We observed DS and H to perform similarly to each other except in few situations where DS provided improved results for all the approaches.
(3) MPS, in many situations, was worse (4) We found
Gaussian assumption based density estimation to be highly unstable. Though it provided extremely competitive results for C10 experiments, sometimes slightly better than the DS based scores, performed very poorly C100. We found this score to be highly unstable as it involves large matrix
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0.12
DNN
200
SNGP
175
175
0.10
100
800
DNN- SRN
1.5
Maxup
Ours
125
1.00
400
400
0.75
0.75
0.50
0.50
0.25
0.2
0.00
0.00
10
15
2.0
2.5
3.0
40
4.
5.0
Corr. Intensityy
Fisher criterion (a) on the embedding space for various degrees of elastic corruption intensity levels for C10 (domain-shift). Higher indicates more compact and separated clusters the feature space which is desirable. Right figures: heatmaps of the profile of entropies the interpolation factor between samples of two classes varies; left-most heatmap: DNN WRN on C10, the entropy is mostly zero; right-most heatmap: Mix-MaxEnt WRN on C10, there a high entropy barrier separating the classes, while the entropy is low close to the class clusters.
Figure 3: First figure:
inversions. We applied the well- tricks such as perturbing the diagonal elements and the low-rank approximation with high variance-ratio, but the results were extremely sensitive such stabilization and there is no clear way to cross-validate these hyperparameters. We provide analyses based on this metric and variants in Appendix Based on these findings, we report all the results in the main paper using D and report the performance using other scores in Appendix E.
OOD results: In the case of SVHN as the OOD dataset (refer Tables and
column), our method either outperforms all the existing approaches (2.54% higher AUROC than the runner-up SNGP in C100 experiments) or it turns out to be a runner-up with the gap of 1.31% compared the top performing and expensive DE. Note, Mix-MaxEnt outperforms all the single model approaches in these experiments as well. In the case of CIFAR as the OOD dataset, DE turns out be the best performing one. Mix-MaxEnt, along with KFAC-LLLA, the runner-up in the case of C1 00 experiments (refer Table 1) while SNGP is the runner-up in C10 experiments (refer Table 2).
5.1.4 WHICH METHOD IS THE BEST PERFORMING ONE?
We presented extensive results using a variety of experimental scenarios and metrics. In most of the experiments, Mix MaxEEn outperformed all the approaches including DE. However, there indeed were a few cases where it was not the best performing one. Following observations further suggest that Mix-MaxEnt is the best performing approach compared the existing single deterministic alternatives. Out of 20 different evaluations presented in Tables and 2:
Mix-MaxEnt outperformed all the approaches (including DE) 11 of them.
Mix-MaxEnt outperformed all the single model approaches in 14 scenarios.
No existing single deterministic baseline outperformed all other approaches in more than scenarios.
5.2
ANALYSING MIX-MAXENT
5.2.1 MIX-MAXENT ENCOURAGES COMPACT AND SEPARATED CLUSTERS
We use the well known Fisher criterion (Bishop, 2006, Chapter 4) to quantify the compactness and separatedness of the feature clusters for various models. Let C₁ denotes the indices of samples for k-th class. Then, the overall within-class covariance matrix is computed as Sw Σk1 St. where
Sa Enec, (ő(x..) (o(x)) -px)) px)T pa Enec. No
@(x..) and d(x..) denote the feature vector.
Similarly, the between-class covariance matrix be computed as SB Σ Ni(u H) u) where + Σk= Ngpk, and N₁ is the number of samples in k-th class. Then, the Fisher
for Mixup we report results using entropy as it provided the hest performance
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criterion is defined as a (ra Note, would be high when the within-class covariance is small and between- covariance is high, thus, a high value of a is desirable. In Figure 3, we compute over the C10 dataset with varying degrees of domain-shift. As the amount of corruption increases, gradually decreases for all the models, indicating that the model is not able to differentiate different classes anymore and is projecting them too close each other. This also explains why the accuracy and the calibration of all the models decreases as the domain-shift increases. However, Mix-MaxEnt consistently provides the best a. This also explains why Mix-MaxEnt performed so well under domain-shife We provide further plots and visualizations in Appendix J.
5.2.2 MIX-MAXENT CREATES A HIGH ENTROPY BARRIER BETWEEN CLASSES
In Figure 3, we provide heat-maps to visually show the entropy barrier. The heat-map is created as follows. We randomly choose 1000 pairs of samples {x,, xj such that y; * yj- For each pair, we synthesize 20 samples Xs using Eq (1) where λ is equally spaced between 0 to 1. The heat map is then created using all the 20K samples. The intensity of each (A, H) bin in the heat-map indicates the entropy peak
number of samples in that bin. It is clearly visible that for standard DNN, although between the two classes exist, the location of the peak is scattered and its intensity is inconsistent. For most of the interpolated samples the entropy is close zero, thus, making the model overconfident even for samples with very high degree of interpolation. However, Mix-MaxEnt clearly creates a high-entropy barrier that spans most of the space between the classes. At the same time, the entropy close to the clean samples is low. Therefore, the model is confident for clean samples and it becomes indecisive as the degree of interpolation increases. We show additional such plots and an alternative visualization of this phenomenon in Appendix F.
5.2.3 ANALYSING DIFFERENT FACTORS IN THE OBJECTIVE FUNCTION OF MIX-MAXENT
The importance of mixing samples between classes We empirically observe that not imposing the constraint y/ * yy does not produce significantly different performance (Table 13 Appendix G). This is because of the fact that the probability of sampling pairs with the same class is extremely low (0.1 for CIFAR-10, 0.01 for CIFAR- 100) compared to sampling pairs from different classes. However, we enforce the constraint that the pairs must belong to the same class y 3/j, the performance degrades significantly (refer Table 14). Therefore, mixing samples between classes is crucial.
The importance of having an interpolation factor close to 0.5 In Table 11, we show that Mix- MaxEnt performance relies on having interpolation factors close to 0.5. Mix-MaxEnt with a << 1 (i.c. making Ao Re 0 or se 1) degrades the OOD detection performance, the data-shift robustness, and the calibration with respect the case >> 1 (i.e. making λo x 0.5).
The importance of maximising the entropy defined over the support of the interpolated points Recall that Mix-MaxEnt (refer Eq (2)) maximises the entropy on the two interpolated classes. The rationale is that the input image will contain features from the two classes, not from any other classes. Therefore, the classifier should assign a non-zero probability only to these classes and there is no reason for it to assign non-zero probability other classes. We empirically check that this is sensitive choice as maximising the entropy over all the classes significantly degrades the performance 1 (and hence Xu 0.5), the entropy regularizer
(refer Table 12). We also observed that, since can be replaced by the Mixup loss with no performance degradation.
6 CONCLUSION
We proposed Mix-MaxEnt, an extremely simple approach that regularizes neural network to be uncertain in regions of the data manifold that are unknown during training. We conducted wide range of experiments to show that Mix-MaxEnt significantly improves the reliability of uncertainty estimates deep neural networks, while also providing a notable boost in the accuracy. A potential extension of our work regards the possibility of mixing features using more sophisticated methods. An interesting future direction would be to elaborate more sophisticated mixing techniques either based on data statistics (e.g. using PCA directions) or latent space geometry that would allow to map out-of- out-off-distrbbl disttributtion points closer to the center of the embedding space, and to further improve the clustering for data-shift examples.

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Ethics Statement. The fact that neural networks be wrong with high confidence poses potential threat to the deployment of deep learning systems into the real-world applications such as driverless cars and medical imaging. Therefore, finding fast and effective methods to train models that can provide reliable uncertainty of utmost importance. Hence, we are not particularly concerned that the research presented in this paper lead harmful applications the contrary, helieve that it can help deploying safer machine learning applications.
Reproducibility Statement. In this paper, we provide extremely simple approach that can be implemented in merely few additional lines of code. We have provided all the details needed to reproduce our and baseline results. All the training hyperparameters and details about the dataset (we only used publicly available datasets) are provided in the manuscript. To facilitate further research on this topic, we will release the code to reproduce all the experiments.
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A THE EFFECT OF BOUNDING LOCAL SMOOTHNESS ON MIX-MAXENT
Let fo = g h(x) be the neural network where h x x ((x) is the feature extractor and g(-) the last linear layer. It is desirable that the predictive distribution is robust to input perturbations that do not alter the label of the image, so that the network is able to preferentially carry over the true signal in the data. In other words, we would like our model to be bi-Lipschitz so that, given an input perturbation, the change in the output is both upper and lower bounded. Of course, these bounds should be reasonable so that the network neither looses its capacity (is trainable), nor too sensitive, which otherwise would cost its generalization ability. It has also been shown that ResNet architectures, because of their skip connections, are by design bi-I.ipschitz if all the linear operators constrained to have a spectral norm of less than one (Behrmann et al., 2019). Recent work (Liu et al., 2020a) built on this finding and showed that applying Spectral Normalization (SN) (Miyato et al., 2018b) standard neural network does improve the quality of its uncertainty estimates.
As an alternative, in this work, we enforce bi bi-Lipschizzness via the application of Stable Rank Normalization (SRN) (Sanyal et al., 2020). Let W be a linear operator with for..... o*} as its ordered set of non-zero singular values. Then, SRN effectively provides a new matrix whose singular values are n - ). where 7) E (0, 1]. Note, if 1 1, SRN boils down SN. Mathematically speaking, along with controlling the largest singular value of linear operator, the SRN objective also allows us to minimize its stable rank (a softer version of the rank operator)*.
We prefer SRN for the following two reasons. (1) It has been shown to control noise sensitivity, and provide better data-dependent empirical Lipschitz constant. (2) It encourages learning more compact embeddings in the feature space (we prove in Proposition A.1), a property that is shown to be important to obtain improved out-domain detection performance (Winkens et al., 2020).
Proposition A.1. Stable Rank Normalization (Sanyal et al., 2020) (SRN) encourages low volume clusters compared to Spectral Normalization (SN).
Proof. Let {v, E R" be a set of zero-mean vectors with empirical covariance matrix as S = 1 +E V.v.". Let Wo be a linear mapping with {a,} as its ordered non-zero singular values (o) being the max). Let the SN and SRN versions of Wo be W₁ and W2, respectively. Then, the volume of the covarinace matrix when each vector v; is mapped using Wo can be obained as det(WW)des) (S)det
ii
Vol(So) (e (2 Σi= (Wo,)((Wva)) (ignoring the multiplicative constant as all the matrices are in the same space). Note, we abuse the notation where the determinant det(-) of a matrix is the product of non-zero singular values. Similarly, the volume of the projected points when the vectors are mapped via W, and W2 can be
II of dS)
obtained as Vol(S,) = o IT of det and Vol(S.) (((-11 I of det(S), respectively. Since 7) E (0, 1] for SRN, it is trivial to note that Vol Vol(S²) < Vo) S Vol(S,). Hence, the clusters obtained using SRN will have the lowest volume.
of
A.1 CONSIDERING DIFFERENT TECHNIQUES TO BOUND LOCAL SMOOTHNESS
In our experiments, we considered two techniques to bound local smoothness: (1) using Spectral Normalization (SN) (Yoshida & Miyato, 2017). (2) using Stable Rank Normalization (SRN) (Sanyal al., 2020). We also considered variant of our method without these two components. We report the performance of these variants in Table 3. It is important to observe that, although in few cases SN can outperform SRN, in most cases they do not exhibit significantly different behaviour. We also observe that, in the case of ResNet50, bounding local smoothness via SN/SRN does not produce improvements (as already visible in Tables 5 and 6). This probably can be attributed to the high capacity of the network and the (relatively) small quantity of training data in the datasets considered.
A.2 THE IMPACT OF SRN ON DATA- ROBUSTNESS
In this section we report additional plots showing the impact of SRN when applied on top of the maximum entropy regularizer. As it can be seen in Figures and 5, for certain kinds of noises significantly improves the robustness to data-shift.
(erank(W) 2 where [ [1g is the spectral norm.
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Methors
Clean (CR
Corrupted BOC
CHAR
SVIIN
Accurny <1)
AdaRCK (1)
Aecoracy (1)
AdaRCR (1)
AUROCAA:
AUPR (1)
ATUROCATT
AUPIR (1)
C:0
Mix Maxtmm
96.99 22 0.17
0.65 0.11
B94 d: 0.2.1
$1.16 2: 1
12.61
1.77
12.52 1.78
87.5 d: 0.67
85.85 A 0.3
94,39
0.72
90.31 2. 0.49
Mix MI
56.80
22 0.19
0.70
0.11
0.24 0.222
1991
1.19
15
0.8
13.7 1. 0.86
87.99 2: 0.89
N5.00
1.21
93.32
1.83
1.39
Mic-Maxtim-SRN
56.74 25 0.00
0.88 0105
0.79 de 0.36
$1.35
1:31
12:1 1: 1.86
12.12 1.88
87:25 1: 0.71
16.19
0.08
94.18
0192
96, 76
0.67
C.00
0.31
0.30
75.50
57.04 0.87
12:45
100
0.336
79.27
85.51
C10 WBN
Mix Mastut
0.63
15.10
10.13
10.08 1.59
89.13
55.12 0.37
96.22 0.45
0.4
Mis-MasFatsSN
:-.00=00077
0.53 = 0.11
0.64 0.0.2
85.15 1:35
8.46 0.59
8.88 0.58
89.59 0.71
84.56 1.11
98.72
0.43
98.33 + 0.20
97.51 0.00
0.53 = 0.12
0.62 0.05
$4.40 0.53
8.00 0.86
8.00 0.84
80.35 + 0.60
83.40 s 0.73
97.22
0.76
98.67 + 0.35
C100 WRN
Mix MaxEnt
+-22
1,67 = 0.99
2: 0.62
59.39 d: 0.72
7:93 a: 0.54
7:98 e: 0.84
51.64 +: 0,48
77.28 0.35
89.32 & 161
94.65 # 0.90
Mis-Mas-Fii-SN
8387 + 0.18
1.48 0.90
141 d 0.32
61 t 0.69
7:42 0.70
7.39 + 0.0.1
R0.83 + 0.16
77.30 x 14
91.08
2.79
95.99 90 1.25
Min-MacFan-SIN
$3.96 + 0.18
1.73 = 0.22
A6 it 0.30
60.70 =t 0.91
8.43 + 1.52
8.40 + 1,52
81.74 + 0.31
78.06 + 0.13
90.38 + 2:40
95.15 + 1.20
Table 3: Evaluation of our method considering a version without any regulariser bound local smoothness (Mix-MaxEnt), using Spectral Normalization MiixMMxXEttSSN) and using Stable Rank Normalization MiX-MMxxEnt-SRNN, The hyperparameters are as in Table 4.
Gaussian Noise
0.9
MaxMaxEnn Mo-MaxEnd 55 Mix MaxEnt-SRN
0.7
Mix-MaxEnt
0.7
Mix Mx-MaxEnt
0.8
0.6
Mix Max-ManEnt-SN
Mix Mx-MaxEEtt.. SN
Mix Ma.EMaxEEntRR SRN
Mix.MaEnn...o SRN
0.7
Accuracy
0.6
0.4
0.4
0.5
0.3
0.3
0.4
02
0.2
0.3
0.2
Corr. Intensity
Intensity
Conr. Intensity
Glass Blur
0.250
Max MaxEnt
0.250
Max-aaxEn Mas-MaxEnt-SM
0.255
Max-MaxEm-Sh
0.225
0.50
Mix MRN
M4x-MaxEnt-SRN
0.200
0.200
Accuracy
0.75
0.175
0.175
0.150
0.70
0.125
0.125
Mox MunEre
0.100
0.100
0.60
Mix Moc-MaxEnt:Si
Mix Maxtaetttti SN
0.055
0.075
0.050
Corr. Intensity
Corr. Intensity
Corr. Intensity
Shot Noise
0.9
MaxMaxEnc
0.6
Mix-MaxEnt
MaxMaxEEt Ma-MaxEnt SN
Mx-MaxEnt-Sh SN
Mis MaxEManEnt-N
0.8
MaxEnt
0.5
Mix Ent SRN
Accuracy
0.7
0.4
AdaECE
0.4
0.2
0.4
0.1
0.
Corr. Intensity
Intensity
Corr. Intensity
Figure 4: Accuracy, ECE and AdaECE for some corruptions that show the remarkable improvements of using SRN for CIFAR-IO-C over not using it, architecture WideResNNe28-10.
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Gaussian Noise
0.06
Max-MaxEnt-SN
MixMaxEnt
20000
MIx-MaxxENt-N
Mix- ManEnt
Mix MakEntt-SR SRN
Mix. MaxEnt:S5 SAN
0.05
18000
Raher
0.04
[5WIL
[|Sb]
0.03
16000
14000
Mix MaxEnt
15
Mix Max.Ma SM
10
Mix Www.MaxEnntSRhe SRN
Corr. Intensity
Corr. Intensitty
Corr. Intensity
Glass Blur
0.0350
22000
Mix MaxEnt
Ma-MaxEm--SH
Mas-MaxEre-SM
Max-Mastré
0.0300
21000
Mix MaxCnt SAN
Mextin SRN
34
0.0250
19000
32
18000
0.0000
17000
0.0155
Mr-MaxEni-SSN
16000
0.0150
25
Intonsity
Corr. Intonsity
Intonsity
Shot Noise
Mix Maxtmtt
Mix MaxEnt
Max-Maxtit-SS
20000
50
Man-Mattt--- SN
MRxMaxEnt-SRN
SRN
0.07
18000
40
0.06
ISWILF
0.05
14000
0.03
Mix Maxfint Mox Maxtine SN Mix-MaxEnt-Shh
0.02
Corr. Intensity
Corr. Intensity
Corr. Intensity
Figure 5: Fisher criterion (and the spectral norm of its factors) for some corruptions that show the re- markable improvements of using SRN for CIFAR-10-C over not using it, architecture Nere 10.
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B KRONECKER-FACTOREN LAST LAYER LAPLACE APPROXIMATION
structural problem of using MLE logistic regression that the produced uncertainties depend on the decision boundary. On the other hand, replacing the MLE logistic regression with a Bayesian logistic regression and estimating the predictive posterior employing a Laplace approximation allows to produce better uncertainties (Kristiadi et al., 2020). However, Bayesian training either requires a modification in the architecture (Liu et al., 2020a) or makes the inference procedure very expensive (Kingma et al., 2015; Gal & Ghahramani, 2016). Since the objective is to utilize the standard MLE training of neural networks, the idea of Kronecker-Factoor Approximation (Kristiadi et al., 2020) is making the network Bayesian at test time with almost no additional cost.
Last Layer Laplace
Let w be the parameters of the of the last layer of neural network, then we seek to obtain the posterior only over w. Let p(w x) be the posterior, then the predictive distribution can be written as:
pyy = k
(3)
where, s is the logit vector and softmax(s:) is the k-th index of the softmax output of the network.
The Laplace approximation assumes that the posterior p(s[D) ~ (s) E), where is a mode of the posterior p( w/ (D) (D) (found via standard optimization algorithms for NNs) and E is the inverse of the Hessian H - (V log p( [w] [D) For the formulations and definitions, including the variants with the terms associated to the bias, we refer to (Kristiadi al., 2020).
For our experiments, we obtain Σ using the Kronecker-factored (KF) approximation (Ritter et al., 2018). Broadly speaking, the KF approximation allows reduce the computational complexity of computing the Hessian by factorizing the inverse of the Hessian as H ~ V X U-", then the covariance of the posterior evaluated a point x takes following form = (ø(x) ((o)))Voo)))) Vo( This procedure can be easily implemented using the Backpack library (Dangel et al., 2020) to compute V and by performing a single pass over the training set after the end of the training, as detailed in the Appendix of (Kristiadi et al., 2020) and clearly exemplified in the code-base of (Hobbhahn et al., 2021).
Let Σg be the covariance matrix of the posterior over the last linear layer parameters for the k- th class obtained using the Laplace approximation around μ. then, given an input x, we obtain on = Φ(x) Σxσ(x) representing the variance of k-th logit Sj. Once obtain the covariance matrix, the Monte Carlo approximation of the predictive distribution (equation (3)) is obtained as:
m
p
softmax(s(i).
m
where, m logit vectors s(t) are sampled from a distribution with mean s and a covariance matrix (depending on the approximation used). Lu et. al (Lu et al., 2020) showed that similar performance can be achieved via the mean-field approximation which provides an approximate closed form solution of the integration in equation (3) involving the re-scaling of the logits and then taking the softmax of the re-scaled logit. The re-scaling is defined as follows:
Sk
SR
(5)
V++
Note, the scaling of the k-th logit depends on its variance (obtained using the Laplace approximation) and hyperparameter X. This approximation is efficient in the sense that it does not require multiple samples as required in the MC approximation (which can become expensive as the number classes and samples grow). In our experiments, we use the MC approximation. since we could not find an obvious way to finettuee A. Additionally, we observe that the mean-field approximation imposes trade-off between calibration and OOD detection performance. Increasing X, indeed, flattens the softmax distribution and improves OOD detection scores; although, as a consequence, harms calibration by making the network underconfident.
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C EXPERIMENT DETAILS
Hyperparameterss For all our experiments we set the batch size to 128 At training time, we apply standard augmentation (random cropping and horizontal flipping, similarly to (Liu et al., 2020a)). The data is appropriately normalized before being fed to the network both at train and test time.
for DNN-SN, DNN-SRN and Mix-MaxEnt the set of SN clamping factors we considered in our experiments are e E (0.5, 0.75, 1.0), the target of stable rank v E {0.3, 0.5, 0.7, (as r = 1 for SRN is the same as applying SN with 1.0).
* for Mixup, we considered the Beta distribution hyperparameters to be (.21, 0.3, 0.4, 0.5). as suggested in the literature (Thulasidasan et al., 2019).
for Mix-MaxEnt we considered the Beta distribution hyperparameters be a e (5, 10, 15, 20, 303. In both cases we set a = /3 in (e,
for the KFAC-LLLA, we took 1000 samples. Although the number might seem quite high, we could not notice significant improvements using a lower number of samples. We tuned the prior variance do needed for the computation of the Laplace approximation minimising the ECE on the validation set. We also tried using the theoretical value og = 1/T (Kristiadi et al., 2020), where T represents the weight decay, but it produced inferior results with respect to our cross-validation procedure.
for Deep Ensembles we use members.
when temperature scaling is applied, the temperature T is tuned on the validation set, minimising the ECE (we considered values ranging from 0.1 to 10, with step size of 0.01). For Deep Ensembles, we first compute the mean of the logits, then scale by the temperature parameter before passing it through the softmax.
All the hyperparameters we chose reported in Table 4. Such hyperparameters have been selected performing cross-va cross-validation with atratffiedsampliin on a 90/10 split of the training set to maximise Accuracy Indeed, it is important to observe that:
optimising the hyperparameters based solely on the ECE can prefer models with lower Accuracy but better calibration. However, Accuracy is commonly considered of primary importance, and any method improving calibration should avoid degrading it.
Optimising considering any of the corrupted experiments and OOD detection metrics would be equivalent to overfittig the test sel (indeed, both corruptions and OOD datasets are assumed to be unknown at training time by all the considered methodologies, and so they should be during the hyperparameter selection procedure).
For all of our experiments we train the methods initialising the networks with 5 different seeds and report the average and standard deviation in percentage for all the metrics. For the hyperpa- rameter search, we trained approximately 350 models. For the final tables (five seeds), we trained approximately 250 models.
Code For the SNGP method we use the official code- base with the suggested hyperparameters and training procedures. The code diverges slightly from the procedure described in their paper, hence the slight differences in the performance. The only modification we performed to the official code-base was to make the inference procedure consistent with the one described in the paper: indeed, in their code they implement 2020), while in their paper they use Monte Carlo Integration with a number of samples equal to the number of members in the ensembles they use as baseline, which provides better calibration. The rationale is that we could not find an obvious way to tune the mean-fieee field approximation hyperparame- ters to improve at the same time both the calibration and OOD detection performance (indeed, the mean-field approximation imposes a trade-off between calibration and OOD detection performance). Additionally, since the standard KFAC-LLLA uses the same Monte Carlo Integration procedure,
mean-field approximation to estimate the predictive distribution (Lu et al.,
$ Foor SNGP and DUQ, we use the hyperparameters suggested their original papers. 'Except for the do of the KFAC-LLIA,, as we could not observe significant differences in Accuracy between hyperparameters optimising the Accuracy and ECE
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Training Set Architecture
Hyp
C10
C100
WRN
R50
WRN
R50
DNN
T
1.3
1.5
1.3
1.4
DNN-SN
C
0.5
0.5
0.5
0.5
T
1.4
1.5
1.2
1.4
0.3
0.3
0.3
0.3
DNN-SR
1.3
1.4
1.2
1.4
DE
T
1.3
1.4
1.1
1.2
SNGP
T
1.4
1.5
Mixup
a
0.3
0.3
0.3
T
0.7
0.8
1.09
1.2
Mix-MaxEnt
a
20
20
10
10
T
1.1
1.3
1.2
1.2
20
20
20
20
Mix-MaxEnt-SN
1.23
1.23
1.09
1.19
0.5
0.5
0.5
0.5
a
20
30
20
10
Mix-MaxEnt-SRN
T
1.2
1.3
1.09
1.09
0.9
0.9
0.9
0.9
samples
1000
1000
1000
1000
KFAC-LLLA
0.6
4
0.1
Table 4: Hyperparameters selected via cross-validation (90/10 split with stratified sampling) for all the methods we trained from scratch. We tuned T and du by minimising the ECE. All other hyperparameters have been tuned to maximise the Accuracy.
Corrupted ECE (4)
CIFARI00
Accuracy (1)
ECE (1)
AdaECE (1)
Accuracy (f)
AdaECE (1)
AUROC
AUPR (t)
AUROC (t)
AUPR (1)
DNN
95.19 = 0.23
1,38 a: 0.19
1,45 0:1.
75.18 0.69
12.31 4: 0.84
12.29 0.5
88.61 4: 0.66
88.05
0.52
93.20
1.98
96.43 1: 0.95
95.20
11.75
11.74-10.44
93.46
DNN-SRN
95.39
0.08
0.64
0.40
88.15
93.54
Deep Ensembles
96.23 + 0.05
1.27 + 0.05
1.28 + 0.03
77.63 + 0.36
13.12 0.32
12.68 + 0.32
91.38 0.21
90.75
0.13
96.90
0.07
98.27 0.09
KFACALLA
95.21 0.26
0.79 0.26
0.69 0.24
75.18 0.89
10.26 0.97
10.23 0.97
89.54
0.41
88.30
0.41
93.13
1.01
0.63
Mixup
96.05 0.15
0.59 =: 0.39
2.17 4: 0.51
78.63 +: 0.72
10.17 + 0.91
10.35 0.95
84.24 4: 2.95
85.35
1,76
89.40
4.35
95.57 + 1.41
Mix-MaxEnl
96.69 0.17
0.65 :: 0.11
0.94 0:21
81.16 1.48
12.61 +: 1.77
12.52 1.78
87.63 1: 0.67
85.85
0.83
94.39
0.72
96.31 0.49
Table 5: ResNet50 trained on C10. The cross-validated hyperparameters are provided in Appendix C.
we opted for the latter for fair comparison. For the KFAC-LLLA we leverage the official reposi- tory? (Hobbhahn et al., 2021) and the Backpack library (Dangel et al., 2020) for the computation of the Kromecker-Factorde Hessian. For the SNGP ResNet50 experiments, implementation. The official implementation for ResNet50 is specifically fine-tuned for ImageNet, and has not been used for experiments CIFAR. We could not make SNGP converge to SOTA accuracy values both on CIFAR-10 10 and CIFAR- 100. All the other methods were implemented by us in PyTorch and the training, cross-validation and evaluation code will be made publicly available upon acceptance of the paper.
tried running the official
D ADDITIONAL RESULTS
D.1 RESNET50 EXPERIMENTS
In Tables 5 and 6 report the experimental results for ResNet50.
D.2 CALIBRATION METRICS WITHOUT TEMPERATURE SCALING
For completeness, we report the calibration metrics over all the methods and considered datasets without temperature scaling (Guo et al., 2017) Table We can observe that temperature scaling
https: / thub. com/ 1921918211 0221911110 11111111111111111111111111111111 11__//__fff_________________
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Methods
Clean
Corrupted
CIFARIO
SVHN
Accurscy (1)
ECE (U)
AdaECE (
Accuracy (1)
ECE (1)
AdaECE
AUPR (7)
AUROC (1)
AUPR (1)
AUROC (1)
DNN
305 0.29
19.80
19.76
75.20
82.45
3.21
DNN-SN
79.27 0.25
5,15 0.12
3.13
50.55
0.47
12.16
79.20 0.17
75.22
0:13
80.78
1.08
88.87
DNN-SRN
78.96 + 0.42
2,98 + 0.24
2,95 +: 0.24
50,48 + 0.37
12.62 + 0.58
12.59 + 0.58
78.77 + 0.14
74.87 = 0.15
82.39
2.83
89.52
1.75
Deep Ensembles
82.09 0.33
3.15 0.10
2,98 0.19
53.91 0.37
12.53 0.31
12.36 0:31
81.93
0.28
77.65
0.34
85.08
1.60
91.49
0.88
KBAC-LLLA
1,30
1.19
0.56
10.37
79.30
75.27
3.84 82.80
Mixup
80.12 #: 0.28
7.49 d: 0.32
7,47 d: 0.35
53.96 d: 0.21
13:57 4: 0.38
13.52 à: 0.38
77.02 4: 0.41
74.40 =: 0.43
76.86
3,40
87.36 :: 1.62
Mix-MaxFnt
81.49 0.31
1.57 0:18
1:53 0.22
57.62
0.93
13.39
79.44 0:33
25,80
88.68
0.69
Table 6: ResNet50 trained on C100. The cross-validated hyperpaamettteeeeeeeeeeeteee are provided in Appendix C.
Methods
Clean
Corrupted
ECE
AdaECE (L)
ECE
AdaECE (L)
INN
3/16 +: 0.20
342 0.39
17.31 # 0.73
17.30 :: 0.73
DNN-SN
2,92 0.11
2900
17.41 1: 1.02
17.40 1.02
DNN-SRN
2.86
Deep Ensembles
2,12
2.10 +:
14.01 # 0.32
13.99 0.32
KFAC-LLLA
0.26
0.76 0:24
11,59 + 0,62
11.52 0.62
Mixup
2.44
Mix Ent (Ours)
1.4 1 0.15
1.42 4: 0.13
12.21 4: 1.44
12.12 1.84
C100 R50
DNN DNN-SN SN
9.51 + 0.58
9.47 + 0.60
25.18 + 1.46
25.17 + 1.47
9.47 + 0.44
9.44 + 0.46
25,06 + 0.73
25.04 +: 0.73
DNN-SRN
9.63 4: 0.3
9.59 023
25.51
0.69
25.50 0.69
Deep Ensembles
6.65 0.12
6.50 4: 0.04
19.80 4: 0.31
19.76 0.31
KFAC-LLLA
1.56 0.09 0.09
1/19 0:24
12.11 =: 1.12
12.18
1,12
Mixup
1.32
7,47 0.35
0.38
2.35
Mix-MaxEnt (Ours)
4.33 + 0.43
4.17 + 0.39
14.06 #: 1.81
14.03 1,80
C1O WRN
DNN
2,30
2.27 1: 0.11
15.94 0.66
15.92 0.66
DNN SN
2,25 0.12
2:21 0.13
15.55 4: 0.23
15.55 à: 0.23
DNN-SRN
0.12
223 0.13
0.M 15.13
15.11 0.44
Deep Ensembles
1.76 1: 0.02
1.74 1:
13.54 0.19
13.52 0.18
SNGPGood
1.62 0.09
1.51 0.01
11.36 1: 0.37
11.33 0.36
KFAC-LLLA
1,06 +: 0.0
1,12
11.69 0.76
11,67 0.76
Mixup
2.0
0.64
0.94
7.95+ 0.97
Mix Mix Ent (Ours)
0.92 1: 0.06
0.71 4: 0.13
8:93 à 0.70
8.87 1: 0.69
DNN
5.34 +
5.30 + 0.42
17:43 + 0.75
17.38 + 0.75
DNN-SN
5,15 0:255:255
4,97 1: 0.24
16.39 0.44
16.35 0.45
DNN-SRN
5.2
15.71
Deep Easembles
0.16
3.2 0.16
13.51 + 0.78
13.47 0.29
SNGPGood
5.68 0.26
5.65 + 0.28
10.97 + 1.52
10.89 1.52
KFAC-LLLA
2.20 0.31
2:30 0.32
8.97 0.21
8.99 0.21
Mixup
3.44 # 1.08
3.60 4: 0.99
16.53 4: 1.52
16.54 4: 1.49
Man (Ours)
2,66 +: 0.07
2/47 +: 0.00
10.54 +: 1.49
10.49 1.49
Table 7:
Calibration metrics for all networks and all datasets, without temperature scaling
leaves the ranking among methods mostly unchanged. Details about the cross-validation procedure used and the temperature T used are in Section C.
E COMPARING DIFFERENT UNCERTAINTY MEASURES
In Section 1.1.3, we reported the OOD detection performance using the DS score, except when damages the performance of a method (e.g. Mixup) or when it does not yield improvements (e.g. KFAC-LLLA), For completeness reasons, in Table we report the performance with respect some common uncertainty measures we defined in Section 5.1.3: (1) Maximum Probability Score (MPS), (2) Entropy (H), (3) Energy Score (cquivalent to Dempster-Shafer (DS).
Feature Space Density Estimation scores also use feature space density estimation based score as an estimate of uncertainty. Assuming that a Gaussian Mixture Model would estimate the density of the feature space accurately, the likelihood
Similar to (Winkens et al., 2020; Lee et al., 2018),
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Medrade
CIFAR
SVIIN
MPS
D6
MPS
AUTOOCAT
AUROCAA)
AUPR
(UROCCT)
R91
4:445
1.75
UNN
5176 9:935
1154. = 0.25
88.10-000 2-035
92.72
96.01 9.3444
00.35
130
MLL85 30 3.41
93 1233.0
2/11
04003 1118
Deop tzacedie
0032 = 0.17
91.38 5-031
98.15 D15
9007 1:0.5
10002
0.14
06.34
DUM
1E-27 = 300
93.41 W/3
07-47
Morp
(Dang)
18.23 8.36
88:33 ( 00.36
57A63 1.0.3
85.05 DUSS
18.30 0..90
16.02 0.00
90.30 0.555
90.35
1.55
14.74 0.72
M6.11 L. 2.49
W0.03 11.9
06.53 0.32
DNON
19-08 2:0.1
75.45 = 14.1
19:3 5-0.10
25.00 =2 DAL
10.18 4:033
75.11 922
15.38-5 2.222
70.36 0: 3.50
XLAA =: 3.31
20100 =
01.33 01.00-20000 2:75
09:35 =0:35:10 1,6
DNN-5R
18.30 9:113
76.20
0.15
18.71 2:014
14.7
16.00:00.05
14.3
80:00 2.66
50.50 206
X2.99 123
20.30 1.75
$1.38 2.35
99:43
Deep Maup
17.00 0:022
0
17/0 0.04
2440
1:033
75,00 0:29
10.04
76.24 8
7.40
8
74.88
18.17
(Dan)
73.2.22
19.44 5:033
75.90
80:33-5 2:0.3
00.05
1441
03.43
$3.00
C10 WIN
DNN-AN
13:20::0:23
05.02 =
18,36 0.3.
93.13 0.34
88.55 0.23
19.95:005
93
00.02 0 064
2: 0.60
07.83 3.22
13.34-22 01-42
17:41
FIVN
16.76 0:0.5
15.15
1.20
45.90
07.08
Enaenble
06966
0.12
91.88
0.12
01.25 0:034
91.13 1:15
61.19 1:311
0.15
96.40 0.55
96.43 B61
V7.53 1,60
16.34 0.3
96.97: 0604
08.40 6.27
S00P
00002 5:000
= 0099
10W1 5.0.0
09.39 0112
9055 2:0.0
00.25
54.17-00 0.300
06.17
0.30
05.25
855
01/66
0.18
54.53
0.25
07:33 0.11
(Dan)
50.27 5:44
09.27 0 0:45
15.35 5.0.0
55.40
53 5775
$9.43
1:030
10.35 0.6
19.30200052 0.52
15.39 B0G
97.23 8: 5.76
18077 8: 2.3
98.00
06.00 D446
DWN
50.68
64
81000 4-0
70 15
11:17 4:0
77.48
79:00
00-50-1-4 4:57
DX0-SBIN
1009 0.632
9199 1115
11:26 0:18
77.36 03368830 030
91.42 1.035
17.52 0.2
83300 0.7
05.30 88 BUT
10.51 1.16
11.96 81 1.12
94.30 1.01
01.15 1J0
ShEP
18-17 8.10
15:13
0.30
15.51 11
73.06
19:99 0.34
75.1 1.
13 16
77.06
1.76
06-76
11 1.00
05.90
106
24.01 1:40
00-05 1777
Moup
1:172
7642:200
1.36
75:39
11.31-21333 1.20
15.05 0.36
77.44-723
71.34
14.55-
10.13
75.59
4.22
(Oan)
91 018
7605 013
76.11 8:11
9 2:1
231
90.38 6: 2.40
95.15 6 1.0
99.82
06.29 1.55
Table 8: OOD detection performance considering Maximum Probability Score (MPS), Desmpter- Shafer/Energy (DS) and Entropy (H) as uncertainty metrics.
that a given sample x belongs to class c can be computed as:
ply = clx) exp(-(ó(x) 2e (o(x) - -He))
((2T) det E.
where, jr. and Ex are the empirical mean and the covariance-matrix of training samples of c-th class.
Using ply = c/ following scores can be computed to estimate the uncertainty:
Maximum Log- -ikelliioohoo:: m(x) max. log p(y c/x). If the score m(x) is low then the sample more likely belongs to OOD.
Belief: This score indicates how likely is it that the sample x belongs to IND and is computed as b(x) E.p clx).
We consider two variants of both the scores mentioned above: (1) tied- cied-covariance (LDA assumption) when all the covariance matrices are tied and there is one common covariance matrix for the entire training data-set; and (2) per-class covariance matrix (QDA assumption) when the covariance matrix is separately computed for each class. Note, in both the assumptions, the mean is always class-dependent.
Note, the scores defined above require inverting matrices, computing logarithms and exponentials. Practically, we observe that these computations introduce numerical instability leading to unreliable uncertainty scores (although a few cases they can achieve better performance). In particular, the main sources of instability are caused by the inversion of the covariance matrices and by the computation of their fog-determinants (hence we evaluate the scores also without considering the nog-ooomalizatinn term). We apply two known procedures make the process numerically stable:
Diagonal perturbation by addition of eI for the computation of the inverse, where is a small positive constant.
Low-Rank Pseudo-Inverse (LRP): preprocess the covariance matrices (see next), compute the precision as the pseudo-inverse of the pre-processed covariance (stabilised further by adding el).
'The causes of such instability can depend on the high dimensionality of the latent space (h- = 640 for WideResNer28-100 h = 2048 for ResNet50) and the few datapoints that can be leveraged for the estimation of the covariances.
21

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Under review as a conference paper at ICLR 2022
Figure 6: DNN (Left), Ours (Center), Mixup (Right). Interpolation experiment on CIFARIO to show embeddings of the linear interpolation of two randomly picked input samples from class (purple) and class 2 (yellow). Red and green samples are classified class 1, and orange and blue samples as class 2. As the color changes from red to green, the predictive entropy increases. Same for the color change from blue orange. Note, DNN classifies interpolated points with very high confidence (low entropy) even if the samples shift drastically from the data. However, Mix-MaxEnt's entropy increases as we go away from the data. Details of this experiment provided in Section 5.2. Observe Mixup scale is different from the one of
and DNN.
We preprocess the covariance matrices with the following procedure. Let's assume A is positive semi-definite matrix, and call Á an approximation of that exhibits better numerical stability to inversion and log-determinant operations. We obtain Á follows:
compute the SVD decomposition of the A as A USVT.
select the top-k SVD components of A that explain the 7% of the variance. Let's call Uk. Σg and V&8 the matrices obtained by considering only such components.
produce an approximation Å A as À = UV
As it can be seen from Tables 9 and 10, the metric can show outsandingly good performance in detecting OOD data in few cases. For instance, in Table 9, consider DNN-SRN tested on SVHN: with the IRP stabilisation and e 97.8 97.84% and an AUPR of 99.06%, which are both superior to the AUROC of 96.1 16%12% and AUPR of 98.10% 10% reported in Table with the DS metric for the same method. However, as clearly visible, the behaviour of these metrics is highly inconsistent across different methods, training and test datasets and stabilisation procedures.
- 08, 7 = 0.95 the the belief b, can achieve an AUROC of
For instance, let's consider some examples from Table 9. It is evident how the m, is extremely sensitive to the choice of E for the DNN method: the OOD performance on SVHN drops from 93.99% AUROC and 92.21% % AUPR to 73.35% AUROC and 65.90% AUPR when using 1e 08 instead of = 1e 07. On the other hand, for the same variation in e, the scores improve for DNN-SN on the same task (inereasing the AUROC from 98.34% to 98.40% and the AUPR from 97.05% to )..12%) Another example of such extreme sensitivity and inconsistency of the metric across various stabilisation techniques and methods can be observed when comparing the be. m. and m. performance using the diagonal perturbation with e 1e 07 or the LRP with same and 0.99 for DNN and DNN-SN when tested on SVHN. For DNN, all the scores increase when adding the LRP. For DNN-SN, the b, increases dramatically (from 76.89% AUROC to 95.35% AUROC, from 88.44% AUPR to 89% AUPR) but at the same time a significant drop observed in m. and m4 (c.g. m, drops from 98.34% AUROC to 94.99%, and from 97.05% AUPR to 92.95%).
The fact the considered stabilisation procedures can produce inconsistent improvements and degra- dations across various methodologies and metrics, and that such variations extremely sensitive to the choice of the stabilisation hyperparameters makes these metrics unreliable. Indeed, there is no stabilisation procedure with specific set of hyperparameters that dominates the others in all the cases, and there is no obvious way to blindly (i.e. without resorting to OOD data at training time) select such procedure and hyperparameters.
22

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Under review as a conference paper at ICLR 2022
Medood
Stablissaionn
AUROC (t)
AUPRI")
AUPRC")
AUROC
AUPR(T)
AUROC
AUPRIT)
IND: CIO, OOD SVMN
10-07
75.78
0.93
$0.30 =: 1.06
40,93 =: 6.32
70.61 6.34
90.51 a: 15.17
$4.75
: 25.13
90.99 :: 297.9
92.21 #: 11.05
1e-07 LRP +0.99
92.20
1.89
96.57 0.66
27.75
4: 4.06
60.10 2.58
94.13 4: 1.44
92.81
1.21
95.46 0.51
93.64 =: 0.69
1e-07 LRP -0.55
94.42
1.43
97.59 2: 0.58
45.57 4: 3.61
70.54
2.43
93.47 4: 1.25
91.82
1.25
93.01
2: 0.58
91.00
2: 0.93
16-07 I.PP -0.85
91.21
2.70
96.11
1:16
71.00
+ 6.71
70.60
56.25
m1 1.79
09.99
1.95
87.55 0.95
$6.00 0.8
DNN
1ox08
61.21
7.28
80:30
8.51
45.09 + 5.31
68.42
2.90
91.0 1.
6.21
$4.80
11.62
73 35
65.90
31.58
C-1e 08 LRP
-0.99
90.34
3.75
95.01
2.90
27:90 01 3.94
00.12
2.23
26.94
21 8.97
70.00
17.71
90.3
0.54
93.53 0.69
1e-00 IRPP? -0:55
94.04
1.45
97.07
0.54
45.99
01 4.26
70.52
2.56
78.95
27.54
76.41
29.15
95300 22
91.10
0.89
Tex08 LXPP 20.55
90.98
188
96.20
0.90
71.00
08 6.76
80.19
5.40
99.38
81 2.18
91:15 =2 2.59
87.57 0..9
36.00
0.67
76.89
12:75
$8.44
12.39
96.68
05:45
0.35
07:03
-ee007 -1/5M
13.22
1.24
( =1195
12.00
DNN
76.65
12.77
11.90
5.97
93.49
98.40
97.12
94.29 0.17
30.13
3.46
91.24
74.47
5.84
0.98
93.74 1.19
1o-88 LRPP -0.05
95.60
181
0.54 96.93 0.62
98.00
57.10 -0 9.67
76.74 6.14
70.59
0::0:::::: 23.63
52.27
25.00
91.32 + 2.50
56.07 5.19
1o-08 LRP -0.35
93.50
1.12
37.34 1 5.18
85.19 5.12
93.69
9: 0.10
87.67
4.19
74,43 14.76
58.76 28.95
C- 07
78.16 :: 15.07
19.11
5.23
57.36
20.24
77.96 =: 10.57
96.79 2: 0.67
95.61
0.59
98.24
0.37
96.96 =: 0.53
Te O7 LKP -0:99
94.24
4.77
97.17
2.58
33.06
16.60
62.09
7.51
94.76 3: 1.08
93.58
0.75
95.75
1.97
96. 10 2.00
Te-77 LRP -1:55
9h.
2.26
08.15
1:15
52.62
19:77
75.03
10:53
99.90 2: 0.7M
41.
0.59
95.15
21 2.32
91.90 2.05
1o-07 LRP -0.55
96.07
2.50
98.26
1.21
71.84
2: 8.46
80.62
5.50
93.66 3: 0.38
92.
0.91
$8.44
3.51
87.36 =2 2.95
INN SRN
<- 18.08
77.87
13.54
89.37
7.35
$7.64 18.29
78.49
9.72
56.64 1: 0.68
95.31
0.81
98.08
0.50
96.74
= 0.72
00
0.05
10
9.90
87
04-
-0.05
97.81
0.04
50.86 9.50
55.77 36.66
1.15
16-08 L.P -0.55
96
2.38
73.52 6.05
81.33 5.65
92.93
90.03
1.45
79.31 14.72
IND: OOD:
LXP -0.95
82.64 1.17
54.27
29.97
0.15
0.42
OT
83.42
0.67
63.92
50.03
90.27
0.70
87.35
DNN
71.17
6.84
60.44
76.45
74.04
14.00
1e-08 LRP -0.09
$1.32
5.47
50.70
0 5.54
46.52 0: 1.58
50.60
1.50
$2.37 2: 6.42
76.12
8.07
56.87
2 0.44
67.25 2: 0.40
16-08 LRP -0.95
82.54
1.39
$3.01
1.24
54.37 2.35
51.71 1.56
63.21
12.65
$1.74
15.96
56.37
2 0.48
87.74 =: 0.45
16-18 J.RP =0.88
81.11 0.8
K1.86 0.66
68.9
1 2.72
90.71 3.80
19.41
0 0.48
55. 40
1.00
56.1 10:00000000 0.70
N7. 08 = 0.52
< c:51e007 1e-07
77.67
1.25
77.11 1: 1.32
74.00 1: 0:80
16:16 0.63
91 19
8: 0.17
91.53
0.21
50:51 1:
50 23 0.34
1e-07 LRP -0.99
82.86
= 0.93
$3.95
= 0.57
50.25
=: 1.22
53.74
0.79
$9.85
21 0.09
90.03
0.13
56.77 21 0.26
87. 10 0.35
1e LRP 0: -0.55
81.96
114
$2.42
= 1.12
57.20 2: 2.77
59,09
2.94
50.08 11 0.14
90.24
0.19
55.21
0.48
87.25 = 0.35
-0.85
83.6
0.96
$4.35
1.13
58.90
90.60
DNN-SN
10-08
1.48
77.14
74.35
76.11
01.19
91.54
0.10 90.50
03.23
1o-08 -0.09
82.83
0.25
50.42
0.06
$6.35
70:31
2.19
0.16
-1195
57.01
71.8
I.RP)
1e-07
74,03
1.80
76.78
74.06 1.66
1.30
91:40
0.22
90.24
1e-07 LRP -0.99
83.49
0.25
48.97
2.65
59.68
89.75
0.30
0.37
87.31 0.30
Te-07 IMMP =0.95
82.01
0.8
K0.05
55.67
3.07
90004
3,67
50.11
1 0.31
501.8
0.55
16.00
8 0.52
87 99 0.41
1o477 IRP =0.35
83.86
0.71
$4.11
0.68
61 5.99
$7:09
5.04
90.45 9 0.15
90.75
0.16
16.78 0.00
83.88
DNN-SRN
=1006
75.63
2.11
77.33
0 2.15
74.86 + 2.70
76.78
2.41
91.10 :0 0.14
91.41
0.20
00.52
0 0.18
90 90 0.36
Ze 08 LXP -0.99
83.29
= 0.25
$4.08
0.57
48.11 :: 1.96
52.09
0.94
88.70
:2 0.40
36.37
2.85
$7.00 0.12
87.47 :: 0.22
0N LRP -0.55
10.11
2.9
M1.77
8 3.23
52.96
6 2.49
50.00
1.88
70,20
13.19
74.44 2: 15.77
55.99 0.44
X7. 00 :: 0.51
10008 LRP
-0.35
83.07
9.80
84.62
0.95
61.59
2 5.35
57.84
4.02
$9.96
2: 0.47
89.73 = 1.29
85.03 2.95
$4.44 :2 6.10
Table 9: OOD detection performance when using Feature Density-based estimation methods. The results are reported for WideResNee28-10, trained on CIFARIO. 10. Stabilisation legend: alone denotes only the addition of el, LRP denotes Low-Rank Pseudoinverse, ? is the fraction of the explained variance.
2.00
400
500
1.00
400
0.75
300
200
100
Figure 7: Left figure: (het)- map for Mixup (a = 0.3) trained using WedeResNer28-10 10 on C0.
23

---

Under review as a conference paper at ICLR 2022
Mathal
Stahilsation
(UROC(()
AUPRITS
AUROC
AUPRC)
AUROC
AUPP((1
AUROC
AUPRI11
IND: DOD: SVIN
1e-07
74.93 4.45
$4.06 +: 2.8%
67. 19 4.76
78.06
2.64
59.55 + 0:97
80.60 3.56
K3.59
9.38
25.00 - 6.2%
1--07 IRP
-0.09
74.13
4.40
$4.09
+ 2.36
68.71 + 2.62
70.00
1.61
19.43 + 5-45
58
10.00
83:28 m 3.16
23.06 = =5200
1o-07 LRP
-0.05
71
4.28
52.73 + 2.46
72.53 9.67
83.23
2.54
64.59 + 5.39
41.18
6.91
81.16
4.13
71.38
=5002
07 LKP
-0.55
71.94
5.91
85.18
2: 3.35
77.36
21 3.46
85.19
2.18
03.04
18.10
41:83 = 17.87
79.45
3.90
09:01
=: 6.7.5
UNN
Te-06
78.97
8.48
14.10
2:46
06. 73 4.67
78.07
2.50
39.0%
0.96
W0.64
3.21
83.78
1:20
75.33 6.25
85.50
64.57
77.23
81:17
67.49
1e-08
84:22
7.55
$6.16
78.33
63.63
15.70
Te
53.49
77.39
1.52
56.06
78.00
77.97
05.06
27.06
78.73
67.24
9.70
69.75
88.33
89.00
56,05
13.94
09
DNN-SN
Ic-06
76:43 7.10
$7.59
70.61 01 9.09
82.51
5.27
$6.84 =: 3.85
80.14
4.25
15.67 8 3.12
77.34
1c-08 LRF
-0.00
$0.94
6.24
$8.95 6 4.01
70.13 12 8.56
80.54 4.00
$2.05
s: 5.01
75.80
5.56
68.23
16.51
51.63 22
1e-08 LRP -0.05
79.52
6.56
$8.36 2:
77.62 6.65
86.46 4.64
$3.20
22: 4.37
75.69
5.05
76.18 5.85
66,05 =: 8,43
1e-16 IMP -0.45
05 6:87
$8:49
11 4.17
81.01 11 5.92
KN 3.91
$2.03:2
1 5.99
73.99
7.56
5.58 75
64.94 :1 7.76
1c-07
$3.40
3.27
50.25
75.35
4.21
84.67
2.74
$5.47 +: 2.90
75.86
5.14
65.54
2.58
27.61 =:
1e-07 LRF 00.00
83.79
2.12
$0.52
2: 1.72
77.51
=: 1.62
85.27
1.34
74.79
10.65
57.64
17:23
78.36
1.72
71.90 1.75
07 LRP > -0.55
83.62 2.49
50.02 0: 2.01
81:47 2: 2.59
89.56
1.94
65.29
14.76
47.26
18:71
17.10
1.37
70.25 1.69
1o47
64:33
1.76
DNN-SRN
1o-08
25.06 4.12
2.63
76.02
23.59
82.66
3933 0:35
72.14
0.9
75:27
41.96
27:00
IND. OOD: C10
0.30
72.63
76.66
73.63 0.11
74,60
78.22
0.13
79.08
79.13
07
0.41
74.56
23.99
0.53
02:37
3.87
27.02
1o-07 LRP -0.05
77.87
1.22
13.08 6: 1.36
13.35 6: 0.78
70.68 0.19
59.26 0: 4.39
57.64
5.76
78.76 2 0.16
79.77 = 0.22
17 LTT 0- .50
76.03
0.52
14.42 d: 0.58
74.47 1: 0.46
71.63 0.32
68.85
12 4.10
65.08
5.02
70.45 0.48
80.31 2: 0.455
DNN
1c:08
15.90
0.32
12.65
2: 0.08
76.70 2: 0.45
73.66 0.12
74.51 1: 0.10
75.14
0.13
70.62
0.17
79.04 2 0.34
e48 TRP =0.0 10
57:41
061
11:00
11 0.49
70.75 1.22
66.00 7.57
68.85
+ 2.35
69
4.43
57 01
10.05
57.55 7.90
1o-08 LRP -0055
77:51
0.66
73.87
+: 0.70
62.18 + 10.6
58.93
10:08
13.25
+ 0.554
76.76
0.76
65:04
9.00
65.91
8.86
le LKP -0.55
77:41
0.75
75.72
3: 0.56
75.52
2: 1.70
70.37
1.21
75.90
11 0.75
76.95
1.35
77.26
1,70
77.73 =
<- 17
15.50
1.04
72.32 3: 0.91
26.24 =: 0.39
73.16
0.84
73.02 3: 0.55
70.98
0.41
79.43 0.27
79.38 = 0.53
1o07
1.75
9.41
77.09 1.07
1o-7
74.12
1.82
72:47
1.30
57.16
5.65
78.10
1o07
0.61
74.86
74.47 1.36
1.57
64.22
74.11
71.01
76,62
0.69
0.67
78.65
74.59
73.82
74.34
2.95
79.89 1.15
1o-07
16 0.54
12:96 0.81
77.18 t 0.58
71.78 0.99
12:56 :: 0:37
36.50
0.20
70.41 0:207
70 14 0.57
1o407 1RP -0.09
68
0.40
15.10-1 0:45
14.75 : 0.95
71.45
0.68
67.65 +: 3:36
63.10
3.75
74.32 0.42
77.03 = 0.68
O7 LKP : -095
19.14
0.52
75.45
2: 0.58
73.60 2: 0.87
70.95
0.53
64.27 2: 3.92
60.71
5.34
78.54 2: 0.47
80.21 =: 0.60
07 LKP -0.5
79.24
0.41
75.52
0.49
75.42 3 0.56
72.23
0.42
50.35
10.11
56.41
7.85
79.78 22 0.54
30.90 2:00::000 0.57
SKN-
Te-08
26.00:1 0335
72.4
8 0.35
71.20 2 1.56
73.81
0.54
71.84 2: 0.35
75 94
0.29
79.36 0.27
79 0.51
1e-06 I.R -0.00
78.56
0.54
74.6
2 0.62
69.75 2.67
64.10
4.41
11.89
21 0.32
75.53
0.45
52.72
13.14
56.92 12:20
le-06 LRP -095
78.72
0.65
75.15
2: 0.92
69.00
2: 1.69
64.39
1.95
72.71
4 0.35
76.26
0.36
52.74
3.53
54.01
=: 1
1c-08 LRP -085
78.28
0.34
75.19
1.00
75.21
0.46
0.83
72.36
96.19
0.54
75,00
73.9
Table 10: OOD detection performance when using Feature bansi estimation methods. The results are reported for WideResNet18-100, 10, trained on CIFARIO0, Stabilisation legend: alone denotes only the addition of cI, LRP denotes Lank Pseudoinverse, y is the fraction of the explained variance.
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Methods
Accuracy (t) ECE
AdaECE ())
C-Acuurcyy
CECE ())
C-AdaECE (L)
CIFAR AUROC (T)
SVEN AUROC (T)
AUPR (1)
AUPR (T)
C10
m0.1
96.36
2,00
2.02
80.00
13.01
12.99
82.70
85.32
88.57
=02
96.54
1.01
1.48
80.41
11.58
11.56
80.39
$3.25
91.42
96.83
-.33
96.92
0.70
81,91
10.11
77.67
30.37
82.81
92.95
79:17
-0.5
97. 14
11866
1.14
82,40
10:77
10.80
78.52
81.35
78.57
90.80
0.6
97.25
0.87
0.91
82.57
9.60
9.64
76.29
78.95
$1.40
91.91
=007
97.10
0.99
82.69
9.15
75.00
75.81
65,30
81.58
À (0.25.0.770 0.76]
97.49
0.69
0.56
85.18
7.44
7.38
87.50
36.09
94.09
96.
X (0.0.25] J0.75,
96.61
1.08
1.49
77.62
14.14
14.24
72.88
75.30
74,55
$8.41
=0.1
N2. 7M
4.75
4.68
$6,00
16.14
16.11
80.00
76.49
M0.04
XX. 36
=02
82.82
5.17
$6.55
17.85
17.78
79.30
76.02
81.52
90.01
28
83.72
4.00
57.87
=05
83.11
3.09
3.10
57.85
15.72
15.68
76.79
74.17
85.11
92.00
06
83.31
3,86
3.64
57,92
16.54
16.49
76.81
74.12
85.05
42.00
a
83.38
3.92
3.80
58.56
15.51
77.58
74.92
83.97
91.28
A (0:25.0.
83.26
4.23
4.14
61.01
12.05
12.05
80.25
76.63
91.82
95.97
A (0.0.25 (0.75,
81.46
5.44
52.84
20.88
20.84
76.31
74,80
68.34
82.50
Table 11: WideResNet28-10 Mix-MaxEn with low a or controlled interpolation factor
F ADDITIONAL VISUALIZATION OF THE MAXIMUM ENTROPY EFFECT OF MIX-MAXENT
In Figure 7 we report plots similar the ones in Section 5.2.2. As already explained, Mixup makes the network less confident both close and far away from the training data, for this reason it results in better calibration (as it alleviates the overconfidence DNN) but worse OOD detection performance (because both IND and OOD samples have higher entropy, hence are more difficult distinguish).
An alternative visualization of the phenomenon can be observed with the following method. We take samples from the first three classes of C10, embed them into the feature space. and visualize them using the approach used in (Müller et al., 2019) (presented in Appendix H for completeness). different classes ('purple' and 'yellow") and create set of interpolated samples {x, = Ax + (1 A)x² ). as varies from 0 to 1 with an interval of 0.005. Projecting these samples into the embedding space (Figure 6) shows that DNN projects most of the samples to either class 1 or class 2, and very few to the region of high uncertainty. However, Mix-MaxEnt projects most the interpolated points with À close 0.5 in high uncertainty regions. Therefore, this plot qualitatively verifies that Mix-MaxEng creates high entropy barrier between classes.
Then, randomly select two samples X₃ and x2 from
G RESULTS OF ABLATIONS IN SECTION 5.2.3
In this Section, we report the results that could not be included in Section 5.2.3 for reason of space. In particular, we report Tables 14, 13, 11 and 12.
We also discuss why with interpolating IND embeddings from different class clusters and introducing maximum entropy regularizer on such samples. However, we observed this approach to be ineffective. A reason could be that neural classifiers are notoriously not bijective and interpolations in the embedding space do not necessarily correspond to inputs that even partially resemble data coming from neither Xr nor Xo (restricting Xo to the manifold of natural images). The interpolated images, instead, bound to contain features from both classes, and hence is reasonable for the network to unsure about which of the two classes is represented in them.
interpolates input images and not their features. We did experiment
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Accuracy (1) (CR(U)
AdnECR
C-Acouracy
C-ECE
C-AdaRCE
AUROCAT)
AUPR (*)
AUROC
AUPR (t)
C10 WRN
=15
95.88
3.15
2.79
25.13
15.50
15.32
85.88
81.51
87.17
89.97
=20
95.90
3.26
2.99
73.81
15.91
15.74
85.51
81.60
91.38
91.90
=30
95.99
2.79
2.05
73.59
16.20
15.9
86.18
82,96
92.55
93-47
C100 WRN
=15
80.57
7.23
7.02
50.08
18.21
18.18
76.68
69.38
72.96
80.91
16.91
Table 12: WideResNet28-10, loss: CE v) - -H (po) where Hpo)) is the entropy over all the labels.
Methods
Accuracy (T)
ECE
AdaECE
C-Acurrcy (1)
CEECE (1)
C-AdaECE (L)
CIFAR AUROC (t)
SVHN AUROC
AUPP (1)
AUPR (1)
-5
97.55
8.90
0.72
83.05
9.58
9.53
87.96
$6.40
94,31
96.93
10
97.19
1.16
1.07
84.29
7.35
7.28
899
89. 26
96.58
98. 38
83.79
8.100
7.94
89.
0 =20
97.31
0.84
83.41
9.89
9.84
88.
96,30
0 =30
97.46
1.12
0.85
85.65
7.50
7.85
89
88. 58.98
96.89
96. 55
-5-
97.26
0.78
0.72
84.66
8.50
8.43
87
85.99
94.00
96.72 74
0= =155, 0.9
0.59
10.61
20.r 0.9
97.33
1.04
0.97
83.70
8.60
8.64
89: 99.53
89.37
97.13
98 62
=00, 0.9
97.44
0.84
0.87
84,45
8.66
8.61
90,
89.51
97.81
98.97
5
83.15
3.76
3.69
58.83
12.98
12.96
80.68
76.86
85.68
92.63
=10
83.22
3.85
60.55
12.24
12.21
81.02
77.63
90.69
95.08
6 20
82.92
4.28
4.13
60.30
12.80
12:77
82.36
78.47
89.48
94.25
=30
82.92
4.70
4,54
60.09
12.62
12.58
81.46
31.71
90.42
95.04
-5,
2.51
61.46
8.63
8.57
81.07
77.39
88.15
93.17
15,7 0.9
84.11
2.45
2.30
61.12
9.40
9.35
81.74
78.35
89.53
94.77
20,7 0.9
83.81
2.55
2.23
60.77
8.95
8,88
81.55
77.81
90.97
98 42
30,7 0.9
83.82
61.28
9.00
8.94
80.49
77.04
88.02
93.30
Table 13: WideResNer28-10, Mix-MaxEnt (+SRN) mixing samples both within and between classes
Methods
Accurscy (1)
ECE (1)
AdaECE ())
C-Acuracy (t)
C-ECE
C-AdaECE (1)
CIFAR AUROC (T) AUPR (*)
SVHN AUPR
AUROC (1)
C10
=0.1
96.49
1.88
1.89
78.45
13:57
13.55
89.39
89.60
96.87
9W.43
=02
96.21
2.08
2.05
77 63
14.38
14.36
89.76
89.72
96.51
9%.11
1.67
1.63
77.91
13.41
13.39
90.47
90.38
96.26
=04
1.74
78
13.12
90.07
a =15
96.59
1.78
1.76
35.85
16.60
16.58
89.15
89.
95.5 25.96
98.13
=20
96.51
1.82
1.82
77.33
14,55
14,53
89.64
89.
91.55
97.37
33
86
C100 WRN
=0.3
82.01
5.58
5.48
53.11
19.72
19.68
$1.04
77.20
84.51
91.43
o 15
$2.92
4.79
52.13
18.01
17.97
80.68
76.98
87.11
92.89
$2.06
5.02
4.89
51.97
16.96
16.91
81.85
78.40
82. 17
90.41
=00
81.73
5.51
5.37
52.65
19.96
19.93
82.05
78. 29.29
79.10
86.11
Table 14: WideResNer28-10, Mix-MaxEnt with only within-class mixing. The missing a hyperpa-
rameters still training.
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H ALGORITHMS
In this section, we report the algorithm box for single iteration of the training procedure of Mix- MaxEnt (Algorithm 1) and the visualization technique used to project the embeddings some the figures of the paper (Müller et al., 2019) (Algorithm 2)
Mix-MaxEnt iteration Given batch B, a Ag is sampled from Beta a)) and for each sample (x1, 31) our algorithm randomly selects another sample (x2, y2) B such that y1 * 1/2 and adds this sample to batch B. Both batches are passed through the network (with a single forward pass, with batch size 2/ B)). The batch B is passed through the usual cross-entropy loss, the batch B is passed through our regularizer. The two terms are summed, and one backpropagation pass is performed.
Visualization Projection Procedure Given three classes 1,2, and their prototypes W1, W2. W3 (e.g. the means of the embeddings of the points of each class), first find an orthonormal basis of the plane passing through the three prototypes by using the standard Gram-Schmidt procedure. Build projection matrix P R by using the vectors of the orthogonal basis (where h is the dimensionality of the embeddings). Use the projection matrix to project any embedding ((x) R to Px R
Algorithm 1: Iteration of the Mix-Max Mnx training procedure on a batch B using the network f
B 0:
An Beta(a, at) foreach (x1, V1) E B do
Randomly select (x2, 12) E Bly1 # 1/2
B B (Aux1 + (1 - X2, duyi (1 )o)82):
end
Impl emented with a single forward pass with double the batch size
ply/x x E B; 0) + f(B):
p E B:0) + f(B):
-= log ply E B)0) Hr.11(1x E B:0));
Update backpropagating the gradient of L:
Algorithm 2: Computing the projection of point ((x) given three class prototypes Wi, W2, W3
Result: Px E 2 projection of ((x) R2
/* Find an orthonormal basis of the plane passing through the three prototypes
Pi < (W2 w1)/|/ will:
P2 W3 W2:
P2 t P2 -ppTp))p:
P2 pe/l|pall:
/* Compute the projection matrix
P + [prpp]":
/x Project the embeddings in 2D
Px = PØ(x);
I EMBEDDING SPACE RELIABILITY PLOTS
A typical technique used analyse the calibration of a model is inspect the reliability plot (Guo et al., 2017), i.e. a histogram that shows the relationship between accuracy and confidence. Drawing inspiration from the visualization technique of (Müller al., 2019) (that we described in Algorithm 2). we design new visualization technique to analyse the calibration of a neural network. Indeed, one of the main disadvantages of the reliability plots is they do not show which the regions of the
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embedding space in which the model is overconfidnt//mmeercoffddda The visualization technique we develop, that we call Embedding Space Reliability Plots, leverages Algorithm to compute the 2D projections of the embeddings of test set. For each test point x, we compute the corresponding accuracy (xx and confidence (x)), and for each of them we compute the quantity Axx) - (x). We associate to each projected point the corresponding difference between the accuracy and confidence, and plot a histogram of this difference. The histogram will reveal in which regions of the latent space covered by the test the model is overconfident (A(x) C(x) << 0) underconfident (A(x) (x) >> 0) or well calibrated (A(x) C(x) = 0). Interesting insights can be extracted from these plots.
We report a few visualizations of this kind of plots in Figure 8. We apply it to the test points of CIFAR-I0-C for some kinds of corruptions. In the figures, we compare DNN versus our method. The red regions of the space represent areas of overconfidence (A(x) C(x) < 0), while green regions of the space represent areas of undereonfidence (A(x) - C(x) > 0). Yellow areas can be either calibrated or contain no data-point. From these plots we observe that DNN has wider over-confidence regions in the space between the three classes, and shows to be underconfident in some of the regions close to the means of the class clusters. Our method alleviates this issue, making the network overall more calibrated both close and far away from the cluster centers.
We believe this to be valuable analysis tool that can be further leveraged by other researchers to better understand the calibration behaviour of their models and to design new algorithms.
J DETAILED ANALYSIS OF THE CORRUPTION EXPERIMENTS
In this section, we report detailed plots that break down the aggregated metric reported Table 2 for CIFAR-10- and WideRes.Ne12- 10. Specifically., in Figures 9, 10 and 11 we show how the Accuracy, ECE and AdaECE vary across all the corruption types and intensity values (horizontal axis) over seeds. As can be seen, in most cases our method achieves better Accuracy than any other method. In terms of calibration, our method is sometimes outperformed by Mixup, but we observe that Mixup exhibits lower Accuracy whenever this happens. As the intensity of the corruption increases, all the metrics deteriorate for all the methods (as expected). However, our method still achieves superior performance in most of the cases also in these circumstances.
To support our claim that better clustering behaviour induces better classification performance, we report the Fisher criterion, ||S | and || (we defined Section 5.2.1) plots for all corruptions and all intensity levels in Figures 12, 13 and 14.
The observed pattern similar for all considered architectures and corrupted datasets, hence we report these plots only for WideResNer28-10 10 on CIFAR- 10-C.
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10.0
-25
-2.5
-5.0
-500
-2
-7.5
-7.5
-4
-10.0
-4
100.0
Elastic corruption, Mix-MaxEnt
Elastic corruption, DNN
10
5
2-
0-
-
-
-2
-2
-10
-10
-4
-4
-2
-4
Fog corruption, Mix-MaxEnt
Fog corruption, DNN
Figure 8: Embedding Space Reliability Plots: a simple way to visualize the confidenee/vvercomfideeee regions in the latent space for specific test set. Red points represent histogram bins which the network is overconfident. Green points represent histogram bins on which the network is underconfident. Yellow histogram bins can either represent well-calibrated histogram bins areas of the projection space that contain no embedding. It is clearly visible that Mix-Maxe MaxEnt produces better-calibrated predictions by reducing the overconfidence far away from the centers of the clusters and by increasing the confidence close to the cluster centers. On the other hand, DNN shows large regions.
and sharp overconfideneenndecconfiii
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Elastic Transform
0.950
0.14
0.14
0.12
DNN-SN
0.12
0.900
Accuracy
0.10
Ours
0.10
Maxu
Ours
0.500
DNN
8
0.00
0.08
SNGP
0.06
0.06
DNN
0.000
ONN-SRN
0.04
0.775
0.02
Intensity
Intensity
Intensity
Gaussian Noise
0.9
DNN
0.5
Accuracy
0.6
0.7
Mixup
Ours
0.4
AdaECE
0.4
0.5
CRN
DNN SNGP
SNGP
02
DNN-SN
0.2
DNN-SN
SWN
0.3
Mixup
Mxxp
Ours
Corr. Intensity
Corr. Intersity
Corr. Intensity
Shot Noise
0.8
0.4
DNN-SN
DNN-SAN
DNIN-SN DNIN- SS-NSNN
Mixup
Maxup
Accuracy
0.3
Ours
AdNECE
0.3
Ours
0.6
SNGP
0.4
Maxp
DAN-SRN
0.1
Ours
0.0
2
Corr. Intensity
Corr. Intersity
Corr. Intensity
Impulse Noise
0.9
DNN
DNN
DNN
SNGP
SNGP
SNCP
0.8
DNN-SN
DNN-SN SN
DNN- SN
DNN-SRN
04
DNN-SRN
0.4
DNN-SRN
0.7
Mixup
Accuracy
AdsECE
0.6
0.5
0.2
0.2
0.4
0.3
1
2
3
5
Corr. Intensity
Corr. Intensity
Corr. Intensity
Defocus Blur
DNN
DNN
SNGP
SNGP
0.9
0:25
DNN-S
0.25
0.20
Mixp
Accuracy
0.8
Ours
AdaECE
0.15
DNN
SNGP
0.10
0.10
SRN
0.6
0.00
0.00
Corr. Intensity
Intensity
Corr. Intensity
Figure 9: (Part of 3) Accuracy, ECE and AdaECE for all corruptions and intensity values of CIFAR-10-C 10-C, architecture WideRessNe188-10, A similar pattern can be observed in all other cases.
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Glass Blur
0.85
DAN
0.30
0.30
0.75
DAN SRN
Accuraty
0.65
Maxp
Ours
0.20
DRNN-SRM
DNN
SNGP
DRN--S
0.20
DNN--SRN
DNNN-SN
DNN
SNGP
015
Mixup
0.55
0.05
Corr. intensity
Corr. Intensity
Corr. intensity
Motion Blur
0.175
DNN.
0.175
DNN
DNN SRN
SRN
Maxu
0.150
Mixup
Accuracy
0.125
Curs
0.125
Clurs
0.100
0.100
0.075
0.075
0.050
0.050
Ours
0.025
Corr. intensity
Corr. Intensity
Corr. Intensity
Zoom Blur
0.25
0.25
0.20
0.20
0.85
Maxp
SAN
Maxu
Accurtty
0.80
0.15
Ours
AdaECE
0.15
Ours
0.75
0.05
Corr. intensity
Corr. Intensity
Corr. Intensity
Snow
0.96
0.12
0.12
UNN-SRN
SAN
SRN
Accurtty
0.90
Mixup
Ours
0.08
Maup
Ours
AMAECE
Ours
0.58
0.06
0.06
0.82
0.02
0.80
Corr. intensity
Corr. Intensity
Corr. Intensity
Frost
0225
0.95
0.200
0.200
0.90
0.775
SAN
0.175
SRN
0.150
Maxp
Mixup
Accuraty
0.85
0.125
Ours
0.125
Ours
0.100
0.100
0.075
0.050
0.050
Ours
0.25
0.025
Corr. intensity
Corr. Intensity
Corr. Intensity
Figure 10: (Part 2 of 3) Accuracy, ECE and AdaECE for all corruptions and intensity values of CIFAR-10- 10-C, architecture WideRessNe28-10.
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Fog
0.175
0.175
0.95
0.150
DNN.
0.150
DNN
DNN: SRN
SRN
0.90
Mxxp
0.125
Maxup
Accuracy
Ours
AdaECE
0.100
0.100
0.075
0.050
0.025
0.025
Ours
Corr. intensity
Corr. Intensity
Corr. intensity
Brightness
DNN
0.05
DNN SN DNN- SRN
DNN-SH
DNN: SAN
0.96
0.04
Mixup
0.04
Mixup
Accuracy
Ours
AdnECE
Ours
0.95
0.03
SRN-
0.93
Ours
Corr. intensity
Corr. Intensity
Corr. Intensity
Contrast
DNN SN
DNN
DNN SRN
DNN SRN
Mixup
Maxup
Accuracy
Durs
AdvECE
Ours
0.7
0.6
0.5
Maxp
DAN SRN
Ourr
0.0
Corr. Intensity
Corr. Intersity
Corr. Intensity
Pixelate
DNN
DNN SNCP
0.35
DNN-SN SN
SNGP
0.35
DWN-SN
0:30
DNN-SRN
DNN-SRN SI-SSNN
0.6
Mixup
Accuracy
0.25
0.25
0.7
0.20
0.20
SNGP
0.6
DWN SRN
DNN--55
0.10
0:10
Mixup
0.00
0.00
Corr. Intensity
Intensity
Corr. Intensity
Jpeg Compression
0.900
CMN
SNOP
0.16
DNN
SNGP
DNN
SNGP
CNN--N
DNN--3N
DRN
DNN-5RN
DNN-SRN
DRNN -30-330
Actracy
0.800
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Figure 11: (Part 3 of 3) Accuracy, ECE and AdaECE for corruptions and intensity values of CIFAR-10-C, architecture Ne12
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Figure 12: (Part I of 3) Fisher criterion, ||Sw and ||S&|[e for all corruptions and intensity values of 10-C,, architecture WideResNer28-110 10. A similar pattern can be observed in all other cases.
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Figure 13: (Part 2 of 3) Fisher criterion, ||Sw || and ||SB|le for all corruptions and intensity values of CIFAR-10-C, architecture Wed8ResNee28-10. 10.
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Figure 14: (Part 3 of 3) Fisher criterion, ||Sw |I and ||SB|le for all corruptions and intensity values of 10-C, architecture Ne18
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Reviews:
[{"review_id": "x9Szz6Je7i", "reviewer": "Reviewer_R5hm", "summary": "", "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": "My impression is that the proposed penalty is a slight variation over mixup, which is the major concern I have about the novelty of this paper. The authors explained the differences on page 5, but I found this paragraph hard to phrase and somewhat ambiguous. Below, I entail the comments I have after reading the paper. \n\nAnalysis and observations: Figure 2 (right) displays the histograms of the distances of the embeddings that correspond to CIFAR10 (for which the model was fitted) and CIFAR100. As illustrated, there is an overlap between the two histograms, where the one that corresponds to CIFAR100 is slightly shifted to the left. Now, I do not understand the intuition that the model “will produce stronger activations if the similarity between the features learned on the IND data and the ones present in the input is high.” This should be better explained, including a demonstration. Also, a table that summarizes the statistics of interest (evaluated on the entire test data) would be informative.\n\nI believe that Figure 3 better illustrates this point. However, following Figure 7, mixup has an effect that is similar to the one presented in the right panel of Figure 3, although the “high entropy barrier” obtained by mixup is smoother compared to Min-MaxEnt.\n\nWhile the data sets used are commonly used in this literature, the distribution shift considered is due to various corruptions in the input image, and therefore reflects the robustness of the fitted model to such deviations. I would suggest studying other types of domain-shifts, such as the one proposed in [1].\n \n[1] Recht, B., Roelofs, R., Schmidt, L., Shankar, V.. (2019). Do ImageNet Classifiers Generalize to ImageNet? Proceedings of the 36th International Conference on Machine Learning.\n\nAlso, the OOD experiments are fairly easy, since CIFAR10 and SVHN are very different data sets. It will be great to explore other cases for which detecting OOD samples is more challenging. For example, using synthetic data can be helpful and informative [2]. In addition, it will be illuminating to compare the proposed method to existing techniques, such as [2].\n\n[2] Liu, J. Z., Lin, Z., Padhy, S., Tran, D., Bedrax-Weiss, T., & Lakshminarayanan, B. (2020). Simple and principled uncertainty estimation with deterministic deep learning via distance awareness. arXiv preprint arXiv:2006.10108.", "summary_of_the_paper": "This paper presents a new penalty function that can be augmented to the common cross-entropy loss, which was shown to improve the accuracy and calibration of deep net classifiers, applied to in-distribution data. This new loss also encourages the model to generate better estimates under domain-shift and better identify out-of-distribution samples.\n\nThe proposed regularizer is tightly connected to mixup. The latter suggests a data augmentation routine that creates new semi-synthetic samples by interpolating two images and their (possibly different) labels. In this paper, the idea is to interpolate between two images from different classes and then maximize the entropy on these samples during training. In a series of experiments, the authors show the advantage of this method over baseline approaches.\n", "summary_of_the_review": "The paper offers an interesting approach to improve robustness to domain-shift and better detect OOD data. The major concern I have is about the novelty of the proposal and its similarity to mixup, especially because the latter was already studied in this context (calibration, OOD, and adversarial robustness). Here, I believe that the authors should do a better job in distinguishing between the two methods (and other existing extensions of mixup).", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "rQomHFPDYkR", "reviewer": "Reviewer_qdpu", "summary": "", "questions": "", "limitations": "", "rating": 6, "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:\n- The paper is clear and easy to understand.\n- The proposed technique is simple and can be easily integrated into other NN architecture. \n- The performance of the proposed model is competitive compared to the baselines.\n- The author conducts experiments on multiple use-cases with multiple datasets.\n\nWeakness:\nI have a few comments, concerns, and questions regarding the paper:\n1) The authors start the paper with an analysis of some of the observations they found on CIFAR and SVHN data. How much of these observations are applicable to other datasets?\n2) The authors made an observation that the OOD and data-shifted samples are mapped by NN to the hypersphere S. Since we are working on high dimensional feature space, a hypersphere is likely to cover more volumes than the area where the data lies. Therefore, this observation should not come as a surprise, shouldn't it?\n3) The technique focuses on developing proxies for OOD samples that lie between different clusters of data. However, most of the OOD samples should lie in the area that is not in between two clusters (at least based on the areas). Has the author considered this case? For example, create experiments on this type of OOD and see if the proposed approach could cope with it.\n4) The closest previous work to this paper is the DUQ paper. However, the author did not properly compare the model with the DUQ (with code base as the reason). All the DUQ numbers mentioned in the paper are based on directly taking the numbers from the DUQ paper. This may cause problems in assessing the fairness of the comparisons, since the setup in the DUQ paper may be a little different from the experiment in this paper.\n5) The comparison of several metrics for evaluating OOD in section 5.1.3 is a bit confusing. What \"metric\" is used to evaluate the performance of the metrics so the author can say that one is better than the others?\n6) As presented by the author, the methods create a high-entropy barrier between two class labels. What if in the classification problems, the transition from one label to another is more or less smooth. For example, in the case of digits 1 and 7 in handwriting recognition. Is applying the proposed approach counter-productive in these cases?\n7) Presentation. The tables are too small to clearly see. Some of the results in the main paper refer to a table in the appendix, which makes it a bit inconvenient to read.", "summary_of_the_paper": "The author proposed a new method for improving uncertainty estimation in deep neural network output. The idea is similar to the mixup paper, but rather than augmenting the data by making a new sample interpolated from two samples having the same label; it creates a new sample that interpolates two samples with different labels. The hope is that these new samples serve as proxies to out-of-distribution data. A new loss function is then proposed by the author that augments the standard cross-entropy loss with an entropy regularization term that encourages the prediction of the new augmented samples to be as uncertain as possible (maximizing the entropy of the prediction). The author then performs experiments to demonstrate the effectiveness of the proposed approach.", "summary_of_the_review": "The paper proposes an interesting approach in uncertainty estimation for deep neural networks. The paper has some weaknesses that I hope the author can fix it soon. For now, I recommend a weak accept.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "WEwrv3Htf_D", "reviewer": "Reviewer_L8Yz", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "The main advantage of this approach is simpleness. The proposed algorithm can be easily used to modify the training. My main concern is the significance of the empirical result.\n\n1. Seems in the compared baselines, mixup tends to be the strongest one, which is quite surprising as mixup is not an algorithm specially designed for robustness and uncertainty. I am not an expert in this area and I am concerned about whether the current SOTA approaches are compared.\n\n2. Cifar-100 is still a kind of a small dataset. For this empirical paper, I would recommend conducting larger-scale experiment on Imagenet with some more challenging natural distribution shifts.\n\nMinor:\n1. What if we mix images from 3 or even more classes? Why do we need to use beta distribution other than others to decide the weight?\n \n\n", "summary_of_the_paper": "This paper proposes a simple approach called mix-maxent to improve the robustness of neural networks. The main approach is to regularize the network to make an uncertain prediction when the data is a interploation of two images with different classes. Experiment shows the good empirical results. ", "summary_of_the_review": "Overall, the main advantage is the simpleness but my concern on the significance of the result making me feel this paper is slightly under the acceptance bar.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "9TBGeioOLQa", "reviewer": "Reviewer_ESue", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "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": "Increasing entropy to improve OOD detection performance is not very novel. For example, the Agnostophobia [1] and Outlier Exposure [2] approaches start from a similar idea. The advantage of the proposed method is the fact that it avoids collecting additional data as required by the Agnostophobia and Outlier Exposure approaches.\n\nHowever, the Mixup [3] approach also uses data augmentation to improve robustness. Similar to the proposed approach, it also does not require collecting extra data. Despite both approaches being based on data augmentation, the authors present some differences between the proposed approach and Mixup. We partially agree with the authors in this case. This is not our main concern regarding the paper.\n\nThe IsoMax loss [4,5,6] and the Enhanced IsoMax (IsoMax+) loss [7] significantly also uses the strategy of improving the OOD detection performance by maximizing the entropy with the advantage of avoiding data augmentation and also massive hyperparameter tuning. For example, ten-fold cross-validation used to search five hyperparameters values requires training the same model 50 times. Rather than a regularization term, the IsoMax loss and the IsoMax+ loss are full losses used to seamlessly and entirely replace the SoftMax loss (combination of the cross-entropy loss, the SoftMax activation, and the last linear layer). Therefore, considering that the paper is proposing a loss regularization term to improve entropy in some cases, it must cite the IsoMax loss and IsoMax+ loss papers. The IsoMax losses work by simply replacing the SoftMax loss.\n\nAdditionally, it needs to change the following sentence because it is no longer valid after the advent of the IsoMax loss and the IsoMax+ loss: _\"While the literature suggests that classifiers whose predictive distributions increase their entropy the further away the test input gets from the training data are desirable, the implementation of models satisfying such a property is not scalable, and can only occur at the cost of performing crude approximations and by non-trivial modifications to the architecture of the neural network (Liu et al., 2020a; Kristiadi et al., 2020). Such modifications sometimes lead to degraded accuracy.\"_\n\nThe SoftMax loss is an extremely easy baseline for the OOD detection task. Hence, the proposed method should combine the proposed method with IsoMax and Enhanced IsoMax (IsoMaxPlus or IsoMax+). This could be done simply by adding the proposed regularization term that was added to the SoftMax loss. By using the code provided with the IsoMax papers, the suggested experiments could be performed by changing two lines of code. Indeed, combining the proposed term with the IsoMax loss variants (i.e., IsoMax and IsoMax+) would be appropriate to allow us to compare with a much-improved OOD detection baseline. \n\nAdditionally, we would evaluate whether IsoMax variants are better than SoftMax loss when combined with the proposed approach from an uncertainty estimation point of view. Adding these additional experiments using an improved baseline loss rather than the SoftMax loss appears essential. One final reason to include IsoMax and IsoMax+ in the experiments is the fact that they are distance-based losses, which the community is starting to agree is the best approach for OOD detection. Hence, it would be great to add one line called \"IsoMaxPlus\" and other \"IsoMaxPlus+Min-MaxEnt\" to Tables 1, 2, 5, and 6.\n\nFinally, considering that the Tables showing experiments with ResNet50 are as important as experiments using WideResNet28-10, to give a better notion of the real generalization power of the approach, we recommend moving Tables 5 and 6 to the main paper rather than being presented in the appendix. Additionally, section 5.1.4 should also consider the results obtained for the ResNet50 model. Considering the ResNet50 results, we believe the solution is attractive but lacks some generalization power (see ResNet50 results in the appendix). Furthermore, it is extremely costly, as we need to train the same model too many times to perform hyperparameter tuning. \n\nThe approach needs to improve the baseline performance of the IsoMax and IsoMax+ losses when combined with them to be valuable. Otherwise, we could simply use the IsoMax+ and improve the OOD detection performance training just once by avoiding the costly hyperparameter tuning.\n\nThe authors could be a bit more explicit about the fact the proposed approach requires increased training times because we need to add augmented data to the training process.\n\n_\"For all the methods, the ECE and AdaECE are computed after performing temperature scaling (Guo et al., 2017) with a **cross-validated temperature parameter**\"_ and _\"when temperature scaling is applied, **the temperature T is tuned on the validation set**, minimising the ECE (we considered values ranging from 0.1 to 10, with a step size of 0.01)\"_ seem contradictory. Considering _\"Such hyperparameters have been selected performing cross-validation with stratified-sampling on a 90/10 split of the training set to maximise\nAccuracy\"_, we believe the second sentence is true. Please, could you clarify?\n\n[1] Reducing Network Agnostophobia: https://arxiv.org/abs/1811.04110\n\n[2] Deep Anomaly Detection with Outlier Exposure: https://arxiv.org/abs/1812.04606\n\n[3] On Mixup Training: Improved Calibration and Predictive Uncertainty for Deep Neural Networks: https://arxiv.org/abs/1905.11001\n\n[4] IsoMax loss (first preprint, 2019): https://arxiv.org/abs/1908.05569v1\n\n[5] IsoMax loss: https://arxiv.org/abs/1908.05569\n\n[6] IsoMax loss (journal version): https://arxiv.org/abs/2006.04005\n\n[7] IsoMax+ loss: https://arxiv.org/abs/2105.14399\n\nDavid Macêdo\n\n\n###########################################################################################\n###########################################################################################\n###########################################################################################\n###########################################################################################\n\n\n===== FINAL RECOMMENDATION POST-REBUTTAL ========\n\nNOVELTY\n\nThe paper presents some level of novelty. It is similar to Mixup, but the differences are relevant. As we said before, this is not our main concern regarding the paper.\n\n**Hence, overall, the novelty is regular.**\n\nEXPERIMENTS\n\nThe paper presents many types of experiments such as uncertainty estimation, domain-shift, and out-of-distribution detection. However, the OOD data initially considered were only CIFAR and SVHN. We did not find results for the other methods for the new OOD data. \n\n**Hence, overall, the experimentation design is ok, but it may be improved regarding OOD detection.**\n\nPERFORMANCE\n\nConsidering WideResNet, the proposed approach generally outperforms other approaches regarding uncertainty estimation and domain-shift. **However, these results does not appear consistent if we consider the ResNet results presented in the appendix. Moreover, it usually performs similar to others regarding OOD detection and commonly does not even outperform regular neural networks in WideResNet.** \n\n**Hence, overall, the performance of the approach is not so great, and the results are not very consistent for different types of models.**\n\nCLARITY\n\n**ResNets are much more largely used and relevant than WideResNets. Unfortunately, the results of the proposed approach are worse on ResNets than WideResNets. We see no reason to present the ResNet results in the appendix.**\n\nWe complained about this in our review. However, the authors did not answer.\n\nSCALABILITY\n\n**The proposed approach has a severe limitation: it appears too expensive to be used in real-world projects.**\n\nTo tune hyperparameters, the authors did 10-fold cross-validation and searched for five values. This means they need to train the same network using the same data (with just a different hyperparameter) 50 times. Considering that they use the same amount of augmented data as the number of training examples, the method also doubles the GPU memory requirements or training time.\n\n**Therefore, overall, the proposed solution is about one hundred times more expensive than training a regular neural network.**\n\nHence, the proposed approach is viable for toy datasets like CIFAR. However, for real-world images such as ImageNet, it is essentially inviable. For example, let's say the cost to train a given big Transformer using a large amount of data is about 1 million dollars. If we plan to use the proposed approach, the total cost will be 100 million dollars. Same for CO2 emissions. \n\n**All this for not very clear or consistent performance improvements.**\n\nWe complained about this in our review. However, the authors did not answer.\n\nRECOMMENDATION\n\n**Considering the above, we recommend REJECTION.**\n\nFUTURE\n\nThe main idea is interesting, but we suggest that the authors remove the hyperparameter dependence and improve the OOD detection performance. We believe that combining the proposed approach with the IsoMax+ may help in this investigation.\n\n\n###########################################################################################\n###########################################################################################\n###########################################################################################\n###########################################################################################\n\n", "summary_of_the_paper": "The paper proposes to add a regularization term to the loss used to train the neural network. The mentioned term imposes high entropy on synthetically generated examples to improve accuracy, uncertainty estimates, and OOD detection performance. In other words, it proposes an \"entropy maximization regularizer.\" The paper also has the strength of performing a significant amount of experiments that includes uncertainty estimation, domain-shift, and out-of-distribution detection.", "summary_of_the_review": "The proposed method is interesting, but the novelty is regular as it is similar to Mixup and the performance is not too impressive when we also consider ResNet50 results. Moreover, the paper completely ignores the IsoMax loss and the IsoMax+ loss that are significantly related solutions. Hence, considering that the IsoMax loss variants already significantly increase the OOD detection performance without requiring costly hyperparameter tuning, this proposed regularization term would be more valuable if the authors show it also increases the ECE, Domain-Shift and OOD detection performance of the IsoMax loss and IsoMax+ loss when combined with them. Finally, it is possible that combining IsoMaxPlus with Min-MaxEnt may produce state-of-the-art results and also make the proposed approach work better (i.e., producing more stable results) when dealing with the ResNet50 model. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "-Pz5XyX2UU1", "reviewer": "Reviewer_iva2", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "Strengths:\n1. Empirical results demonstrate meaningfuly improved accuracy in both clean and (non-adversarially) corrupted data compared to the baseline methods.\n2. The method can be applied without modifying the network architecture, and it seems to perform well without being sensitive to HP tuning.\n\nWeaknesses:\n1. Limited evaluation to support claims (\"Mix-MaxEnt consistently provides much improved classification accuracy, better calibrated probabilities for in-distribution data, and reliable uncertainty estimates when exposed to situations involving domain-shift and out-of-distribution samples\"):\n\n1.a. In the context of classifier calibration, the comparison to Mixup seems to be limited given the similarity between the methods and the calibration results in the paper. Specifically, ECE/AdaECE results of C100-C and C10-C for Wide-ResNet (Tables 1 and 2) show conflicting results (this appears to persist in ResNet50 experiments as well). Moreover, the authors report the average results over all corruption types and intensities, however, Figure-10 shows that Mixup ECE is more consistent when facing severe corruption (>3), this could suggest a tradeoff between the two variants. Finally, the ablation study should be improved to clearly emphasize the importance of the proposed modifications including a clear summary of the empirical results. I would suggest to dedicate a section of the paper for a deep analysis comparing the two methods instead of scattering results and discussions in different sections.\n \n1.b. Evaluation is limited to ResNet architectures and small image classification datasets. As a general regularization method, the authors should provide a wider range of architectures and datasets to demonstrate the method is broadly applicable (at least in the context of supporting better accuracy/calibration claims).\n\n2. OOD results are not very impressive compared to the baselines (e.g., AUROC is comparable to vanilla DNN in Tables 1 & 2). The authors should also justify why they do not compare against other methods that have trained on proxy distributions to increase uncertainty on OOD [e.g., Dan Hendrycks, Mantas Mazeika, and Thomas Dietterich. Deep anomaly detection with outlier\nexposure, 2018, Weitang Liu, Xiaoyun Wang, John D. Owens, and Yixuan Li. Energy-based out-of-distribution\ndetection, 2020 ]. The authors also do not provide a sufficient explanation to avoid comparison to Mukhoti, Jishnu, et al. \"Calibrating deep neural networks using focal loss 2020.   \n\n3. Some aspects of the method are overlooked. For instance, the importance/impact of regularization strength is not discussed in the paper (i.e., the balance between CE and ERL, there is an implicit balance when setting a specific ratio of mixed and clean data in a batch). Also, the use of beta distribution is not properly motivated when preferring $\\lambda$ values that are close to 0.5.\n\nGeneral remark:\nThe paper can improve segnificantly in terms of readability, flow and focus. For example, some paragraphs contain highly technical details or remarks that can be moved to the appendix. Moreover, some important results are deferred to the appendix and presented in an inconvinient way and with limited discussion (e.g., missing a summary for ablation study results, appendix G. tables are missing baseline results etc.). In contrast, some content that made it to the main paper is arguably beneficial to readers (e.g., SR,SRN given the results are comparable to vanilla DNN).", "summary_of_the_paper": "The authors propose a method for regularizing DNN classifiers during training by encouraging the classifier to have high entropy on embeddings of mixed samples (from two different classes) as a proxy for the unknown portion of the data distribution (i.e., the portion of the in-distribution that is not represented by the training samples). The objective of the method is to produce better deterministic uncertainty estimates without compromising accuracy.\n\nThe suggested approach is similar to the previously proposed Mixup strategy, except regularization is explicitly applied to the mixed samples and labels while the optimization objective of clean samples is unchanged. Moreover, the distribution of $\\lambda$ is set to produce values close to ~0.5 (i.e., mixture farther away from the class clusters).", "summary_of_the_review": "The general idea, insight from empirical analysis and results appear promising. However, important aspects of this work are not properly addressed to support the main claims (limited evaluation, comparison to relevant work, general writing quality). The authors are encouredged to consider revising the paper accordingly and re-submit.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a new regularizer, based on entropy maximization of samples near the decision boundary, to improve the calibration of neural networks while maintaining their accuracy. 

The method seems simple, sufficiently novel, and has promising results. However, based on the review process (described below), I feel the paper needs to significantly improve its evaluation and presentation before it can be accepted.  

The review process summary:

* Two reviews were eventually weakly positive about the paper: without major concerns, but not enthusiastic.

* One review (L8Yz) was not sufficiently informative. 

* One review (ESue) raised many points. I disagreed with most of these points, following the authors' discussion. However, a few points seemed valid, such as the not-so-impressive performance for OOD detection, which the authors did not address.

* I therefore asked for an additional review (iva2). The review concluded the paper is interesting and potentially useful, but requires another round of revision before it can be accepted, mainly because of its clarity and missing comparisons. I agree with these conclusions.
Decision: Reject