Improving Dirichlet Prior Network for Out-of-Distribution Example Detection
Authors: Jay Nandy
Venue:  (2020)
Keywords: predictive uncertainty
distributional uncertainty
Dirichlet distribution
out-of-distribution detection
deep learning

Abstract: Determining the source of uncertainties in the predictions of AI systems are important. It allows the users to act in an informative manner to improve the safety of such systems, applied to the real-world sensitive applications. Predictive uncertainties can originate from the uncertainty in model parameters, data uncertainty or due to distributional mismatch between training and test examples. While recently, significant progress has been made to improve the predictive uncertainty estimation of deep learning models, most of these approaches either conflate the distributional uncertainty with model uncertainty or data uncertainty. In contrast, the Dirichlet Prior Network (DPN)  can model distributional uncertainty distinctly by parameterizing a prior Dirichlet over the predictive categorical distributions. However, their complex loss function by explicitly incorporating KL divergence between Dirichlet distributions often makes the error surface 
ill-suited to optimize for challenging datasets with multiple classes. In this paper, we present an improved DPN framework by proposing a novel loss function using the standard cross-entropy loss along with a regularization term to control the sharpness of the output Dirichlet distributions from the network. Our proposed loss function aims to improve the training efficiency of the DPN framework for challenging classification tasks with large number of classes. In our experiments using synthetic and real datasets, we demonstrate that our DPN models can distinguish the distributional uncertainty from other uncertainty types. Our proposed approach significantly improves DPN frameworks and outperform the existing OOD detectors on CIFAR-10 and CIFAR-100 dataset while also being able to recognize distributional uncertainty distinctly.

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IMPROVING DIRICHLET PRIOR NETWORK FOR OUT- OF-DISTRIBUTION EXAMPLE DETECTION
An Anonymous authors
Paper under double-blind review
ABSTRACT
Determining the source of uncertainties in the predictions of Al systems are im- portant. It allows the users to act in an informative manner to improve the safety of such systems, applied to the real-world sensitive applications. Predictive uncer- tainties can originate from the uncertainty in model parameters, data uncertainty or due to distributional mismatch between training and test examples. While re- cently, significant progress has been made to improve the predictive uncertainty estimation of deep learning models, most of these approaches either conflate the distributional uncertainty with model uncertainty data uncertainty. In contrast, the Dirichlet Prior Network (DPN) can model distributional uncertainty distinctly by parameterizing prior Dirichlet over the predictive categorical distributions. However, their complex loss function by explicitly incorporating KL divergence between Dirichlet distributions often makes the error surface ill-suited to opti- mize for challenging datasets with multiple classes. In this paper, we present an improved DPN framework by proposing a novel loss function using the standard cross- cross-entropy loss along with a regularization term to control the sharpness of the output Dirichlet distributions from the network, Our proposed loss function aims to improve the training efficiency of the DPN framework for challenging classification tasks with a large number of classes. In our experiments using syn- thetic and real datasets, we demonstrate that our DPN models can distinguish the distributional uncertainty from other uncertainty types. Our proposed approach significantly improves DPN frameworks and outperform the existing OOD de- tectors on CIFAR-10 and CIFAR-100 dataset while also being able to recognize distributional uncertainty distinctly.
INTRODUCTION
Deep neural networks (DNNs) have achieved impeccable success to address various real world tasks (Simonyan Zisserman 2014a Hinton et al., 2012; Litjens et al. 2017). However, despite impres- sive, and ever-improving performance in various supervised learning tasks, DNNs tend to make over-confident predictions for every input. Predictive uncertainties of DNNs can be confronted from three different factors such model uncertainty, data uncertainty and distributional uncertainty (Malinin & Gales) 2018). Model or epistemic uncertainty captures the uncertainty in estimating the model parameters, conditioning on training data (Gal, 2016). This uncertainty can be explained away given enough training data. Data or aleatoric uncertainty is originated from the inherent com- plexities of the training data, such as class overlap, label noise, homoscedastic and heteroscedastic noise (Gal 2016). Distributional uncertainty or dataset shift arises due to the distributional mis- match between the training and test examples (Quionero- Canroo--anneaaaa et al.) 2009 Malinin & Gales, 2018). In this case, as the network receives unfamiliar out-of-distribution (OOD) test data, it should not confidently make predictions. The ability to separately model these three types of predictive uncertainty is important, it enables the users to take appropriate actions depending of uncertainty. For example, in the active learning scenario, distributional uncertainty indicates that the classifier requires additional data for training. On the other hand, for various real-world appli- cations where the cost of an error is high, such as in autonomous vehicle control, medical, financial and legal fields, the source of uncertainty informs whether input requires manual intervention.
the source
Recently notable progress has been made to detect OOD images. Bayesian neural network based approaches conflate the distributional uncertainty through model uncertainty (Hernandez-Loohan &

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However, since obtaining the true posterior distribution for the model parameters are intractable, the success of these approaches depends on the chosen prior distribution over parameters and the nature of approximations. Here, the predictive uncertainties can be mea- sured by using an an ensemble of multiple stochastic forward passes using dropouts from a single DNN (Monti-Carke Dropout MCDP) (Gal & Ghahramani 2016) by ensembling results from multiple DNNs (Lakshminarayanan et al. 2017) and computing their mean and spread. On the other hand, most of the non-Bayesian approaches model their distributional uncertainty with data uncertainty. These approaches can explicitly train the network in multi-task fashion incorporat- ing both in-domain and OOD examples to produce sharp and flat predictive posteriors respectively (Lee et al. 201 Sa. Hendryeks et al., 2019). However, none of these approaches can robustly de- termine the source of uncertainty. Malinin & Gales) (2018) introduced Dirichlet Prior Network (DPN) to distinctly model the distributional uncertainty from the other uncertainty types. A DPN classifier aims to produce sharp distributions to indicate low-order uncertainty for the in-domain examples and flat distributions for the OOD examples. However, their complex loss function, using the Kullback-Leible (KL) divergence between Dirichlet distributions, results in the error surface to become poorly suited for optimization and makes it difficult efficiently train the DNN classifiers for challenging datasets with large number of classes (Malinin & Gales 2019).
Adams 2013
Gal 2016).
In this work, aim to improve the training efficiency of the DPN framework by proposing a novel loss function that also allows the distributional uncertainty to be modeled distinctly from both data uncertainty and model uncertainty. Instead of explicitly using Dirichlet distributions in the loss function, we propose to apply the standard cross-entropy loss on the softmax outputs along with a novel regularization term for the logit (pre-softmax activation) outputs. We train the models in multi-task function by leveraging both in-domain training images and OOD training images. The proposed loss function can be also viewed from the perspective of the oon-Bayesian frameworks (Lee et al. 201 8a] Hendrycks et al.) 2019) where the proposed regularizer presents an additional term to control the sharpness of the output Dirichlet distributions. In our experiments, we demonstrate that proposed regularization term effectively control the sharpness of the output Dirichlet distributions from the DPN to detect distributional uncertainties along with making the framework scalable for more challenging datasets. We also find that the OOD detection performance of our DPN models often improves by augmenting Gaussian noises with the OOD training images to train the DPN. Our experimental results CIFAR-10 and CIFAR-100 suggest that our proposed approach significantly improves the performance of the DPN framework for OOD detection and out-performs the recently proposed OOD detection techniques.
RELATED WORKS
In Bayesian frameworks, the predictive uncertainty of a classification model, trained on a finite dataset, Dm {x,, y L. ~ P.n (x. y), is expressed in terms of data (aleatoric) and model (epis- temic) uncertainty (Gal 2016). For an input x", the predictive uncertainty is expressed as:
p(uel=", = pluclaa, 0) ((n) de
Here, x and y represents the images and the corresponding class labels, sampled from an underlying probability distribution Pin (2, y). Here, the data uncertainty, p(will=,, 0) is described by the pos- terior distribution over class labels given model parameters, 0 and model uncertainty, p(.)D.... is given by the posterior distribution over parameters given the data, D.,..
The expected distribution for predictive uncertainty, p(w)|=", Din)) is obtained by marginalizing out 0. However, true posterior for p( O (i..) is intractable. Approaches such as Monte-Carke dropout (MCDP) (Gal & Ghahramani 2016), Langevin Dynamics (Welling & Teh 201T), explicit ensem-
bling (lakshminarayanan al. (017) approximate the integral in eq. as:
M
p(wela" D...)
Σ
p(welze" xx" ((())
((m) (())
M
where, ((m) is sampled from an explicit or implicit variational approximation, q(0) of the true pos- terior p(0 (D..). Each p(we|=", ((m) represents categorical distribution, f [ 14d] = [p(y- = 21),. p(y = wx)] over the class labels and the ensemble can be visualized as a collec- tion of points on the simplex. While for confident prediction, the ensemble is expected to sharply
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(a) Confident prediction
(b) Data uncertainty
(c) Distributional uncertainty
Figure I: Desired behavior of a DPN to indicate the three different uncertaintics.
appear in one corner of the simplex, the flatly spread ensembles cannot determine whether the un- certainty is due to data distributional uncertainty. Furthermore, for standard DNNs, with millions of parameters, it becomes even harder to find an appropriate prior distribution and inference scheme estimate the posterior distribution of the model. Dirichlet Prior Network (DPN) introduced to explicitly model the distributional uncertainty by parameterizing a Dirichlet distribution over simplex (Malinin & Gales 2018). More discussions about DPN presented in section 3.1
Alternatively, BanesBae frameworks derive their measure of uncertainties using the predictive posteriors obtained from DNNs. Lee et al. (2018a) and Hendryycks et al.. (2019) introduce new components in their loss functions to explicitly incorporate OOD data for training. De & Taylor 2018 append an auxiliary branch onto a pre-trained classifier derive the OOD score. [Shalev al. 2018 uses multiple semantic dense representations network. Several recent works such as (Lee et al., 2018b Liang et al. 2018) have demonstrated that by tweaking the input images during inference using adversarial perturbations can enhance the performance of DNN for OOD detection (Goodfellow et al. 22014b) However, their discriminative scores are achieved by tailoring the parameters for each OOD distributions during test time, which is not possible for real-world OOD examples. Hein et al. 2019) propose an adversarial training (Madry) al. 2018)) like approach produce lower confident predictions for OOD examples. However, while these models can identify the total predictive uncertainties, they can not robustly determine whether the source of uncertainty is due to an in-domain input region of class overlap or an OOD example far away from the training distribution.
the target label to train the OOD detection
PROPOSED METHODOLOGY
This section first describes the DPN framework and the difficulties of the existing modeling tech- niques to scale DPNs for challenging datasets. We then present our improved version DPN by proposing novel loss function to address these difficulties while allowing to model the distribu- tional uncertainty distinctly from the model and data uncertainty.
3.1 DIRICHLET PRIOR NETWORK
A DPN for classification directly parametrizes a prior Dirichlet distribution over the categorical out- put distributions simplex (Malinin & Gales 2018). For in-domain examples, DPN attempts to produce sharp Dirichlet in one corner of the simplex, when it confident in its predictions (Figure
Tap. It should produce sharp distribution in the middle of the simplex to indicate the data (low- order) uncertainty for the in-domain example with high degree of noise or belongs to a class over- lapping region (Figure Tb). In contrast, for OOD examples, DPN should produce Dirichlet that spreads over the simplex or across the edge of the simplex to indicate the distributional uncertainty (FigurqIc). Here, the data uncertainty is expressed by the coint-estimate categorical distribution μ while the distributional uncertainty is described using the distribution over the predictive categorical i. p(p1ar a=" O). The overall predictive uncertainty expressed as:
plwek wx" ,D) = ((walk) p(u)==" .0) p(O(D) du d0
(3)
This expression forms a three layered hierarchy of uncertainties: a large model uncertainty, p(0 would induce large variation in distributional uncertainty in p(μ| 0) and large degree of un- certainty for la leads to higher data uncertainty. DPN framework is consistent with the existing approaches, where additional layer of uncertainty is included to capture the distributional un- certainty. For example, marginalization of in Eqn. 3 will reproduce Eqn. [ while loose the control over the sharpness of the output Dirichlet distributions. The marginalization of 0 produces the expected estimation of data and distributional uncertainty given model uncertainty as:
p(wel>", D) p(wele)) If plulæ" 0) p(0 du p(woll) p(ulx", D)du
(4)
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marginalizing 0 is eq. is also intractable. Since the model uncertainty is reducible given large training data, for simplicity, here we assume dirac delta estimation, @ for 0:
However, similar eq. eq.
D(0 5( 0) p(ul=" D) = p( x;
(5)
Constructing DPN. DPN constructs a Dirichlet distribution as a prior over the categorical distributions, which is parameterized by the concentration parameters, a 01, O.K
I(an) Dir(ula) II [(a)
K
K
II
ac > 0, 00
a
(6)
where, ag is called the precision of the Dirichlet. A larger value of ag produces sharper distribu- tions to indicate low order uncertainties (fig Ta and Tbp. A DPN, fa produces the concentration parameters, a and the posterior over class labels, p(we x= ; )), is given by the mean of the Dirichlet.
a = fo(x") p( 0) = Dir(ula)
p(uel= 0) p(a.ll) p(ul>>; 0) du
ak
standard DNN with the softmax activation function can be represented as a DPN where the con- centration parameters are a e².(#"); z<(x") is the pre-softmax (logit) output corresponding to the class, e for an input x". The expected posterior probability of class label we is given as:
(ac))
p(wel="; ;)
(8)
ax
Σ e=c(#*)
However, the mean of the Dirichlet is now insensitive to any arbitrary scaling of ag- Hence, the precision of the Dirichlet, ogg degrades under the standard cross-entropy loss.
Malinin & Gales (0188) instead introduced new loss function that explicitly minimizes the KL
divergence between the output Dirichlet and target Dirichlet to produce predefined target preci- sion value for the output Dirichlet distributions. For in-domain examples, the target distribution is chosen to be sharp Dirichlet, Dir(u (.y,, focusing on their ground truth classes. While for OOD examples, a flat Dirichlet, Dir Djir(11 (pa @x) selected that spreads over the whole simplex.
[(0)= = RP... KL. ( [p(1// 0)] + Br... (T[Di((ul@)]]p((i 0)]
(9)
where, P... and Pout the underlying distribution of in-domain and OOD training examples spectively. However, learning the model using sparse 1-hot continuous distributions for class labels, which are effectively a delta function, is challenging due their complex loss function (eq. [9) Here, the error surface becomes poorly suited for optimization using the prok-propagttton algo- rithm (Malinin Gales 2 Malinin & Gales (2018) tackle this problem by using label smooth- ing (Szegedy et al. (016)) or teacher-student training (Hinton et al.) (015a) to redistribute small amount of probability density to each corner of the Dirichlet. This technique found to work well for datasets with a fewer number of class labels. However, for more challenging datasets with large number of classes, even these techniques cannot efficiently redistribute the probability densities at each corner and results in the target distribution to tend to delta function. Hence, it becomes dif- ficult to train the DPN to achieve competitive performances. Malinin & Gales] (2019) have recently proposed to reverse the terms within the KL divergence in eq. to improve the training efficiency of DPN models. This approach still requires to explicitly constrain the precision of the output Dirichlet distributions using appropriately chosen
for training.
3.2 IMPROVED DIRICHLET PRIOR NETWORK
We now propose an improved technique to model DPN by proposing a novel loss function using the standard cross-entropy loss along with regularization term control the precision of the output Dirichlet distribution from the network. As we have seen in equation g the precision of a Dirichlet distribution produced by the standard DNN is given as Σ exp <xx") Hence, we can control the sharpness of the distribution by designing a regularization term that increases the sum of logit (pre- softmax) outputs for the in-domain examples to produce sharp distributions to indicate their lower uncertainties. For the OOD examples, the regularization term aims to decrease the sum of logit (pre- softmax) outputs produce flat distributions to indicate distributional (higher-order) uncertainties. Hence, instead of explicitly constraining the precision the output Dirichlet with a specific hyper- parameter, we allow the network appropriately produce the precision values for different input.
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a (50, 50, 50) (b) a= (1.5, 1,5, 1.5) a = (0.99, 0.99, 0.99)
Figure 2: Dirichlet distributions become flatter as the precision is reduced (from (a) to (b)) and the densities move to the edges the concentration parameters become fractional i.e oe (0. 1).
In this paper, propose the regularization term as R Lem1 (z)= to control the sharp- ness of the output Dirichlet distribution along with the standard cross-entropy loss for classification. Here, the sigmoid function is chosen for the regularization term as it always produces a value within the range of (0, 1) for any 2e (x). For in-domain training examples, the loss function is given as:
Ain Σ (z.
(.. = E.... log 0)
(10)
K
For OOD training examples, the loss function is given
Lone (0) Erow. p(wpl 0))
Aout
Σ sigm)
(11)
K
where, 4 denotes the uniform distribution over the class labels. He is the cross-entropy function. We train the network in multi-task fashion with the overall loss function as:
min ((0) (0) + (l),
>=
(12)
9 In this loss function, we have incorporated three user-defined hyper-parameters: Ain Aous and in Eq. 10 Eq. II and Eq. 12 respectively. balances between the loss values for in-domain examples and OOD examples. The hyper-parameters Ain and Aout controls the sharpness of the output Dirichlet from DPN. By choosing Ain > O, we enforce the network produce positive logit values for in-domain examples that lead to producing sharper Dirichlet distributions (Fig. (a)). We choose Xin > Aout to ensure that the density either spread over the the simplex or across the boundary (Fig Tcp. The choice of Xin Xout > O will lead the network to produce comparatively flatter Dirichlet distributions for OOD examples (Fig. 2p). In contrast, by choosing Aout < 0, we enforce the network to produce negative values for z. (x and hence fractional values for a. 's (i.e ag E (0, 1)) for OOD examples. This will cause the densities of the Dirichlet to be distributed the edges of the simplex and produces an extremely sharp distribution, shown in Fig 2b.
The proposed loss function in Eq. 12 is also very closely related to non- Ban-Baysiinn approaches, where by choosing Ain. Aout to zero re-obtain similar loss functions as proposed by Lee et al. (2018)); Hendrycks et al. (201)). However, by setting Xin. Aout to zero, we lose control over the precision of the Dirichlet distribution that distinguishes distributional uncertainty from data uncertainty.
Our multi-task loss function (eq. 12 requires training samples from the in-domain distribution, P., well as from OOD Pout- However, since Pout is unknown, [ee et al. (2018a) propose to synthetically generate the OOD training samples from the boundary of in-domain region, Po, using generative models such as GAN (Goodfellow al., 20144). Alternatively, different, easily avail- able, real datasets can be used as OOD training examples. In practice, the latter approach is to found to more effective for training the OOD detectors and has been applied for our experiments on vision datasets (Hendrycks et al.
2019)
EXPERIMENTAL STUDY
We demonstrate the effectiveness of the DPN framework using the proposed loss function by con- ducting two sets of experiments. First, we experiment on a synthetic dataset. Next, we present comparative study of our proposed framework with the existing approaches on CIFARIO and CI- 100, and show the advantages over the original DPN framework (Malinin & Gales 2018).
4.1 SYNTHETIC DATASET
We construct simple dataset with three classes where the instances sampled from three different isotropic Gaussian distributions as shown in Figure B(a). We select isotropic co-variances, o²1 with 4, ensure that the classes are overlapping. We train small DPN with 2 hidden layers of 50
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(b)
(a) Synthetic Data
(c) Max Probability
(e) Entropy
(g) D-Ent
(i) >> e=c(=")
Figure 3: Visualizing the uncertainty measures different data-points for DPN(3 = 0.0) the top row and
for DPN = 0.5) in the bottom row.
nodes each for the synthetic dataset. For our loss function, we set À as 1.0. We choose both positive and negative values for Aout in our experiments. Here, Xin and Xnat chosen Xin. 3) and Aout (cllas
B). We train two different sets of DPN models using 3 = 0.0 (i.e positive Aout) and 0.5 (i.e negative Aoua), denoted DPN(3 3 0.0) and DPN 0.5), respectively (See
Appendix B for additional details).
An uncertainty measure can be computed as the probability of the predicted class max probability, marP max, p(w, x², DD)), in the expected predictive categorical distribution, N(welx". Din)
(Figure (h) and ((c) for DPN (3 0.0) and DPN(B 0.5), respectively).
Entropy of the predicted distribution, H(p(wala "x", Din)]= = Σ p(we/=", Din) In p(wee/"" la", Din) can be also applied total uncertainty measure that produces low prediction (Figure and for DPN = 0.0) and DPN = 0.5) respectively).
when the model is confident in its
Max probability and entropy are the most frequently used uncertainty measures used by the existing OOD detection models. However, since the predicted distribution is obtained by marginalizing μ (eq. 3) these mea- sures cannot capture the sharpness of the output Dirichlet for DPN. Hence, they cannot distinguish hetween misclassified examples and OOD examples. This observation also indicates the limitation of the existing non- DPN approachesto differentiate between data distributional uncertainties (Malinin & Gales] 2018). A DPN framework address this limitation by using the differential entropy (D-Ent) as an uncertainty measure that pro- duces lower scores for sharper Dirichlet distributions (eq. 13) Appendix presents the expression to compute D-Ent scores for given data point.
Hlp(ulæ", D...) = Don))= - p(|la>", Dan) In p(p/="" Din)
(13)
SK-
Figure 3(t f) 3nd demonstrate that D-Ent distinguish between in-domain and OOD examples. DPN 3 0.0) produces smaller positive logit values, resulting
Dirichlet distributions for OOD examples. Hence, we obtain relatively lower D-Ent scores for OOD examples compared to the in-domain examples (fig (d)).
Subsequently, D 0.5) produces negative logit values, resulting in sharp Dirichlet distributions across the edge of the simplex for OOD examples (Fig. ()). As the in-oomiin confident predictions, indicating that the output Dirichlet
see in Figure g]g). the D-Ent scores for
OOD examples are even smaller distributions for OOD examples are even sharper.
Since we explicitly constrain the logit outputs to produce smaller values for OOD examples, it is meaningful to define the sun of the exponential of logits, Σ c=*((=)) as new measure predictive uncertainty. In Figure d) and h)) we visualize this uncertainty measure for our DPN(8 = 0.0) and DPN = 0.5), respectively.
We found that our DPN models produce very high logit values. Hence, we scaled down these values by factor of 100 before computing this uncertainty measure for better visualization. We can see, ΣK ca((=") produces high scores for all in domain data points to distinguish them from OOD examples.
4.2 EXPERIMENTS ON CIFAR- 10 AND CIFAR-100
Experimental setup. In this section, we present our experiments CIFAR-10 and CIFAR-100 datasets (Knzhevsky 2009 CIFAR-10 images belong 10 different classes while CIFAR-100 is more challenging
dataset, containing 100 image classes. Both of these datasets contain 32 natural colored images of 50, 000 training and 10, 000 testing examples. In Appendix[A] we have also experimented on TinyImageNet (TIM)
that consists of 64 x 64 natural images from 200 classes.
The CIFAR-10 classifiers are trained using CIFAR-10 10 training images as in-domain and CIFAR-100 training images as OOD dataset (Simonyan Zisserman 2014b). We VGG-16 architecture for this case. For

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Table I: Training details for our proposed DPN models
Classilcationn Task
Impui Shape
4Classes
Details Training data
(XD
Details of
CIFAR-LO
32 32
10
CIFAR- 10 training set (50,000 images)
CIFAR- 100 training set (50,000 images)
In- In-Domai:: CIARR T0 Test sot
(16,000 Imagee))
00D: TIM. LSUN,,
In- Domuin: CIFAR- TÜ lest set (10,000 Images)
CIFAR-100 training set (50,000 images)
CIPAR-I0 10 training set (50,000 images)
CIFAR- 100
32 x 32
100
OOD: TIM, LSUN,
CIFAR-100, we CIFAR-100 training images in-domain and CIFAR-10 training images as the OOD dataset. Here, we consider DenseNet with depth = 55 and growth rate = 12 (Huang et al. 2017).
We present in-domain misclassification detection and OOD detection experiments to evaluate the performance of our models. For our misclassffication detection experiments, we take the in-domain test set and attempt to distinguish the correctly classified examples from misclassified examples (see Table - The OOD detection experiments attempt distinguish the in-domain test images from unknown OOD images. In Table we present smaller set of results where we consider TinylmageNer (TIM) as the OOD dataset and attempt to distinguish them from in- no-oomain test data points. In Appendix [A] we present an expanded version of this comparative table for wide range of OOD examples. Note that, the OOD test images are selected from datasets different from the datasets used for training. The description of our training and test datasets presented in Table [ Please refer to Appendix foor further details.
Evaluation of Predictive Uncertainty Estimation. To evaluate the performance of our model for misclassi- fication detection, we consider the misclassified examples as the positive class and correctly classified exam- ples as the negative class. For the OOD detection task, we treat the OOD examples as the positive class and in-domain examples
negative class. The detection performance for these tasks are measured using two metries; area under the receiver operating characteristic curve (AUROC) and area under the preciston-recall curve (AUPR) (Hendryeks & Gimpel] 2016). The AUROC can be interpreted as the prohability of an OOD example to produce a higher detection score than an in-domain example (Davis & Goadrich) (0006). Hence, a higher AUROC is desirable, and uninformative detector produces an AUROC N 50%. The AUPR is more informative when the positive class and negative class have greatly differing base rates. It can take these base rates into account Manning & Schütze 1999).
Mathoak
AUROC
ALPR
D-fir
MCDP
03.6
03.6
CIFAR-10
ODIN
91.1
94.1
91.6
34.8
94.0
DPN
92.2
92.1
90.9
52.7
51.0
45.5
92.5
D 0.0.0
200)
91.7
91.3
3011
37.0
35.7
312
31.5
DPN 0.0.0
91.3
90.8
85.5
53.9
35.0
35.3
29.1
29.7
93.6
DPN
0.0. 0.05;
98.4
93.2
91.4
91.8
44.7
42.5
35.1
36.4
93.7
d -.5.e : (.0)
92.0
91.6
80.9
60.3
36.8
34.8
31.5
17.0
94.1
DPN -.....ri--0....-11 : 3.01)
91.3
90.8
87.7
61.9
37.1
35.3
28.9
17.0
94.1
DPN.zidd PPW../dd-G..6.111 Du06)
93.0
92.8
90.1
58.5
39.8
38.3
29.8
32,8
94.1
Baszlino
67.0
MCDP
87.7
87.7
CIFAR-100 100
ODIN
79.5
86.0
85.0
55.5
TIPN 0.0.0
(U)
W5.5
15.4
254
477
41.8
1 0.0.a
001;
86.9
86.2
75.1
75.6
61.5
58.5
43.1
43.1
75.6
1 0.0.0
0.55;
87.3
86.7
76.7
76.4
64.6
61.7
45.8
43.
757
d -..5. : (.0)
86.7
85.6
73.5
73.1
61.3
57.3
41.4
41.9
760
DP -0.5.. : 3.01)
86.5
85.7
74.1
74.
61.0
57.5
41.
43.2
76.2
1 0.5.6 (005)
87.6
87.0
26.3
77.1
68.6
61.
45.4
46.1
76.3
Table 2: Comparative results of misclassified image detection for CIFAR-10 and CIFAR-100.
Hyper-parameterss for Training Loss. Unlike Liang et al. [2018): Lee et al. (018a), and similar McMalinnn & Gales] (2018); [Hendrycks et al. (2019), we do not require to tune any parpempareettrss during testing for different datasets. In other words, the OOD test examples remain unknown to our DPN classifiers, as in a real-world scenario. During training. multiple DPN models for CIFAR-10 and CIFAR- classifiers using both positive and negative values for Aout- We choose Ain and Noue as Ain = (1 - 3) and Aout = (gea - 3). We train two different sets of DPN models using 0.0 (i.e positive dous) and 8 0.5 (i.e negative )out), denoted as DP : 0.0) and ( 0.5), respectively.
set À 0.5 (eq. [2]. similar to Hendrycks et al. (2019). We train
Detecting the source of uncertainty. [1] DPN model with Aous > O: By choosing a positive value for Aouts we enforce the network to produce flat Dirichlet distributions over the simplex for OOD examples. Hence, the D-Ent measure should produce higher scores for OOD examples and lower scores for in-domain examples. Therefore, given a test example with high scores for both entropy as well as D-Ent, it indicates distributional uncertainty. In contrast, if the test example achieves high entropy scores and lower D-Ent score, it indicates data uncertainty. Our ( 0.0) models achieve high AUROC and AUPR scores using D- Ent to detect OOD examples for both CIFAR-10 10 and CIFAR-100 (see Table 3) while achieves lower AUROC and AUPR scores for D-Ent while detecting misclasification for CIFAR- 100 (see Table Z). However for CIFAR-
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Under review as a conference paper at ICLR 2020
Mathods
AUROC Σ Ee
AUPR
Max.P
Enc.
D D.E.
Max
Enc.
Boc
Bascline
583
85.4
851
86.7
MCDP
88.5
89.2
84-7
86.1
CIFAR--OO 10
ODIN
96.4
95.8
11M
OE
9%.0
98.0
97.5
97.9
DPN
96.3 1
94.3
94-6
94.0
94.0
94.2
DPN 30. 100)
97.5
97.7
97.6
97.6
07.3
97.6
97.5
97.5
DPN 00.0 0.1
98.5
98.5
98.4
05.5
9%.4
98.5
98.5
98.4
DPN
:00...
0405
97.1
97.4
97.6
97.77
97..1
97.5
97.7
97.8
DPN (0.....
:00)
98.7
98.8
96.7
0.3
998.6
95.7
92.8
32.5
99.0
99.1
96.3
6.8
98.5
08.9
90077
32.2
DPN..)(3:0...
97.9
98.2
98.2
30.9
97.9
98.2
98.1
54.1
Biscline
14.9
75.3
CIFAR- 100
ODIN
78.9
81.0
TIM
OB
88.0
9013
35.1
K5.4
DPN a :001)
96.5
97.4
97.1
98.9
DPNworld a
96.7
98.2
DPNwor13 a.5.a 200)
98.9
99.1
99.5
31.8
DPN a.5.0: :001)
99.7
DPN 0005)
59.8
Table 3: Comparative results of OOD example detection for CIFAR-10 and CIFAR-100. Expanded version of this table along with wide range OOD datasets are provided in Appendix
10, our DoN (8 : 0.0) models fail to produce low AUROC and AUPR scores under D-Ent measure for misclassification detection. This indicates that these models achieve high D-Ent scores for both misclassified and OOD examples and make it difficult to distinguish distributional uncertainty from data uncertainty for CIFAR-10.
Note that, similar to our (B8 0.0). DP (Malinin Gales 2018 also produces high AUPR and AUROC scores under D-Fntt measure for for both misclassiteed examples and OOD examples for CIFAR-10 model (Table 2 and 3). This indicates that both DPN and our D (8 0.0) models may be ineffective detect distributional uncertainty for CIFAR-IO. 10. In contrast, our DPN, DPN..f (8 0.5) models are always found to be effective to identify the source of uncertainties.
[2] DPN model with Aout <. O: In contrast, if we choose negative value for Aouts our DPN network produces very sharp Dirichlet distributions across the edge of the simplex for OOD examples. Hence, it produces very low D-Ent scores. Notably, these distributions are even sharper than the Dirichlet distributions obtained for confidently predicted examples. Hence, for high entropy and low D-Ent scores. As we can see, our DPN..ff (B : 0.5) models produce very low AUROC and AUPR scores for D-Ent measure to detect OOD examples (Table while achieve relatively higher AUROC and AUPR scores for detecting misclassified examples (Table (
example, we can detect distributional uncertainty if it achieves
Data Augmentation with white-noise for Training, After training our DPN models using clean training images, we also fine-tune them using noisy OOD training images for few epochs. We add minor Gaussian noise sampled from isotropic Gaussian distributions (N0, o²I) with our OOD training images. Here, the idea is to add minor perturbation without distorting the features of the OOD training images. It further exposes the network into the out of distribution space. Note that, the DPN models are fine-tuned only using OOD training images. The OOD test examples remain unknown. In practice, this technique may often further improve the OOD detection performance of our DPN models as we choose 0.01 (see Table However, for larger perturbation, performance degrades for the DPN (a = 0.05).
Comparative Study, In Table 2 and 3 we compare the performance of our approach with several baselines, such as standard DNN (Hendrycks & Gimpel (2016). MCDP (Gal & Ghahramani 2016). DPN (Malinin & Gales 2018), 3 ODIN (Liang et al. 2018), and OE (Hendrycks et al. 2019). We use the same architecture DPN models for all the competitive models. OE models are trained using the of in-domain and OOD
training images with their proposed loss function (Hendrycks et al. 2019). Note that, since non-DPN methods
do not explicitly model the logit outputs, the D-Ent or > fc measures not meaningful (Malinin & Gales 2018).
We also compare our results with the existing DPN framework, namely DPN Dir Malinin & Gales (0118) for 10. Note that, the DPN framework fails to scale for the CIFAR- 100 dataset that consists of 100 classes. Due to the unavailability of codes and results (under the same settings), we could not compare model with Malinin Gales (2019) Overall, our DPN models significantly improved the performance of the DPN framework and consistently out-performed the existing OOD detection models. In addition, it is able to distinguish distributional uncertainty from other uncertainty types.
CONCLUSION
In this paper, we propose a novel framework to improve the training efficiency of DPN models for challenging classification tasks with large number of classes. We also propose novel regularization term that allows con- trols the sharpness of the Dirichlet distrihutions. We show that the proposed regularizer can be easily integrated
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Under review as a conference paper at ICLR 2020
with the standard cross-entropy loss function. Our experiments synthetic and real datasets demonstrate that our proposed framework can efficiently distinguish the distributional uncertainty from other uncertainty our DPN models can be often improved by
types. We demonstrate that the OOD detection performance training with noisy OOD examples. Experiments show that our proposed approach significantly improves DPN frameworks, and outperforms the existing OOD detectors on both CIFAR-I0 and CIFAR-100 datasets.
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A
EXPANDED RESULTS
Table 4: Results of OOD image detection for CIFAR-10. Description of these OOD datasets provided in Appendix
D
Methods
AUROC
AUPR
Max.P
Ent.
Σ e
D-Ent
Max.P
Ent.
Σ cx
D-Ent
Baseline
73.0
73.3
70.4
71.3
MCDP
74.0
74.0
70.9
71.2
Gaussian
ODIN
72.6
70.1
OE
81.9
82.0
82.5
82.6
DPN
sofs
0:0,0
82.3
82.4
82.7
82.7
83.9
84.1
84.0
84.0
DPN (8
0:0,0
96.0
96.2
96.4
96.4
96.3
96.7
96.7
96.7
DPN (8 0.0,0 0.0,
89.2
89.6
90.8
90.7
91.6
92.0
93.0
92.9
DPN (3 (3 0.5.0:00:0000
82.9
83.0
83.3
37.3
83.8
83.9
82.6
48.7
DPN n
0.01)
95.8
96.0
94.6
13.0
96.4
96.7
90.9
35.1
DPN
f (8: 0.5, 0.05)
0.005
90.2
90.6
92.4
28.1
92.4
92.8
94.0
47.3
Baseline
88.8
89.4
85.1
86.7
MCDP
88.5
89.2
84.7
86.1
ODIN
94.4
93.8
TIM
OE
98.0
98.0
97.9
97.9
DPN Dir
94.3
94.3
94.6
94.0
94.0
94.2
0.00
97.6
97.7
97.6
97.6
97.5
97.6
97.5
97.5
(8 (3 0
98.5
98.5
98.4
98.5
98.4
98.5
98.3
98.4
so (3: 0:0,0:
97.1
97.4
97.6
97.7
97.2
97.5
97.7
97.8
DPN (3 0.5,
98.7
98.8
96.7
6.8
98.6
98.7
92.8
32.5
DPN (3 0.5, 0.01) 0.01)
99.0
99.1
96.3
6.8
98.5
98.9
90.7
32.2
DPN n (3 0.5, 0.05)
97.9
98.2
98.2
30.9
97.9
98.2
98.1
54.1
Baseline
90.2
91.0
86.6
88.6
MCDP
90.3
91.1
86.8
88.9
ODIN
96.6
96.2
LSUN
OE
97.9
98.0
97.6
97.7
DPN
94.4
94.4
94.6
93.3
93.4
93.3
DPN m0 0.0,
97.7
97.8
97.7
97.7
97.3
97.4
97.3
97.3
DPN DP 0.0, a: 0.011
98.7
98.7
98.6
98.6
98.5
98.5
98.3
98.3
DPN (3 0
97.3
97.7
97.9
98.0
97.2
97.6
97.7
97.8
DPN (8 (8: 0.5,
98.8
98.8
97.1
6.2
98.6
98.6
93.8
32.4
DPN (8 0.5, 0.00) 0.01)
99.2
99.2
96.7
5,5
98.7
98.9
91.8
31.8
DPN (8 0.5, 0.05)
97.9
98.2
98.4
33.0
97.7
98.1
98.3
56.0
Baseline
89.4
90.0
95.6
96.2
MCDP
89.3
90.2
95.6
96.2
Places 365
ODIN
95.4
98.5
OE
97.9
98.0
99.4
99.4
[o 0.0)
97.7
97.8
97.7
97.7
99.3
99.3
99.3
99.3
DPN
0.01)
98.8
98.8
98.7
98.7
99.6
99.6
99.6
99.6
DPN
0 0.0,
97.3
97.6
97.8
97.8
99.2
99.3
99,4
99.4
DPN (8 0:5.0
98.7
98.7
96.7
6.7
99.6
99.6
97.9
60.7
DPN (8 0 0.00)
99.2
99.3
96.5
5.4
99.6
99.7
97.3
59.5
DPN .o ( 0.5, :.05)
97.9
98.2
98.3
29.5
99.4
99.5
99.5
77.1
Baseline
88.6
89.0
74.9
77.0
MCDP
87.6
88.0
73.7
75.8
Textures
ODIN
94.9
90.5
OE
99.2
99.7
98.6
98.5
DPN
sofi f 0.000000000000
99.3
99.3
99.5
99.3
98.8
99.0
99.0
99.0
DPN
0.01)
99.5
99.5
99.3
99.4
98.9
98.9
98.6
98.7
DPN
vofs 0:00
99.4
99.5
99.5
99.6
99.0
99.1
99.2
99.3
DPN so 0
99.5
99.5
96.7
4.6
99.1
99.1
85.6
21.3
DPN (8
0..00:000000
99.4
99.5
96.5
5.5
98.0
98.8
84.3
21.4
DPN (8
(8: 0.055
99.1
99.3
98.7
24.7
98.3
98.7
97.3
38.3
B EXPERIMENTAL DETAILS ON SYNTHETIC DATASETS
The three classes of our synthetic dataset are constructed by sampling from three different isotropic Gaussian distributions with means of (-4,0 0). (4, 0) and (0, 5) and isotropic variances of a = 4. We sample 200 training data points from each distribution for each class. We also sample 600 OOD training examples from an uniform distribution of (([115, [-].[-13, [-13, 17)).

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Under review as a conference paper at ICLR 2020
Table 5: Results of OOD image detection for CIFAR-100. Description of these OOD datasets are
provided in Appendix C2
Methods
AUROC
AUPR
Max.P
Ent.
Σ Detlle))
D-Ent
Max.P
Ent.
Eetler
D-Ent
Baseline MCDP
73.2
74.0
67.7
68.5
74.4
74.7
68.9
69.3
Gaussian
ODIN
80.4
76.2
OE
72.9
74.0
64.6
64.9
DPN
(3: 0.0, :.00)
71.7
72.6
71.9
71.9
63.6
63.9
61.7
61.5
DPN
0.000
94.4
95.2
97.6
96.4
95.3
96.0
97.9
94.0
DPN
so 0.0,0
96.1
96.7
98.7
97.2
97.0
97.5
98.9
93.5
DPN 58 0.5,
91.9
92.8
96.3
37.6
91.8
92.4
95.8
50.8
DPN (
98.3
98.6
99.6
3.4
98.6
98.8
99.7
31.0
DPN fo
(8: 0, 0.05)
96.9
97.4
99.6
9.3
97.7
98.1
99.7
33.6
Baseline
74.9
76.3
71.1
73.1
MCDP
78.9
81.0
75.4
78.0
ODIN
83.8
81.4
TIM
OE
86.5
88.0
82.8
83.0
DPN
ooft (3 0.0,0
88.9
90.3
91.1
90.7
86.1
86.4
83.4
83.6
DPN
0.0000000000
96.5
97.4
98.8
98.0
97.1
97.8
98.9
95.2
DPN
(3
0.0,
95.8
96.7
98.0
96.7
96.4
97.2
98.2
92.9
DPN
005,00000::0000000
98.9
99.1
99.5
5.0
98.9
99.0
99.5
31.8
DPN
(3
00,,0
99.1
99.3
99.7
5,9
99.2
99.3
99.7
32.4
DPN
0.5,0
95.8
96,9
98.8
41,8
96.4
97.4
98.9
59.8
Baseline
78.9
80.4
74.2
75.9
MCDP
83.2
85.4
79.0
81.5
ODIN
87.8
84.6
LSUN
OE
90.6
91.6
86.5
86.1
DPN
(3 0.0
91.6
92.6
93.3
92.6
88.3
88.3
87.3
85.4
DPN soft
DPN
(8.00 0.0.
98.9
99.2
99.7
98.9
99.0
99.3
99.7
96.1
DPN
:8 0.05)
96.0
96.8
98.2
96.8
96.6
97.2
98.4
93.0
DPN soft
0..
99.7
99.7
99.9
1.4
99.6
99.7
99.8
30.8
DPN (B 0.0.
99.7
99.7
99.9
2.8
99.7
99.7
99.9
31.1
DPN
o 0.5, 0.005)
96.8
97.5
99.1
37.1
97.3
98.0
99.2
56.2
Baseline
76.6
78.1
90.2
91.0
MCDP
80.9
83.1
92.2
93.3
Places365
ODIN
86.4
94.7
OE
DPN.
0.0,
92.6
96.0
95.9
94,9
DPN (3
0.0,
97.8
98.4
99.3
98.5
99.4
99.8
98.6
DPN
(3 0:0,
95.9
96.7
98.1
96.7
98.9
99.5
97.6
DPN
(3: :.0.
99.7
3.4
99.9
DPN
o0 0..
99.5
99.8
60.3
DPN.. 035.0
96.9
98.8
40.0
99.2
99.7
80.9
Baseline
60.0
60.2
43.4
MCDP
64.3
Textures
ODIN
48.9
73.6
74.8
58.4
DPN.oft
0.00 (8000,00000
80.5
81.8
85.0
83.7
66.9
71.3
67.2
(8
0.0.
86
92.9
90.9
90.9
84.1
DPN
0.0,)
89.2
90.7
94.5
93.1
84.5
DPN
0..0
91.2
96.2
10.6
22.4
DPN. (3: (3
0
92.2
93.6
97.4
8.9
89.9
91.6
96.9
22.1
DPN 0.5, 0.05)
86.7
88.9
96.6
29.6
83.7
86.8
96.1
35.1
We train neural network with 2 hidden layers with 50 nodes each and relu activation function. The network trained for 2, 500 epochs using stochastic gradient descent (SGD) optimization with constant learning rate 0.01.
C EXPERIMENTAL DETAILS ON CIFAR-10 AND CIFAR-100
C. TRAINING DETAILS
For our experiments CIFAR-10, train VGG-16 model with CIFAR-10 the in-domain and CIFAR- 100 the OOD training data (Simonyan & Zisserman (2014b)). For CIFAR-100, we train DenseNet with
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Under review as a conference paper at ICLR 2020
Table 6: Results for misclassification detection for TinyImageNer
Methods
AUROK
AUPR
Acc.(%)
Max.P
Ent.
Σ etclaa))
D-Ent
Max.F P
Ent.
Σ ctcle))
D-Ent
Buseline
85.4
85.2
78.3
77.8
57.8
Tiny TinyImigeNe
MCDP
85.8
77.9
OE
85.2
85.4
77.8
77.9
57.7
85.6
85.6
81.7
80.0
18.8
78.6
73.5
73.5
57.8
D ::000:::0000000000000
85.7
85.8
81.8
80.2
79.0
78.7
73.6
73.6
57.8
o :.00.
85.6
80.3
73.6
( ::5.0
85.9
86.3
81.8
77.6
79.7
80.1
73.6
72.7
57.5
of (3: 0.5.0
85.8
86.2
81.9
77.5
79.5
80.1
73.7
72.8
57.5
of 0.5. :.005)
85.7
86.1
81.8
77.9
79.2
80.0
73.7
73.1
57.6
Table 7: Results of OOD image detection for TinylmageNet classifiers. Description of these OOD datasets are provided Appendix C.2
Dreat
Methods
AUROC
AUPR
Max.P
Ent.
Σ (en(t")
D-Fnt
Max.P
Ent.
Σ Seti(e))
D-Ent
Baseline
62.9
61.7
51.4
50.0
MCDP
63.8
63.6
52.2
51.8
OE
98.9
99.0
99.0
99.1
STL-10
98.4
98.3
98.9
98.8
98.6
98.8
99.1
99.0
DPN (8
0.011
98.1
98.5
98.6
98.6
98.4
98.6
98.9
98.8
(3:
97.0
98.1
97.9
97.9
97.5
97.7
98.4
98.3
DPN
(8 0.5,
98.6
98.7
99.0
4.3
98.8
98.9
99.2
28.3
DPN
28 :
98.3
98.4
98.8
5.2
98.5
98.7
99.0
28.8
D 0.5, 0.05)
97.5
97.7
98.4
7.6
97.8
98.0
98.6
30.3
Baseline
64.9
65.6
59.7
59.5
MCDP
65.9
66.1
60.3
60.2
OE
98.2
98.5
98.7
99.0
LSUN
oo (3 0:000
97.8
98.2
98.9
98.8
98.4
98.7
99.2
99.2
DoN 0.0,
97.3
97.7
98.7
98.6
98.1
98.4
99.0
99.0
DPN
(0:f:(3::0000 0.05)
95.2
95.9
97.7
97.5
96.5
97.0
98.3
98.1
2 0.5,
98.2
97.6
99.1
6.8
99.1
98.9
99.4
35.0
DPN (3 0.5.... 0.01)
97.6
98.0
98.8
9.1
98.3
98.6
99.1
36.8
DPN 2o (3: 0.5, 0.05)
96.1
96.6
98.0
14.4
97.1
97.1
98.6
40.8
Baseline
67.8
68.9
85,8
86.0
MCDP
68.5
68.9
86.1
86.2
Places365
OE
95.8
96.5
98.9
99.1
DPN [3: 0.0,
95.0
95.8
97.3
97.1
98.7
98.9
99.3
99.3
DPN
0.0,
94.1
94.9
96.7
96.5
98.5
98.7
99.1
99.1
DPN
0.0,0
90.5
91.8
94.4
94.1
97.4
97.4
98.5
98.5
DPN (8 0.5,0
95.6
96.2
97.6
17.0
98.9
99.0
99.4
69.6
DPN mn (3 0.5, a: 0.05)
0.5, 0.001)
94.4
95.1
96.8
21.3
98.6
98.7
99.2
72.1
DPN
91.6
92.6
95.0
30.8
97.8
98.0
98.7
77.0
Bascline
68.8
70.9
51.0
53.3
MCDP
69.1
69.7
51.0
51.6
Textures
OE
87.5
89.1
85.6
87.5
DPN soft 0.0,0
85.1
86.8
88.4
89.0
82.6
84.7
86.0
86.6
DPN
DPN.of (8 0.00,:::000
83.9
85.8
87.6
88.2
81.1
83.3
85.0
85.6
DPN DPN.ofx
(8 0.0,0:
81.4
83.5
85.6
86.2
77.3
79.9
81.9
82.6
DPN
(8 a
85.7
87.0
88.1
45.0
83.3
84.8
85.8
46.6
DPN (8 8 0.5, a: 0.01)
84.5
85.8
87.1
47.5
81.6
83.3
84.4
48.4
DPN
sf 0,5, 0.05)
82.3
83.8
85.3
53.0
78.5
80.4
81.8
52.0
depth = 55, growth rate (Huang et al. (2017)) We trained multiple DPN../f models using our proposed loss functions with different hyper-paramen parametrrss
12 and CIFAR-100 as the in-domain and CIFAR-10 as the OOD training data
CIFAR-10, 10, we use VGG- network. Here, we use CIFAR-10 training images (50. 000 images) as our in-domain training data and CIFAR-106 100 training images (50, 000 images) as our OOD training data.
For CIFAR- 100, Densenet(5S, 12) is trained using the same setup proposed by Huang et al. (2017). Here, CIFAR-101 1-00 training images (50, 000 images) our in-domain training data and CIFAR-10 training images (50, 000 images) as OOD training data.
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For TinylmageNet (TIM), we use the VGG-16 network. Here, TIM training images ( (100,000 images) our in-domain training data and ImageNet-25K images (25, 000 images) as our OOD training data. ImageNet- 25K is obtained by randomly selecting 25, 000 images from the ImageNet datasct (Deng ct al. 2009).
After training the models with clean in-domain and OOD images, we further fine-tune the models using noisy OOD training images for 50 epochs with the learning rate of 0.0001. The noises are chosen form an isotropic Gaussian distribution, (O, o² I). We have experimented with three different values of {0.0, 0.01, 0.05) introduce different level of noises.
Table 8: Details of training and test datasets
Details of Training
Classilication Task
Input Shape
#Classe
In- In-Domain
OOD
Details of Test data.
In-Domain: CIEAR-10 10 test
CIFAR-10
32 x 32
10
CIFAR-IB 10 training set (50,000 images)
CIFAR-100 training set (50,000 imagos)
(10,000 Images)
OOD: LSUN etc.
In Domutn: CIARR 100 test set
CIFAR-100 training
CIFAR-10 10 training
CIFAR-100
100
(10,000 Images)
(50,000 images)
(50,000 images)
0OD: TIM. LSUN, ctc.
VinylmageNet (TIM)
TinyImageNet training set (100,000 images)
ImageNet-25K
In-Domain: TIM
64 x 64
200
(25,000 randomly sampled
(10,000 Images)
images from ImageNet)
STL-10. LSUN, etc.
C.2 OOD TEST DATASETS
We use a wide range of OOD dataset to evaluate the performance of our proposed OOD detection models. For CIFAR-10 and CIFAR-100 classifiers, these input test images are resized to 32 x 32, while for Tiny ImageNet classifiers, we resize them to 64 x 64. In this task, we attempt distinguish the unknown OOD images from the corresponding in-domain test images from different classification tasks. We compute different uncertainty measures these images for this purpose. For our evaluations, we use following OOD images described in the following.
1. Tiny ImageNet (TIM) (Li et al. (2017). This is a subset of Imagenet dataset. We use the validation set, that contains 10, 000 test images from 200 different image classes for our evaluation during test time. This dataset is used as an OOD test dataset only for CIFAR-10 and CIFAR-100 classifiers. Note that, for metteeeeeet classifiers, this is the in-domain test
2. LSUN Yuu et al. (2015)). The Large-scale Scene UNderstanding dataset (LSUN) contains images of
10 different scene categories. We its validation set, containing 10, 000 images, as unknown OOD test set.
Places 365 (Zhou et al. 2017)) The validation of this datasct consists of 36500 images of 365 scene categories.
4. Textures (Cimpoi 2014) contains 5640 textural images the wild belonging 47 categories.
5. STL-10 contains 000 images of natural images from 10 different classes (Coates al. 2011).
6. Gaussian Noise. This is an artificially generated dataset obtained by adding Gaussian noise to the in-domain test images. The Gaussian noises are sampled from isotropic Gaussian distribution, (00,0*1) with a 0.05.
C.3 DETAILS OF COMPETITIVE SYSTEMS
We compare the performance of our models with standard DNN as baseline model (Hendryeks & Gimpel (2016)). the Bayesian framework, monti-carlo dropout (MCDP) (Gal & Ghahramani (2016)). DPN per using the loss function proposed by Malinin & Gales) 20188), non-Bayesian frameworks such as ODIN (Liang al. (2018) and outlier exposure (OE) by Hendrycks et al. (2019). We use the same architecture as DPN for the competitive models. For DPN Diesehlet, we could not reproduce the same performance as given in Malinin
& Gales 2018) and hence use their reported results for CIFAR- 10 for our comparison.
For MCDP, we use the standard DNN model with randomly dropping the nodes during test time. The predictive categorical distributions are obtained by averaging the oulputs for 10 iterations.
ODIN applies the standard DNN models trained only using in-domain training examples for OOD detec- tion. During testing phase, it perturbs the input images using FGSM adversarial attack (Goodfellow et al.
22014b Jand softmax activation function by incorporating the temperature hyper- parameter (Hinton et al. (015b)). The maximum Prohability score is then applied for their uncertainty measure. They propose to different paraeepas for different OOD examples. However, in practice, the source of expected OOD)
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Under review as a conference paper at ICLR 2020
examples camnot be known. Hence, for comparisons, always the perturbation size 0.002 and the temperature 1000.
OE models are trained using the proposed loss function by Hendrycks et al. (0119. Here, we use the same training set up as applied for our DfN.oft models: CIFAR-IO classifiers are trained using CIFAR-10 training
images as in-domain examples and CIFAR-100 training images OOD examples. For CIFAR-100, the OE models trained using CIFAR-10 training images OOD examples.
D DIFFERENTIAL ENTROPY MEASURE FOR DIRICHLET PRIOR NETWORK
Differential Entropy of Dirichlet distribution be calculated follows (Malinin & Gales 2018)
H(p(u/'" D.n)]] = [ p(ula=, Dia)) In p(pla"" Din)) SK
K (aa - -
(14)
> IaT()))) InT((a))
e=l
Note that, de function of x". and y denotes the Gamma digamma functions respectively.
15


Reviews:
[{"review_id": "rkeS5u3TYH", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This work studies the predictive uncertainty issue of deep learning models. In particular, this work focuses on the distributional uncertainty which is caused by distributional mismatch between training and test examples. The proposed method is developed based on the existing work called Dirichlet Prior Network (DPN). It aims to address the issue of DPN that its loss function is complicated and makes the optimization difficult. Instead, this paper proposes a new loss function for DPN, which consists of the commonly used cross-entropy loss term and a regularization term. Two loss functions are respectively defined over in-domain training examples and out-of-distribution (OOD) training examples. The final objective function is a weighted combination of the two loss functions. Experimental study is conducted on one synthetic dataset and two image datasets (CIFAR-10 and CIFAR-100) to demonstrate the properties of the proposed method and compare its performance with the relevant ones in the literature. The issue researched in this work is of significance because understanding the predictive uncertainty of a deep learning model has its both theoretical and practical value. The motivation, research issues and the proposed method are overall clearly presented. \n\nThe current recommendation is Weak Reject because the experimental study is not convincing or comprehensive enough. \n\n1.\tAlthough the goal of this work is to deal with the inefficiency issue of the objective function of existing DPN with the newly proposed one, this experimental study does not seem to conduct sufficient experiments to demonstrate the advantages (say, in terms of training efficiency & the capability in making the network scalable for more challenging dataset) of the proposed objective function over the existing one; \n2.\tTable 1 compares the proposed method with ODIN. However, as indicated in this work, ODIN is trained with in-domain examples only. Is this comparison fair? Actually, ODIN's setting seems to be more practical and more challenging than the setting used by the propose methods. \n3.\tThe evaluation criteria shall be better explained at the beginning of the experiment, especially how they can be collectively used to verify that the proposed method can better distinguish distributional uncertainty from other uncertainty types.\n4.\tIn addition, the experimental study can be clearer on the training and test splits. How many samples from CIFAR-10 and CIFAR-100 are used for training and test purpose, respectively? Also, since training examples are from CIFAR-10 and CIFAR-100 and the test examples are also from these two datasets, does this contradict with the motivation of “distributional mismatch between training and test examples” mentioned in the abstract?  \n5.\tThe experimental study can have more comparison on challenging datasets with more classes since it is indicated that DPN has difficulty in dealing with a large number of classes. \n\nMinor:\n\n1. Please define the \\hat\\theta in Eq.(5). Also, is the dirac delta estimation a good enough approximation here?\n2. The \\lambda_{out} < \\lambda_{in} in Eq.(11) needs to be better explained. In particular, are the first terms in Eq.(10) and Eq.(11) comparable in terms of magnitude? Otherwise,  \\lambda_{out} < \\lambda_{in} may not make sense.\n3. The novelty and significance of fine-tuning the proposed model with noisy OOD training images can be better justified.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "SkgcPvqAYB", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes an improved DPN framework with a novel loss function, which uses the standard cross-entropy loss along with a regularization term to control the sharpness of the output Dirichlet distributions from the network.The proposed loss function aims to improve the training efficiency of the DPN framework for challenging classification tasks with large number of classes \n\nThe proposed improved DPN is very incremental. It only adds a simple regularization term to the standard cross-entropy loss. The regularization term is the precision of the Dirichlet from the DPN. The technical novelty and contribution is not significant. \n\nThe paper claims the proposed loss function allows distributional uncertainty to be modelled separately from data uncertainty and model uncertainty, and the proposed framework can improve efficiency. However these claims lack of sufficient support. \n\nMoreover, determining the source of uncertainty is just a mean of achieving better classification model, not the goal. The experimental results are not very convincing in improving classification over OOD examples due to the lack of comparison with state-of-the-art related works. Note many domain adaptation methods can handle OOD examples. ", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I made a quick assessment of this paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory."}, "extra_text": {}}, {"review_id": "HylRSrBc9B", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a novel loss function using the standard cross-entropy loss along with a regularization term on logits for training the Dirichlet prior network. The benefit of using Dirichlet prior network is that it can distinguish the in-domain noisy data and completely out-of-domain data. For in-domain noisy data, the Dirichlet distribution should be sharp but in the middle of the simplex, while for the OOD data, the Dirichlet distribution should be flat. The Dirichlet prior network is proposed by Malinin & Gales (2018), the new method in this paper overcomes the challenge of training the network based on the KL divergence which cannot work well for dataset with large number of classes. The paper is well written and easy to follow. Here are some comments and questions: \n\n- In the proposed loss function, could you explain what is the reason that you choose to use sigmoid(z_c(x)) instead of \\sum exp(z_c(x))? As you mentioned in the paper,  \\sum exp(z_c(x)) suggests the sharpness of the distribution. Shouldn’t using \\sum exp(z_c(x)) be more direct than the sigmoid(z_c(x))?\n\n- The methods requires OOD dataset for training. The authors used the same OOD dataset for training and test. One concern is that what if the OOD dataset for test is not available at training. What is the alternative plan? How does that perform?\n\n- In Table 1, for Gaussian in-domain dataset, why is \\sum exp(z_c(x)) able to distinguish Gaussian in-domain from original in-domain images? If I understand correctly, Gusaain in-domain should have a sharp distribution which means the differential entropy (D. Ent) is small (as illustrated in Figure 2(d)), and the sum of the exponential of logits (\\sum exp(z_c(x))) should be large. For the original in-domain, it should also have a sharp distribution with small differential entropy and large sum of the exponential of logits. Then I don’t understand why in the table, the AUROC and AUPR for the measure \\sum exp(z_c(x) are very high values while that for the measure D. Ent are very low.\n\n- Could you also compute the sum of the exponential of logits for the synthetic data, since it is the only metric that is evaluated in the real data experiments but not in the synthetic data?\n\n- Could you clarify how you compute the differential entropy of the Dirichlet distribution given an input x? Did you use \\alpha_c = exp(z_c(x))? \n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}]

Author rebuttal:

Meta-review: In this work the authors build on the Dirichlet prior network of Malinin & Gales, replacing the loss function and adding a regularization term which improve training in the setting with a significant number of classes.   Improving uncertainty for deep learning is a challenging but very important problem.  The reviewers of this paper gave two weak rejects (one is of low confidence) and one weak accept.  They found the paper well written, easy to follow and well motivated but somewhat incremental and not entirely empirically justified.  None of the reviewers were willing to strongly champion the paper for acceptance.  Unfortunately as such the paper falls below the bar for acceptance.  It appears that the authors significantly added to the experiments in the discussion phase and hopefully that will make the paper much stronger for a future submission.
Decision: Reject