Are all outliers alike?  On Understanding the Diversity of Outliers for Detecting OODs

1 INTRODUCTION . Deep neural networks ( DNNs ) have achieved remarkable performance-levels in many areas such as computer vision ( Gkioxari et al. , 2015 ) , speech recognition ( Hannun et al. , 2014 ) , and text analysis ( Majumder et al. , 2017 ) . But their deployment in the safety-critical systems such as self-driving vehicles ( Bojarski et al. , 2016 ) , aircraft collision avoidance ( Julian & Kochenderfer , 2017 ) , and medical diagnoses ( De Fauw et al. , 2018 ) is hindered by their brittleness . One major challenge is the inability of DNNs to be self-aware of when new inputs are outside the training distribution and likely to produce incorrect predictions . It has been widely reported in literature ( Guo et al. , 2017a ; Hendrycks & Gimpel , 2016 ) that deep neural networks exhibit overconfident incorrect predictions on inputs which are outside the training distribution . The responsible deployment of deep neural network models in high-assurance applications necessitates detection of out-of-distribution ( OOD ) data so that DNNs can abstain from making decisions on those . Recent approaches for OOD detection consider different statistical , geometric or topological signatures in data that differentiate OODs from the training distribution . For example , the changes in the softmax scores due to input perturbations and temperature scaling have been used to detect OODs ( Hendrycks & Gimpel , 2016 ; Liang et al. , 2017 ; Guo et al. , 2017b ) . Papernot & McDaniel ( 2018 ) use the conformance among the labels of the nearest neighbors while Tack et al . ( 2020 ) use cosine similarity ( modulated by the norm of the feature vector ) to the nearest training sample for the detection of OODs . Lee et al . ( 2018 ) consider the Mahalanobis distance of an input from the in-distribution data to detect OODs . Several other metrics such as reconstruction error ( An & Cho , 2015 ) , likelihood-ratio between the in-distribution and OOD samples ( Ren et al. , 2019 ) , trust scores ( ratio of the distance to the nearest class different from the predicted class and the distance to the predicted class ) ( Jiang et al. , 2018 ) , density function ( Liu et al. , 2020 ; Hendrycks et al. , 2019a ) , probability distribution of the softmax scores ( Lee et al. , 2017 ; Hendrycks et al. , 2019b ; Tack et al. , 2020 ; Hendrycks et al. , 2019a ) have also been used to detect OODs . All these methods attempt to develop a uniform approach with a single signature to detect all OODs accompanied by empirical evaluations that use datasets such as CIFAR10 as in-distribution data and other datasets such as SVHN as OOD . Our study shows that OODs can be of diverse types with different defining characteristics . Consequently , an integrated approach that takes into account the diversity of these outliers is needed for effective OOD detection . We make the following three contributions in this paper : • Taxonomy of OODs . We define a taxonomy of OOD samples that classify OODs into different types based on aleatoric vs epistemic uncertainty ( Hüllermeier & Waegeman , 2019 ) , distance from the predicted class vs the distance from the tied training distribution , and uncertainty in the principal components vs uncertainty in non-principal components with low variance . • Incompleteness of existing uniform OOD detection approaches . We examine the limitations of the state-of-the-art approaches to detect various types of OOD samples . We observe that not all outliers are alike and existing approaches fail to detect particular types of OODs . We use a toy dataset comprising two halfmoons as two different classes to demonstrate these limitations . • An integrated OOD detection approach . We propose an integrated approach that can detect different types of OOD inputs . We demonstrate the effectiveness of our approach on several benchmarks , and compare against state-of-the-art OOD detection approaches such as the ODIN ( Liang et al. , 2017 ) and Mahalanobis distance method ( Lee et al. , 2018 ) . 2 OOD TAXONOMY AND EXISTING DETECTION METHODS . DNNs predict the class of a new input based on the classification boundaries learned from the samples of the training distribution . Aleatory uncertainty is high for inputs which are close to the classification boundaries , and epistemic uncertainty is high when the input is far from the learned distributions of all classes ( Hora , 1996 ; Hüllermeier & Waegeman , 2019 ) . Given the predicted class of a DNN model on a given input , we can observe the distance of the input from the distribution of this particular class and identify it as an OOD if this distance is high . We use this top-down inference approach to detect this type of OODs which are characterized by an inconsistency in model ’ s prediction and input ’ s distance from the distribution of the predicted class . Further , typical inputs to DNNs are high-dimensional and can be decomposed into principal and non-principal components based on the direction of high variation ; this yields another dimension for classification of OODs . We , thus , categorize an OOD using the following three criteria . 1 . Is the OOD associated with higher epistemic or aleatoric uncertainty , i.e. , is the input away from in-distribution data or can it be confused between multiple classes ? 2 . Is the epistemic uncertainty of an OOD sample unconditional or is it conditioned on the class predicted by the DNN model ? 3 . Is the OOD an outlier due to unusually high deviation in the principal components of the data or due to small deviation in the non-principal ( and hence , statistically invariant ) components ? Figure 1 demonstrates different types of OODs which differ along these criteria . Type 1 OODs have high epistemic uncertainty and are away from the indistribution data . Type 2 OODs have high epistemic uncertainty with respect to each of the 3 classes even though approximating all in-distribution ( ID ) data using a single Guassian distribution will miss these outliers . Type 3 OODs have high aleatoric uncertainty as they are close to the decision boundary between class 0 and class 1 . Type 4 and 5 have high epistemic uncertainty with respect to their closest classes . While Type 4 OODs are far from the distribution along the principal axis , Type 5 OODs vary along a relatively invariant axis where even a small deviation indicates that the sample is an OOD . Limitations of Existing Detection Methods . We empirically demonstrate the limitations of existing OOD detection methods on a two-dimensional ( 2D ) half-moon dataset with two classes . As shown in Figure 2 , we consider three clusters of OOD samples : cluster A ( black ) , B ( brown ) and C ( red ) . Figure 2 ( right ) shows the 2D penultimate features of the classifier . Different approaches differ in their ability to detect different OOD types as illustrated in Figure 3 . • Figure 3 ( a ) shows that the Mahalanobis distance ( Lee et al. , 2018 ) from the mean and tied covariance of all the training data in the feature space can not detect OODs in the clusters B and C corresponding to class-conditional epistemic uncertainty and aleatoric uncertainty , respectively . It attains the overall true negative rate ( TNR ) of 39.09 % at the 95 % true positive rate ( TPR ) . • Figure 3 ( b ) shows that the softmax prediction probability ( SPB ) ( Hendrycks & Gimpel , 2016 ) can not detect the OODs in cluster A corresponding to high epsitemic uncertainty . The TNR ( at 95 % TPR ) reported by the SPB technique is 60.91 % . • Figure 3 ( c ) shows that class-wise Principal Component Analysis ( PCA ) ( Hoffmann , 2007 ) can not detect OODs in cluster C corresponding to high aleatoric uncertainty . We performed PCA of the two classes separately in the feature space and used the minimum reconstruction error to detect OODs . This obtained overall TNR of 80.91 % ( at 95 % TPR ) . • Figure 3 ( d ) shows that K-Nearest Neighbor ( kNN ) ( Papernot & McDaniel , 2018 ) nonconformance in the labels of the nearest neighbors can not detect OODs in clusters A and B with high epistemic uncertainty . The overall TNR ( at 95 % TPR ) reported by this technique is 15 % . These observations can be explained by the focus of different detection techniques on measuring different forms of uncertainty . This motivates our integrated OOD detection method . 3 INTEGRATED OOD DETECTION METHOD . Complementary information about different OOD types can be used to detect a wider range of OODs . Figure 4 shows the improvement in the TNR of the OOD detector composed with information about different classes of OODs on the two half-moons dataset . Non-conformity in the labels of the nearest neighbors captures OODs in cluster C. Mahalanobis distance from the tied in-distribution detects OODs in cluster A. Reconstruction error from the PCA of the 2 class distributions captures OODs in cluster B. Softmax scores further strengthens the OOD detection by reporting OODs in cluster C that are undetected by the other three methods . The integrated OOD detection approach , thus , uses the following attributes , each specialized in detecting a specific type ( or a combination of types ) of OODs : 1 . Mahalanobis distance from the in-distribution density estimate that considers either tied ( Lee et al. , 2018 ) or class-wise covariance estimate . This attribute captures the overall or classconditional epistemic uncertainty of an OOD . Our refinement to also use class-wise covariance significantly improves detection of OODs when coupled with PCA approach described below . 2 . Conformance measure among the variance of the Annoy ( Bernhardsson , 2018 ) nearest neighbors calculated as the Mahalanobis distance of the input ’ s conformance to the closest class conformance . Our experiments found this to be very effective in capturing aleatoric uncertainty . This new attribute is a fusion of nearest-neighbor and Mahalanobis distance methods in literature . 3 . Prediction confidence of the classifier as the maximum softmax score on the perturbed input where the perturbation used is the same as ODIN approach ( Liang et al. , 2017 ) . This boosts the detection of high aleatoric uncertainty by sharpening the class-wise distributions . 4 . Reconstruction error using top 40 % of PCA components where the components are obtained via class conditional PCA of the training data . This boosts the detection of high class-wise epistemic uncertainty by eliminating irrelevant features . This fusion of attributes from existing state-of-the-art detection methods and new attributes was found to be the most effective integrated appraoch capable of detecting the different types of OODs . We evaluated it on several benchmarks as discussed in Section 4 with ablation study in Appendix . 4 EXPERIMENTAL RESULTS . Attributes forming the signature of the OOD detector used in the experiments The signature of the OOD detector used in the experiments is the weighted sum of four attributes , one from each of the following four categories : 1 . Distance from the in-distribution density estimate : We use mahalanobis distance of the input with respect to the closest class conditional distribution . The parameters of this distance are chosen from one of the following two categories : • empirical class means and tied empirical covariance of training samples • empirical class means and empirical class covariance of training samples 2 . Reconstruction error : We perform class conditional PCA empirically from the training samples . We use minimum reconstruction error of the input from the top 40 % eigen vectors of the class conditional eigen spaces . 3 . Prediction confidence of the classifier : We use maximum value of the temperature scaled softmax scores ( S ) on the perturbed input . Perturbations to the input ( x ) are made according to the following equation ( Liang et al. , 2017 ) x̃ = x− sign ( −∇xlogSŷ ( x ; T ) ) ( 1 ) The values of the magnitude of noise ( ) and the temperature scaling parameter ( T ) are chosen from one of the following three categories : • = 0 and T = 0 • = 0 and T = 10 • = 0.005 and T = 10 4 . Conformance measure among the nearest neighbors : We compute an m-dimensional feature vector to capture the conformance among the input ’ s nearest neighbors in the training samples , where m is the dimension of the input . We call this m-dimensional feature vector as the conformance vector . The conformance vector is calculated by taking the mean deviation along each dimension of the nearest neighbors from the input . We hypothesize that this deviation for the in-distribution samples would vary from the OODs due to aleatory uncertainty . The value of the conformance measure is calculated by computing mahalanobis distance of the input ’ s conformance vector to the closest class conformance distribution . Similar to the distance for the in-distribution density estimate , the parameters of this mahalanobis distance are chosen from the following two categories : • empirical class means and tied empirical covariance on the conformance vectors of the training samples • empirical class means and empirical class covariance on the conformance vectors of the training samples The value of the number of the nearest neighbors is chosen from the set { 10 , 20 , 30 , 40 , 50 } via validation . We used Annoy ( Approximate Nearest Neighbors Oh Yeah ) ( Bernhardsson , 2018 ) to compute the nearest neighbors . The weights of the four attributes forming the signature of the OOD detector are generated in the following manner . We use a small subset ( 1000 samples ) of both the in-distribution and the generated OOD data to train a binary classifier using the logistic loss . The OOD data used to train the classifier is generated by perturbing the in-distribution data using the Fast Gradient Sign attack ( FGSM ) ( Goodfellow et al. , 2014 ) . The trained classifier ( or OOD detector ) is then evaluated on the real OOD dataset at the True Positive Rate of 95 % . The best result , in terms of the highest TNR on the validation dataset ( from the training phase of the OOD detector ) , from the twelve combinations of the aforementioned sub-categories ( one from each of the four attributes ) are then reported on the test ( or real ) OOD datasets . Datasets and metrics . We evaluate the proposed integrated OOD detection on benchmarks such as CIFAR10 ( Krizhevsky et al. , 2009 ) and SVHN ( Netzer et al. , 2011 ) . We consider standard metrics ( Hendrycks & Gimpel , 2016 ; Liang et al. , 2017 ; Lee et al. , 2018 ) such as the true negative rate ( TNR ) at 95 % true positive rate ( TPR ) , the area under the receiver operating characteristic curve ( AUROC ) , area under precision recall curve ( AUPR ) , and the detection accuracy ( DTACC ) to evaluate our performance . DNN-based classifier architectures . To demonstrate that the proposed approach generalizes across various network architectures , we consider a wide range of DNN models such as , ResNet ( He et al. , 2016 ) , WideResNet ( Zagoruyko & Komodakis , 2016 ) , and DenseNet ( Huang et al. , 2017 ) . Comparison with the state-of-the-art . We compare our approach with the three state-of-the-art approaches : SPB ( Hendrycks & Gimpel , 2016 ) , ODIN ( Liang et al. , 2017 ) , and Mahalanobis ( Lee et al. , 2018 ) . For the ODIN method , the perturbation noise is chosen from the set { 0 , 0.0005 , 0.001 , 0.0014 , 0.002 , 0.0024 , 0.005 , 0.01 , 0.05 , 0.1 , 0.2 } , and the temperature T is chosen from the set { 1 , 10 , 100 , 1000 } . These values are chosen from the validation set of the adversarial samples of the in-distribution data generated by the FGSM attack . For the Mahalanobis method , we consider their best results obtained after feature ensemble and input preprocessing with the hyperparameters of their OOD detector tuned on the in-distribution and adversarial samples generated by the FGSM attack . The magnitude of the noise used in pre-processing of the inputs is chosen from the set { 0.0 , 0.01 , 0.005 , 0.002 , 0.0014 , 0.001 , 0.0005 } . CIFAR10 . With CIFAR10 as in-distribution , we consider SVHN ( Netzer et al. , 2011 ) , TinyImagenet ( Deng et al. , 2009 ) , and LSUN ( Yu et al. , 2015 ) as the OOD datasets . For CIFAR10 , we consider two DNNs : ResNet50 , and WideResNet . Table 1 shows the results . SVHN . With SVHN as in-distribution , we consider CIFAR10 , Imagenet , and LSUN and as the OOD datasets . For SVHN , we use the DenseNet classifier . Table 1 shows the results . Key observations . We do not consider pre-processing of the inputs in our integrated OOD detector . Even without input pre-processing and with the exception of CIFAR10 OOD dataset for SVHN indistribution trained on DenseNet , we could perform equally well ( and even out-perform in most of the cases ) as the Mahalanobis method on its best results generated after pre-processing the input . We also consider a Subset-CIFAR100 as OODs for CIFAR10 . Specifically , from the CIFAR100 classes , we select sea , road , bee , and butterfly as OODs which are visually similar to the ship , automobile , and bird classes in the CIFAR10 , respectively . Thus , there can be numerous OOD samples due to aleatoric and class-conditional epistemic uncertainty which makes the OOD detection challenging . Figure 5 shows the t-SNE ( Maaten & Hinton , 2008 ) plot of the penultimate features from the ResNet50 model trained on CIFAR10 . We show 4 examples of OODs ( 2 due to epistemic and 2 due to aleatoric uncertainty ) from Subset-CIFAR100 . These OODs were detected by our integrated approach but missed by the Mahalanobis approach . These observations justify the effectiveness of integrating multiple attributes to detect OOD samples . Additional experimental results in the appendix . We also compare the performance of the integrated OOD detector with the SPB , ODIN and Mahalanobis detector in supervised settings , as reported by the Mahalanobis method for OOD detection ( Lee et al. , 2018 ) . These results include experiments on CIFAR10 , SVHN and MNIST as in-distribution data and Imagenet , LSUN , SVHN ( for CIFAR10 ) , CIFAR10 ( for SVHN ) , KMNIST , and F-MNISTas OOD data across different DNN architectures such as ResNet34 , WideResNet , DenseNet , and LeNet5 . All these results , along with the ablation studies on OOD detectors with single attributes are included in the Appendix . In almost all of the reported results in the Appendix , our OOD detector could outperform the compared state-of-the-art methods with improvements of even 2X higher TNR at 95 % TPR in some cases .

Summary: The authors explore the different kinds of outliers and show that the methods previously proposed detect different kinds of OOD and not a single one can detect them all. The authors propose an interesting study of the different kind of outlier on synthetic data which  illustrates well the different characteristics of the outlier types. The authors then propose to combine different methods to increase the OOD detection rate. Experiments are conducted on 3 images classification datasets using different deep neural networks. For each dataset, samples from other databases are introduced as outliers and must be detected. The combination method yield better detection rates than baseline methods in almost all configurations.