Exploring Covariate and Concept Shift for Detection and Confidence Calibration of Out-of-Distribution Data

1 INTRODUCTION . Out-of-distribution ( OOD ) detection is a fundamental research area important for downstream tasks with open-world assumptions such as continual learning ( Smith et al. , 2021 ) , open-set learning ( Yu et al. , 2020 ) and safety-critical applications such as self-driving ( Bojarski et al. , 2016 ) . However , common OOD detection benchmarks1 are not well categorized . This prevents us from more systematically studying of OOD detection . A recent attempt ( Winkens et al. , 2020 ; Fort et al. , 2021 ) to provide a more granular characterization of OOD data introduces the concept of near OOD ( Winkens et al. , 2020 ; Fort et al. , 2021 ) . Near OOD detection is posed as a more challenging problem than far OOD detection because near OOD datasets usually share similar semantics , and style , e.g. , all natural images of similar environment , as the training dataset . Existing works characterize OOD in terms of difficulty of OOD detection such as Confusion Log Probability ( CLP ) ( Winkens et al. , 2020 ) . However , this characterization only conveys difficulty of OOD detection across a single dimension and does not expose intrinsic characteristics of OOD data . In this paper we provide a more systematic categorization of OOD data , which motivates a more robust OOD detection method and helps us reflect on some existing approaches . Specifically , there are at least two dimensions along which we can characterize the spectrum of OOD data . Intuitively , out-of-distribution data refers to data that are sampled from distributions different from the training distribution . This is well known in machine learning as distribution shift ( Moreno-Torres et al. , 2012 ) . There are two dominant shift types : covariate shift2 and concept shift . The former usually 1Texturess ( Cimpoi et al. , 2014 ) , SVHN ( Netzer et al. , 2011 ) , Place365 ( Zhou et al. , 2017 ) , LSUNCrop/Resize ( Yu et al. , 2015 ) , and iSUN ( Xu et al. , 2015 ) 2Covariate shift is often associated with model calibration ( Ovadia et al. , 2019 ; Chan et al. , 2020 ) refers to change in style , e.g. , clean to noised and natural images to cartoon , and the latter refers to change in semantics , e.g. , dog to cat . Even though existing works Hsu et al . ( 2020 ) have acknowledged the different types of OOD datasets and the difficulty of detecting them , their methods do not distinguish them and do not explicitly disentangle them . To study these two distribution shifts , we propose to create a benchmark with multiple magnitude of distribution shift in each dimension . Specifically , we use corrupted CIFAR10C/CIFAR100C ( Hendrycks & Dietterich , 2019 ) , originally developed for testing robustness , with varying degrees of severity to represent increasing covariate shift ; we also introduce a new benchmark , CIFAR100 Splits , which divides the CIFAR100 dataset into 10 splits with increasing conceptual shift from CIFAR10 classes , measured by semantic similarity in a word embedding space ( Pennington et al. , 2014 ) . We show that the increasing conceptual difference measured by semantic similarity in language translates to a spectrum of OOD datasets under gradual concept shift measured by the difficulty of OOD detection in the image space . To address these two distribution shifts , we need to approach it from two aspects : representation and modeling , analogous to the role of entropy and Bayesian inference in uncertainty quantification ( Depeweg et al. , 2018 ) where entropy is a score function that represents uncertainty and Bayesian inference is a method that models uncertainty ( Hüllermeier & Waegeman , 2021 ) . To represent these two shifts , we can derive score functions which output scalars indicating the severity of distribution shift . Specifically we derive two score functions by expanding and decomposing the KL divergence between a predictive distribution from a classifier and a discrete uniform distribution . Arising from the decomposition is a covariate shift score which is a function of feature norms and a concept shift score which is a function of feature angles . To model these two shifts , and motivated by the covariate score ( a function of norms ) and the concept score ( a function of angles ) , we propose to directly improve the sensitivity of norms and angles to distribution shift through a geometric perspective ( Tian et al. , 2021 ) . Specifically , we adopt the Geometric Sensitivity Decomposition proposed in Tian et al . ( 2021 ) , originally developed for model calibration , which decomposes norms and angles into a variance component and a scalar offset , and improve it by parametrizng the scalar offsets as standalone networks . Our approach improves detection of OOD data under both covariate and concept shifts . While OOD detection and calibration3 has been studied separately in the literature , they share the same underlying motivation : confidence/uncertainty should be low/high when distribution shift occurs . Our method to improve OOD detection naturally yields a powerful calibration function in the family of intra class-preserving functions ( Rahimi et al. , 2020 ) . This class of functions provides enough expressive power to calibrate complex decision boundaries in neural networks and proves to be effective in calibration on in-distribution data . Therefore , we apply a post-calibration ( Guo et al. , 2017 ) method on a validation set as described in those works . Additionally , thanks to the improved sensitivity to distribution shift , our model also is on par with state-of-the-art calibration performance on distribution shifted data as well . To summarize our contributions : • We propose to characterize the spectrum of OOD data in terms of covariate and concept shift and unify the notion of near OOD in different literature ( Sec . 3.1 ) . Additionally , we construct a new benchmark ( e.g. , CIFAR100 Splits ) to represent gradual distribution shift in each direction ( Sec . 4.1.2 and 4.1.3 ) . • To represent distribution shifts , we analytically derive two OOD score functions based on KL-divergence to capture both shifts more effectively ( Sec . 3.2 ) . • To model distribution shifts , we propose Geometric ODIN to improve the sensitivity of neural networks during training ( Sec . 3.3 ) and calibration during inference ( Sec . 3.4 ) . This method achieves state-of-the-art performance on both detection ( Sec . 4.1 ) and calibration ( Sec . 4.2 ) of OOD data . 2 RELATED WORK . Out-of-Distribution ( OOD ) detection methods can be largely divided into two camps depending on whether they require OOD data during training . Hendrycks et al . ( 2018 ) ; Thulasidasan et al . ( 2020 ) ; Roy et al . ( 2021 ) leverages anomalous data in training . Our method belongs to the class of methods that do not assume the availability of OOD data during training . Hendrycks & Gimpel ( 2016 ) uses the maximum softmax probability ( MSP ) to detect incorrect predictions and OOD data . Lee et al . 3OOD detection focuses on concept-shifted data while calibration focuses on covariate-shifted data . ( 2018 ) proposes to use Mahalanobis distance by fitting a Gaussian mixture model ( GMM ) in the feature space . Mukhoti et al . ( 2021 ) uses log density of the GMM model instead . Liu et al . ( 2020b ) uses an energy score as the uncertainty metric . ODIN ( Liang et al. , 2017 ) uses a combination of input processing and post-training tuning to improve OOD detection performance . Generalized ODIN Hsu et al . ( 2020 ) ( also Techapanurak et al . ( 2019 ) ) includes an additional network in the last layer to improve OOD detection during training . There are other interesting OOD detection approaches without OOD data such as using contrastive learning with various transformations ( Winkens et al. , 2020 ; Tack et al. , 2020 ) , training a deep ensemble of multiple models ( Lakshminarayanan et al. , 2016 ) and leveraging large pretrained models ( Fort et al. , 2021 ) . They require extended training time , hyperparameter tuning and careful selections of transformations , whereas our method does not introduce any hyperparameters and has negligible influence on standard cross-entropy training time . Model Calibration methods can also be largely divided in two categories : 1 ) training time calibration using augmentations ( Thulasidasan et al. , 2019 ; Jang et al. , 2021 ) , using modified losses ( Kumar et al. , 2018 ) ; 2 ) post-hoc calibration ( Guo et al. , 2017 ; Kull et al. , 2019 ; Kumar et al. , 2019 ; Rahimi et al. , 2020 ) . Recently , Rahimi et al . ( 2020 ) formally generalizes a family of expressive functions for calibration , the intra order-preserving functions . This class of functions has more representation power to calibrate more complex decision boundaries in neural networks . Our proposed method belongs to this class of functions . However , all these works focus on calibration of in-distribution data . To obtain better calibration on OOD data , on which the confidence of a model needs to decrease accordingly , sensitivity and deterministic uncertainty modeling is explored by SNGP ( Van Amersfoort et al. , 2020 ) and DUQ ( Liu et al. , 2020a ) . Our proposed model not only belongs to the family of intra order-preserving functions , which ensures good in-distribution calibration , but also improves sensitivity to distribution shift , which improves out-of-distribution calibration simultaneously . Please refer to Appendix 6.8 for a more detailed discussion on related works . 3 METHOD . 3.1 MOTIVATION : COVARIATE AND CONCEPT SHIFT . We follow Moreno-Torres et al . ( 2012 ) for the formal definition of distribution shift , covariate shift and concept shift . LetX ∈ RD denote the covariate which is the input and Y ∈ R denote the output label . Distribution shift happens when the training joint distribution is not equal to the testing joint distribution Ptr ( X , Y ) 6= Ptst ( X , Y ) . Covariate shift appears when Ptr ( Y |X ) = Ptst ( Y |X ) and Ptr ( X ) 6= Ptst ( X ) . Concept shift appears when Ptr ( Y |X ) 6= Ptst ( Y |X ) and Ptr ( X ) = Ptst ( X ) . However , in the image domain , it is rare that such concept shift occurs without changes in P ( X ) . We therefore modify the equality in concept shfit to Ptr ( X ) ≈ Ptst ( X ) where superficial , low-level statistics may be retained . Specifically , covariate shift happens when the testing data is non-semantically different from the training data and concept shift happens when the testing data is semantically different from the training data . We illustrate a spectrum of OOD dataset in Fig . 1 . In this example , CIFAR10 is the training dataset . CIFAR100 represents concept shift because CIFAR100 has a non-overlapping label space , i.e. , Ptr ( Y |X ) 6= Ptst ( Y |X ) , but similar style with CIFAR10 . The corrupted CIFAR10C dataset ( Hendrycks & Dietterich , 2019 ) represents covariate shift because it has the same labels but different style , i.e. , Ptr ( X ) 6= Ptst ( X ) , compared to CIFAR10 . SVHN represents both covariate and concept shift due to its non-overlapping label space and very different style . As shown in Fig . 2 , the first goal of this paper is to derive score functions to represent the shift in either P ( X ) or P ( Y |X ) . We denote the score function that reflects change from Ptr ( X ) to Ptst ( X ) as the covariate shift score function : g ( x ) : RD → R and the score function that reflects change from Ptr ( Y |X ) to Ptst ( Y |X ) as the concept shift score function : h ( y , x ) : Y × RD → R. The second goal is to improve the sensitivity of these scores to their corresponding distribution shift . Specifically , we propose a geometrically inspired out-of-distribution detection method with only indistribution data ( Geometric ODIN ) . The third goal is to show that , unlike prior works which treat OOD detection ( Mukhoti et al. , 2021 ) and calibration ( Van Amersfoort et al. , 2020 ; Liu et al. , 2020a ) separately , our proposed Geomeric ODIN naturally leads to an expressive calibration function in the family of intra order-preserving functions ( Rahimi et al. , 2020 ) , which ensures good calibration .

Summary: This paper proposes a method for detecting two types of distributional shifts: covariate shifts in the input space $\mathcal{X}$ (due to input corruption) and semantic shifts (due to test data falling outside the support set of ID classes, $ y_\text{test} \notin \mathcal{Y}$). The idea is based on the decomposition of KL-divergence between softmax prediction and a uniform vector. Furthermore, the authors propose Geometric ODIN to improve OOD detection and calibration, outperforming strong baseline on CIFAR10, CIFAR 100, and SVHN datasets.