Finding Competence Regions in Domain Generalization
Authors: Jens Müller
Stefan T. Radev
Robert Schmier
Felix Draxler
Carsten Rother
Ullrich Koethe
Venue: Accepted by TMLR ()

Abstract: We investigate a "learning to reject" framework to address the problem of silent failures in Domain Generalization (DG), where the test distribution differs from the training distribution. Assuming a mild distribution shift, we wish to accept out-of-distribution (OOD) data from a new domain whenever a model's estimated competence foresees trustworthy responses, instead of rejecting OOD data outright. Trustworthiness is then predicted via a proxy incompetence score that is tightly linked to the performance of a classifier. We present a comprehensive experimental evaluation of existing proxy scores as incompetence scores for classification and highlight the resulting trade-offs between rejection rate and accuracy gain. For comparability with prior work, we focus on standard DG benchmarks and consider the effect of measuring incompetence via different learned representations in a closed versus an open world setting. Our results suggest that increasing incompetence scores are indeed predictive of reduced accuracy, leading to significant improvements of the average accuracy below a suitable incompetence threshold. However, the scores are not yet good enough to allow for a favorable accuracy/rejection trade-off in all tested domains. Surprisingly, our results also indicate that classifiers optimized for DG robustness do not outperform a naive Empirical Risk Minimization (ERM) baseline in the competence region, that is, where test samples elicit low incompetence scores.

Published in Transactions on Machine Learning Research (06/2023)
Finding Competence Regions in Domain Generalization
Jens Müller
Informatics for Life, Heidelberg University, Germany
Stefan T. Radev
stefan.radevS3@@maai.
STRUCTURES Cluster of Excellence, Heidelberg University, Germany
Robert Schmier
robert. sobert.schmiirreeee bosch. com
Bosch Center for Artificial Intelligence, Renningen, Germany Heidelberg University, Germany
Felix Draxler
felis. drazler@kur wareeerreBrruuunbeiieeeeeeerrr
Heidelbery University, Germany
Carsten Rother Heidelberg University, Germany
Ullrich Köthe
wrr.ich.kotthcBBouuuumm--hhhhii- de
Heidelbery University, Germany
Reviewed on OpenReview: https: operrevie.. net/ ftee/fou forum? id= TSyOvuuQFN
Abstract
We investigate "learning to reject" framework to address the problem of silent failures in Domain Generalization (DG), where the test distribution differs from the training dis- tribution. Assuming mild distribution shift, we wish to accept out-of-distribution (OOD) daía from new domain whenever & model's estimated competence foresees trustworthy responses, instead of rejecting OOD data outright. Trustworthiness is then predicted via proxy incompetence score that is tightly linked to the performance of classifier. We present comprehensive experimental evaluation of existing proxy scores as incompetence scores for classification and highlight the resulting trade-offs between rejection rate and accuracy gain. For comparability with prior work, we focus on standard DG benchmarks and consider the effect of measuring incompetence via different learned representations in a closed versus an open world setting. Our results suggest that increasing incompetence scores are indeed predictive of reduced accuracy, leading to significant improvements of the average accuracy below suitable incompetence threshold. However, the scores are not yet good enough to allow for a favorable accuracy /rejecccrreeoooonn trade-off in all tested domains. Surprisingly, our results also indicate that classifiers optimized for DG robustness do not outperform a naive Empirical Risk Minimization (ERM) baseline in the competence region, that is, where test samples elicit low incompetence scores.
1 Introduction
Although modern deep learning methods exhibits excellent generalization, they are prone to silent failures when the actual data distribution differs from the distribution during training (Sanner et al., 2021; Yang et al., 2021). We address this problem in a "Icarning reject" framework (Flores, 1958; Hendrickx et al., 2021: Zhang et al., 2023): Given pre-trained model and potentially problematic data instances, can we determine if the model's 's responses still trustworthy?

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A major goal of this work is to explore the above question in settings where wish to make predictions a test set from new domain following potentially different distribution than the one available during training. This setting is referred to as Domain Generalization (DG) and assumes that we have access to multiple domains (also known as data sets or environments) during training. The generalization task asks to provide accurale predictions for a new domain, usually subject to a mild distribution shift (e.g., from one hospital to the next). Still, from a data-centric perspective, almost all instances in the new domain are out-of-distributiinn (OOD). Following the rationale of DG, we do not want to reject all OOD instances outright, but only those for which the estimated model competence falls below some acceptance threshold.
Response:
(accept) elephant
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Response: (reject) bird
Photo
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Domains
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Figure 1: The main principle behind incompetence scores for improved domain generalization: We reject instances above the incompetence threshold, which is located at the 95% quantile of the training distribution.
Since we do not have access to the distribution of the test data during training, we can neither determine out-of-domain competence directly (Xic al., 2006), nor define the acceptance threshold in Bayes-optimal way (Chow, 1970). Instead, we investigate proxy scores that are negatively correlated with competence: We call them incompetence scores and they should monotonically decrease as model's accuracy increases (see Section 3). For a simple example of such score, we may consider the distance of a new data point to the nearest neighbor of the training data in a model's lcarned feature space. In this case, we expect the per- formance to drop with increasing distance. Interestingly, our experiments demonstrale that the monotonicity property typically holds for well-known choices of these scores.
Setting threshold to delineate competence region inevitably results in a trade-off between accuracy and coverage: The more instances add to the competence region, the worse the accuracy, and vice versa. Identifying the optimal (task-dependent) trade-off in DG is difficult due to the differences between the training and the (unknown) test distributions. Thus, we find it pertinent to explore this trade-off for different thresholds across DG tasks (see Section 4.3).
The concept of incompetence underlying the present work strongly linked to previous rescarch on classi- fication with reject option (e.g., Bartlett & Wegkamp, 2008) and selective classification (e-g., Geifman & El- Yaniv, 2017). Common to these concepts is the idea to accurately predict errors based on proxy quantity. Since we are interested in the task-dependent competence of pre-trained classifiers, we concentrate solely on post-hoc OOD detection methods as proxy scores for incompetence. And although the current work focuses on classification tasks, our approach should also be worth pursuing in regression lasks, since feature-based incompetence scores seem equally applicable in this case.
In the following, we present comprehensive experimental evaluation of incompetence scores in variety of DG tasks. For comparability with prior work, we focus standard data sets from the DG literature
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(Gulrajani & Lopez-Paz, 2020) and consider the closed vs. open world setting (i.e., new appearances of known classes vs. hitherto unknown classes) as well as the effect of measuring incompetence through different data representations. Further, we investigate whether state-of- tfetttrt-rrtt classifiers that are optimized specifically for domain shift robustness exhibit more accurate competence regions than naively trained oncs. Finally, we investigate whether it possible to estimate an incompetence threshold, such that a classifier guaranteed recover its ID accuracy the corresponding competence region under domain shift. In summary, we make the following contributions:
1. We demonstrate empirically that accuracy decreases as incompetence scores increase and highlight the resulting trade-offs between rejection rate and accuracy gain (see Section 4.3)
We find that both feature- and logit-based scores are competitive in the closed world, whereas baature-based approaches work best in the open world setting (see Section 4.4 and Section 4.5)
3. We propose an approach to determine an incompetence threshold from TD data and demonstrate its utility for most domain shifts considered in this work (see Section 4.6)
4. We observe that robust classifiers do not outperform naive baseline in terms of generalization performance in the elicited competence regions (see Section 4.4)
Training Domains
Test Domain (Cartoons)
Photo
Sketch
Elephant
Classes
Horse
Porson
Lowest Scores
Highest Scores
Figure 2: An incompetence score is able to sort so that higher incompetence scores result in lower classification accuracy. (Left) Example images from the training domains. (Right) Images from the test domains resulting in lowest and highest incompetence scores (using a Deep-KNN scoring function) in the feature space of a baseline ERM classifier. Green and red frames denote correctly and incorrectly classified images, respectively. Higher incompetence scores correlate with a decrease in the classifier's accuracy.
(OOD) images from the PACS data set,
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Related Work
2.1 OOD Detection
Dealing with anomalous (i.e,, ant-of-distribution: OOD) instances that differ from those contained in the training set (i.c., our proxy for the in-distribution; ID) is widely discussed and conceptually overloaded topic in the machine and statistical learning literature (Aggarwal & Yu, 2001; Yang el al., 2021; Shen el al., 2021; Han et al., 2022; Yang et al., 2022). OOD detection addresses the problem flagging unusual data points which could undermine the reliability of machine learning systems (Yang et al., 2021); OOD generalization addresses the need make predictions even when the test distribution is completely unknown known be different than the training distribution (Shen et al., 2021).
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In this work, we are interested in analyzing established domain-robust classifiers Thus, we focus on OOD detection methods that do not modify the classifier architecture or training. Such methods are called post- hoc detection (Yang et al., 2021), as they do not intervene on the downstream classifier. In this work, we utilize established post-hoc methods that rely on various aspects of model output such as the softmax output (e.g. Hendrycks & Gimpel (2016)), logit outpul (e.g. Hendrycks et al. (2019); Liu el al. (2020)), or intermediate featuro-outputs (e.g. Aggarwal & Aggarwal (2017); Sun et al. (2022); Wang et al. (2022)).
Post-hoc OOD scores have been shown to perform well across a variety of OOD detection benchmarks (Yang et al., 2022). Previous work analyzed post-hoc OOD detection scores to predict the accuracy of classifier on novel inputs (Techapanurak & Okatani, 2021) or to detect ID failure cases (Xia & Bouganis, 2022). In addition, Techapanurak & Okatani (2021) compute an aggregated OOD-score over an entire ID data set to predict the global accuracy of a classifier on OOD data. Differently, we aim to predict the likelihood of error from individual incompetence score values and show that this approach provides us with a finer control over the trade-off between coverage and accuracy (sce Scction 4.6).
Despite the large volume of literature focusing OOD detection and generalization (e.g., Ruff et al. (2021)), there are no extensive studies applying OOD scores to domain generalization (DG) benchmarks. Thus, one of the main goals of this work was to provide such a comprehensive analysis on the utility of OOD scores for improving DG.
2.2 Domain Generalization
The goal of domain generalization (DG) is to train models that generalize well under covariate shifts (Muan- det et al., 2013; Zhou et al., 2022), such as adversarial attacks (Goodfellow et al., 2014) or style changes (Gatys et al., 2016), for which the label space remains unchanged during testing (Yang et al., 2021). In DG settings, we assume that we have access to different environments or data sets (e-g., art and sketch images) and the goal is to make good predictions in completely unknown environments (e.g., real world images).
Compared to Domain Adaptation (DA; Wang & Deng, 2018), where we have unlabeled data from the test domain, the DG problems assume that we have no knowledge about the lest domain(s). Consequently, it is not possible to train the algorithm using unlabeled test data as in self-training (Xiaojin, 2008). However, a recent study has demonstrated that a model can be effectively adjusted during test time (Chen et al., 2023). Moreover, it has been shown that classifiers can assign high likelihoods under domain shift even when they are plainly wrong, which makes it hard to detect failure cases (Nguyen et al., 2015; Nalisnick et al., 2019). Thus, proxy "incompetence" OOD scores appear to be good candidates for spotlighting such failures. However, to the best of our knowledge, there are no extensive studies which atlempt to quantify the competence of domain-robust models in the context of DG.
Many benchmark data sets in DG have been established, on which researchers can study generalization performance beyond a single training environment (Gulrajani & Lopez-Paz, 2020; Koh et al., 2021). In this work, we consider the main data sets contained in the DomainBed benchmark (Gulrajani & Lopez-Paz, 2020). We additionally distinguish between closed world sctting, where only instances of known classes are encountered in the test domain, and an open world setting, where instances of unknown classes are also present in the test domain. We believe the open world setting to be of practical interest, even though typical DG problems are formulated under a closed world assumption (Zhou et al., 2022).
Domain generalization approaches can roughly be classified into two categories: learning- based methods and augmentation-bast methods (Li et al., 2021). The current work primarily focuses on learning- bassing-basod approaches that seek to learn features that remain invariant under domain shift. According to Ben-David et al. (2006), there is theoretical evidence to suggest that features that remain invariant across domains enable accurate predictions in cases of distributional shifts. As result, various algorithms have been proposed with the goal of learning invariant features (Muandet et al., 2013; Arjovsky et al., 2019; Müller et al., 2021; Shi et al., 2021). However, it is not clear which DG methods can achieve consistently robust performance across different data sets. On the one hand, it has been suggested that a strong standard classifier trained with empirical risk estimation (ERM) performs favorably across multiple DG data sets (Gulrajani & Lopez-Paz, 2020: Korevaar et al., 2023). On the other hand, some DG methods have been shown to outperform an ERM baseliee on several benchmark data sets (Koh et al., 2021). Here, we complement the existing literature
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by examining whether the competence regions of different DG classifiers differ in terms of the achieved improvements in accuracy.
2.3 Selective Classification
Inference with reject option (aka selective classification, Geifman & El- El-Yaniv, 2017; E1- El-Yaniv et al., 2010) enables classifiers to refrain from making prediction under ambiguous or novel conditions (Hendrickx et al., 2021). The reject option has been extensively studied in statistical and machine learning (Hellman, 1970; Fumera. & Roli, 2002; Grandvalet et al., 2008; Wegkamp & Yuan, 2012). The origins of these approaches can be traced back until at least the 50s of the last century, as demonstrated by works such as Chow (1957; 1970); Hellman (1970). However, selective classification has only recently gained attention in the context of deep neural networks (Geifiman & EI- Yaniv, 2017).
Zhang et al. (2023) outline the three main reasons why a reject option could be reasonable choice in any practical application: 1) failure cases; 2) unknown cases; and 3) fake inputs. For instance, Kamath et al. (2020) train natural language processing (NLP) models for selective question answering under domain shift. Varshney et al. (2022) investigate the utility of MaxProb (a common OOD detection score) a rejection criterion across several NLP data sets. Ren et al. (2022) use the Mahalanobis distance as OOD detection method to filter inputs to NLP models for conditional text generation and Mesquita et al. (2016) showcase the reject option for catching software defects.
The main challenge selective classifiers face is how to reduce the error rate by "rejecting" instances for which reliable prediction can be made, while keeping coverage (i.e., the number of "accepted" instances) high possible (Condessa et al., 2017; Nadeem et al., 2009; Chow, 1970). And while the theoretical characteristics of the resulting tradc-off have been systematically studied (EI-Yaniv et al., 2010; Wiener & El-Yaniv, 2011), the empirical utility of OOD "rejection scores" for ensuring robust performance in the DG setting remains largely unclear. In this work, we perform an extensive evaluation of this trade-off across a wide variety of thea-- OOD scores, domain-robust classifiers, DG data and environments.
3 Method
We denote with arbitrary classifier with vector of trainable parameters (e.g., neural network weights) which we typically suppress for readability. To evaluate a classifier, we consider its accuracy, which we denote as Adist, based on inference queries from some reference distribution x ~ Pdist (x).
3.1 Incompetence Scores
The goal of an incompetence score se: RP R is to indicate whether classifier c familiar with some input x E X. We consider familiarity with the input to be equivalent to competence. The fundamental principle of this work is that instances cliciting a high incompetence score are intrinsically hard to predict and vice versa. Due to the close conceptual connection between competence and familiarity or incompetence and OOD, we employ OOD scores as proxy for incompetence. In particular, we employ post-hoe methods that compute an OOD score taking into account the classifier.
In our subsequent experiments, we compute the incompetence scores via number of post-hoe methods. The post-hoc methods used in this paper can be grouped into the following categories:
Feature-based: Virtual-logit Matching (ViM; Wang et al., 2022), Deep-KNN (Deep-KNN; Sun al., 2022);
Density-basedd Gaussian mixture models (GMM), minimum Mahalanobis distance between features and class-wise centroids (Mahalanobis; Lee et al., 2018);
Reconstruuction-bsse.. reconstruction error of PCA in feature space (Aggarwal & Aggarwal, 2017);
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Logit-based: energy score (Energy; Liu et al., 2020), maximum logit (Logit; Hendrycks et al., 2019): maximum softmax (Softmax; Hendrycks & Gimpel, 2016), and energy-react (Energy-React; Sun et al., 2021).
Note, that we interpret higher scores indicative of incompetence (e.g., we consider the negative of the maximum softmax and the maximum logit).
3.2 Admissible Incompetence Scores
Detecting out-of-comppteecce means checking whether some given incompetence score (x(x) E R falls below some threshold a (classified as in-competence) or above (classified as out-of-conm competence). We consider scores sz(z) that depend on the classifier с and the input I at hand.
The threshold a trades off accuracy (how well does the classifier perform on accepted data) with coverage (how many samples does the score accept). In this section, we describe how a useful (ideal) incompetence score should affect downstream classification as a function of the threshold a. In particular, consider the subset of input space where the classifier is deemed competent given fixed threshold a:
Xa(a)= (x:: (c(a) S aa).
(1)
We use the ID) data to determine suitable threshold for the competence region, for instance we later pick a = a95 such that 95% of the ID data is in Xo(a9s). We consider the accuracy Aoo of the classifier on the unknown test domain restricted to the competence region X²(aa) as a function a.
We summarize the above description in the fundamental criterion of this work: An admissible incompe- tence score must assign low incompetence to those regions where the downstream accuracy is high. We formalize this as follows:
Criterion 3.1. An incompelence score sc(x) is called "admissible" if the downstream accuracy Aco is increased for any distribution that undergoes o mild covariate shift.
decreases monotonically
This monotonic trend requires that the incompetence score sa(x) is closely related to the performance of the classifier. Such connection allows to make predictions the downstream accuracy function of as Proposition 3.1. Given classifier co(x) and its corresponding in-distribution PID. Then, for test distri- bution of interest PooD and a corresponding admissible score sz(x) as in Criterion 3.1:
(a) If there is threshold a" R such that for all S a- ID and OOD have the same support and classifieation accuracy, then, Aoon(a) 2 And for < a².
(b) In the limit of + co, we find that AooD(a) + AOoD.
The first statement describes the behavior of (ao for small a and the second for large a. We observe this behavior empirically in Figure and give the proof Appendix A.7
4 Experiments
In our experiments, we analyze the effect of an incompetence threshold a (see Section 3.2) on DG perfor- 1 In the following, we first describe our experimental protocol. Then, we analyze the competence region in dependence on the threshold a and show that the competence region behaves as predicted in Proposition 3.1. Finally, carry out an extensive investigation of the competence region for closed and open world settings, where we show the utility of the concept for various incompetence scores and point out current weaknesses.
As an introductory example to the competence region, we consider Figure which depicts the experimental procedure on the PACS data set for standard classifier trained with Empirical Risk Minimization (ERM;
We provide access our code under https:// http://ggghthhhtte http:ttth.ce//tttviia///ttttttte wtw.eeancceettiiiiiiiaii.o

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Figure 3: The accuracy of the ERM classifier on OOD data (a) as the competence region is enlarged by increasing the incompetence threshold a. As predicted by our monotonicity criterion (Criterion 3.1), the accuracy starts off AooD(a) 2 AId and then falls off monotonically with a. At the same time, the fraction of data the classifier is applied to increases. The classifier accuracy and fraction of considered data can easily be traded off using this figure.
Vapnik, 1999). We train the classifier on the domains Art, Photo, and Sketch, and apply the trained classifier in the unknown Cartoon domain. The samples in the test domain are ordered by the predicted incompetence score se(x). As expectod, the classifier still performs well on Cartoon samples with low incompetence scores (9 out of classified correctly in the example), but the accuracy drops for high scores (only 2 out of correct classification). Qualitatively, the score correctly notices that images with significantly different characteristics are much harder to classify. In the following sections, we quantify this behavior eystematically for number of different classifiers, incompetence scores, data sets, and domain. But first, we give details on our experimental setup.
4.1 Experimental Setup
We consider all combinations of nine pre-trained classifiers ca(x), varying both in archilecture and training, nine OOD post-hoc scores se(x) as incompetence scores on a total of 32 DG tasks from six different DG data sets The pre-trained classifiers are obtained as follows. We train various stato-of-the-art classifiers from DG literature, namely Fish (Shi et al,, 2021), GroupDRO (Sagawa et al., 2019), SD (Pezeshki et al., 2021), SagNet (Nam et al., 2021), Mixup (Yan et al., 2020) and VREx (Krueger et al., 2021). Furthermore, we train three different neural network architectures with ResNet based architecture which we denote by ERM (He et al., 2016), a Vision Transformer (Dosovitskiy et al., 2020) and a Swin Transformer (Tiu et al., 2021). Training details and hyperparameter settings are listed in Appendix A.5.
miniimiiionnnnn (Vapnik, 1999): A
These models are trained on six domain generalization data sets from the DomainBed repository (Gulrajani & Lopez-Paz, 2020): PACS (Li et al., 2017), OfficeHome (Venkateswara et al., 2017), VLCS (Fang et al., 2013), TerraIncognita (Beery et al., 2018), DomainNet (Peng et al., 2019) and SVIRO (Cruz et al., 2020). Each DG data set consists of four to ten different domains from which we construct different DG tasks: We train a classifier on all but one domain. The one left out during training is then the OOD test domain where the competence region is evaluated. As an example consider the DG task behind the carlier example in Figure 2: If we train a model on the domains Photos, Art images, and Sketches, the DG task asks for an accurate model on the domain Cartoons which constitute the OOD test domain (see Figure 2). Overall we
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consider 32 DG tasks which result in 288 trained networks. We then compute the incompetence scores of each trained network In Section 3.1, we describe the process of calculating the incompetence scores.
For each DG task, we distinguish four data sets. For the ID distribution, we consider a training set, a validation set for hyperparameter optimization, and a test set that has no influence on the optimization process for the subsequent evaluation. The classifiers are trained on the ID training set. We compute the scores for the ID distribution on the ID validation set and the ID accuracy on the ID test set. The OOD apply all post-hoc methods to
test set is given by the DG task (e.g., as in Figure 2). After training, the penultimate feature layer the ouput (logits) layer of the classifier, as is typical in the OOD detection literature. If the post-hoe method needs to fit the data (as for instance with GMMs), we fit the score function on the ID training data.
4.2 Competence Threshold
In this section, we analyze the performance of the classifiers as a function of the threshold a which determines their competence region (see Equation in Section 3). To this end, we compute the incompetence scores on the ID validation data set and on all OOD data samples.
Figure 3 depicts the resulting score distributions and accuracy (a) as a function of the threshold a for a single classifier (ERM). Here, we consider four incompetence scores on one of the DG tasks provided by PACS and Terralncognita, respectively. We find that the considered incompetence scores fulfill the requirement for competence detector in Criterion 3.1 that the accuracy must decrease monotonically as the threshold a increases. We then find the theoretical results in Section 3 confirmed: For low a, the accuracy AooD(a) is high, and even exceeds the average accuracy on the ID data AId (see Proposition 3.1 (a)). It eventually decreases until (ao + AooD for large a (see Proposition 3.1b).
Figure 3 also depicts the fraction of ID and OOD data that is considered (i.e., not rejected) as we increase the incompetence threshold a:
Coverage dist (ist(a)) (a) = [Deist n Xa(a)] [Daist]
For instance, can compare the methods at the 0g5 which includes 95% of the ID data (vertical gray line in Figure 3). Here, Logit keeps a significantly larger fraction of test data compared to the other incompetence scores. However, this results in a lower accuracy in the competence region AOOD (aas).
Unfortunately, due to the nature of the DG problem, the accuracy curve (a) in Figure is not accessible during inference, which makes it difficult to choose a suitable threshold a. In Figure 3 we can observe that ViM and KNN achieve at the 95% percentile (w.r.t. the ID validation set) accuracy that is comparable to the ID accuracy, rendering the predictions in this competence region very accurate and trustworthy. GMM and Logit obtain very high accuracies in the competence region a larger drop in accuracy at the 95% percentile (w.r.t. the ID distribution). We show the accuracies Aood(a) for different threshold values a for all data sets and DG tasks in Appendix A.2.
for small values, but exhibit
4.3 Accuracy vs. Coverage Trade-Off
We illustrate the trade-off between accuracy and coverage for the VIM, GMM and Logit score for all domains in PACS and TerraTncognita Figure 4. Here, we consider the empirical average over all classifiers. All scores behave relatively monotonically in the sense that increased coverage results a roduction in accuracy.
First, it is evident that GMM (red curve) shows non-competitive accuracy-coverage trade-off. Further, while the Logit score (green curve) exhibits slightly favorable accuracy coverage trade-off across the PACS domain, a clear winner for TerraIncoguita does not emerge. Overerall, ViM (blue curve) performs better than the Logit score in terms of accuracy in the competence region elicited via a threshold al the 95th percentile of the score ID distribution. This indicates that the ViM score demonstrates greater selectivity and classifies more samples as outliers comparison to the Logit score. However, for all cases considered, we conclude that Logit and ViM exhibit a similar accuracy coverage trade-off. Note that the curves in Figure 4 are not accessible when we need to set the threshold.
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Figure 4: The expectation over the different domain robust classifier's accuracy on OOD data as a function of the coverage of the competence region. The 95th percentiles refer to the validation set of the ID distribution and also represent averages over all classífiers (and are thus sometimes off-curve). The domains are ordered in canonical sequence in Table in Appendix A.3.
4.4
Extensive Survey
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-14 [:31-0]
82 (67-96)
2 [0-4]
-10 [-27-13)
93 [62-98]
Mahalonobis
1 [0-12]
-8 1:22-4
50-96
10-7]
-17 [-2-0]
91 [75-95]
[-13]
-11
[-28-]] 14]
93 [73-99
GMM
[0-13]
-8
2-1-4]
550-16
[0-7]
-18 [-2--0]
92 [76-915
-12
[-28-14]
85 [63-90]
PCA
[-1110
-12 [21-3]
78 57-97
0 [0-7]
-18 [-2-0]
93 [78-96]
0 [-1-2]
-12
[-28-14)
88 (64-99)
Incognita ID-Gein
DomainNet
SVIRO
OOD-GGia
Coverage
OOD-Gain
ID-Gem
Coverage
GOD+Gain
ID-Gain
Coverage
Deep-KNN
[1251]]
[:36-4]
13:52
[0-6]
[-50-7
[70-93]
[-1--0
[5-64]
VIM
28 19-51
-13 [-55-3)
41 [7-57]
2 [0-7]
-8 [-50-6]
90 [68-97
[1-30]
[0-0]
19 [6-62]
Softmax
4 [1-12]
-38 [-54-21]
85 68-96
3 [0-5]
-8 (-52-5
94 [77-98]
4 [0-24
[-10-)
60 [28-88 [3-83
Logit
[1-18]
[-55-29]
600-08
[0-5]
-8 [-61-6]
93 [77-97
[0-21]
[-19-0]
67 [40-87
Energy
5 [1-19]
-33 [-55
55-96
[0-5]
-9 [-51-5]
94 [79-58]
[-2-21]
[-24-0]
67 [40-87]
Energy-Rese React
5 [1-19]
-33
[-55 21
85 55-98
2 [0-5]
-9 [-61-5]
93 [79-98]
2 [-2-21]
0 [-24-0]
67 [42-88]
Mahulonobis
[-1-18
-33
[-56- 11]
62 [7-94]
-1 [-3-5]
-11 [-53-4]
92 [77-97]
[-1-26]
[-19-0]
20 [5-95]
GMM
[-1-38
[7-84]
33-4
-12 [-33-3]
93 [79-98]
[-1-28)
[-11-0]
[5-67]
PCA
[-126
-38
[-53-21]
35-99
3-2
-12 [33-22]
92 (64-98]
[-1-15)
(-26-0
87 [47-99)
Table 1: Accuracy on competence region of OOD domain for different domain generalization data sets and
incompetence scores. As the threshold for the competence regions, we choose the 95% percentile of the ID validation set. For all metrics, a higher value means better performance (1). All displayed values are medians over different domain roles and classifiers, brackets indicate 90% confidence interval.
In the following, we evaluate all nine incompetence scores on all six DG data sets using the nine classifiers. Since each data set features 32 different DG tasks, we perform total of 32 9 2592 experiments. For each experiment, we obtain accuracy curves as in Figure as a function of a. To summarize and compare the performance of each score on cach data set, we need deal with the tradc-off between accuracy and coverage. Thus, we measure accuracy (agss the score agg, such that 95% of ID validation data fall below this threshold, that is FracIp(095) 95%. As mentioned in Section 4.2, choosing in DG is
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1.0
0.8
Aooo(a95)
0.6
0.4
0.2
ERM
Mixup
Fish
VREx
SD
SagNet
Group DRO
0.0
PACS
VLCS
OfficeHome Data set
Terralncognita
DomainNet
Figure 5: Accuracy in the competence region for different DG algorithms and OOD scores different data sets. The chosen threshold corresponds to the 95th percentile of the ID-distribution scores. Note: VREx failed to converge with the hyperparameters used on the DomainNet data set.
notoriously difficult, since we have no access the test domain(s) during training. The following quantities provide useful summary statisties for comparing our results across all experiments:
OOD-Gain = AooD(ags) - AOoD: The performance gain in the OOD domain by considering only the data in the competence region (a(0gs).
2. ID-Gain= (oo) Am: Expresses the performance gap between the accuracy on OOD data in the competence region (a(sss and the accuracy on the entire ID data Anp-
3. Coverage CoveragCoop (09s) given by Equation 2: The proportion of OOD data that falls within the competence region.
For each quantity, a higher value indicates better performance (t). Note that the coverage of the competence region alone is not informative. A naive approach that includes all samples in the competence region would achieve the largest competence region but would fall short in terms of OOD-Gain ID-Gain.
Table summarizes the results from extensive sweep over classifiers, data sets, and incompetence scores. The displayed values are the medians over different domain roles and classifiers, Overall, we observe that in the competence region, higher accuracy is achieved compared to the naive application on all OOD data instances. This confirms that incompetence score and accuracy are indoed tightly linked. However, for most DG data sets and incompetence scores, we are not able to replicate the ID accuracy. This indicates that we cannot naively expect the classifier attain the same accuracy as observed in the ID distribution in the 95% percentile ags. Further important findings arc:
In general, feature-based (Deep-KNN, ViM), well logit-based incompetence scores (Softmax, Logit, Energy, (nercy-Reacc) obtain significantly higher accuracy on OOD data (higher OOD-Gain) by filtering the data to the competence region (agg)) than the density- and reconstruction- approaches (Mahalanobis, GMM, PCA).
The feature-based scores achieve significant performance boost TerraIncognita. TerraInco contains DG tasks that suffer from particularly huge drop in accuracy from ID to OOD distribution (see Appendix A.6).
The proportion of OOD data that falls inside the competence region (i.e., coverage) smallest for feakure-based methods, but they also provide the highest accuracy across all DG data sets.
It important to note that the specific threshold investigated, the accuracy in the competence region remains unaffected by the DG algorithms (see Figure 5). Based on this observation, the incremental utility of
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0.8
Softmax
Logit
VIM
Deep KNN
0.6
OOD Gain
0.4
0.2
0.0
0.0
0.05
0.1
0.15
0.2
0.25
Fraction of open world samples
Figure 6: Performance of Logit and Softmax scores (logit-based) against Deep-KNN and ViM caasture-based for an increasing fraction of open world data (unknown classes) in the test domain. The performance gain on the OOD data (OOD-Guin, higher is better) for the logit-based methods is less pronounced compared lo ViM and Deep-KNN.
DG algorithms specifically designed for the purpose of domain robustness becomes uncertain. If we assume that DG algorithms successfully achieve their primary goal of learning domain-invariant features, then we can speculate that they create a more valuable competence region. However, this is not in line with our observation in Figure 5: We observe that no domain-robust classífier is consistently and significantly more accurate the competence region than the simple baseline classifier (ERM). In Appendix A.6, we also show that the same result holds without restrictions on the competence region. Furthermore, we explore in Appendix A.2 thresholds close to the 95th percentile and demonstrated that the relative performance of the scores remains quite consistent.
4.5 Open World Performance
In this section, we study how different incompetence scores shape the competence region when instances of unknown classes present in the ID distribution. Accordingly, for each domain in PACS, VLCS, Office- Home, and TorraIncognita data sets, we create matching "open world domain containing only instances of unknown classes. In lotal, we create 16 open world domains. For example, if we evaluate model on the PACS Sketch domain, we create an open world domain containing only sketches of classes that are not in the PACS data set. We describe the procedure for creating the open world domains in detail in Appendix A.4. In the following, we restrict our analysis to the 16 domains for which an open world twin exists.
We enrich the existing test domains with 0%, 5%, 10%, 15%, 20%, and 25% instances with unknown classes. A good incompetence score should mark instances of unknown classes with a high value and therefore render them outside of the competence region Xe(ags). In this case, the OOD-Gain would increase more open world instances find their way into the test set. In Figure 6 we observe that this behavior is achieved particularly well for the ViM score. The Logit and Softmax scores are less successful in delineating unknown class instances from the competence region and therefore the OOD-Gain is less pronouneed.
Indeed, to test whether this observation holds statistically across all classifiers, we fit hierarchical lin- ear regression (Stephen & Anthony, 2002) on OOD Gain with Classifier, Percentage Open World, and Incompetence Score, well as their interactions fixed factors, together with Data Set and Test Domain random factors (to account for the fact that the same classifier is evaluated in multiple data sets and test domains). The statistical results confirm the general trends visible in Figure 6. First, we find significant main effects of Percentage Open World (i.e., overall OOD Gain increases with an increasing number of open world instances) and Incompetence Score (i.e., ViM and Deep-KNN achieve a higher overall OOD Gain). Importantly, the only significant interaction revealed by the hicrarchical regression model suggests that ViM able achieve the largest OOD Gain as the fraction of open world samples increases. Note, that the same trend is present for Deep-KNN, but it fails to achieve statistical significance due to its high
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0.4
Logit (transformed)
ViM (transformed)
0.2
ID-Gain
0.0
-0.2
-0.4
PACS
Home
VLCS
TerraIncognita
DomainNet
SVIRO
1.0
0.8
Coverage
0.6
0.4
0.2
0.0
PACS
Home
VLCS
TerraIncognita
DomainNet
SVIRO
Data Set
Figure 7: ID-Gain and Coverage for Logit and ViM if transformed as described in Section 4.6. The threshold is set such that the ID-Gain should be at least 0. Top row: ID-Gain for Logit and VIM due different data sets. Boltom row: Coverage for Logit and VIM across different DG data sets. Medians and quantiles for all boxplots are computed over different domain roles and classifiers. The threshold is set as the ID accuracy.
variability (see Figure 6). Moreover, none of the effects involving the factor Classifier turn out to be significant, predictors of OOD Gain, suggesting that the results are largely classifeerinddepnnden.
In the closed world setting, differences between logit- and feature-based scores are for most DG data sets small (see e.g. Table 1). However, we have shown that is very relevant in the setting where instances of unknown classes occur. In Appendix A.4 we show the open world behavior for all incompetence scores.
4.6 Estimating the Incompetence Threshold
Choosing the 95% percentile of the ID distribution as incompetence threshold can be considered as weighting the trade-off between accuracy and coverage towards coverage only 5% of ID data are rejected. We now seek a slightly different incompetence threshold which puts more weight on the accuracy. The question we want to address is whether we can set threshold such that certain accuracy achieved in the competence region? This question of high practical relevance, but also particularly challenging for reasons. First, many scores used so far have out-of-the-bor connection to the accuracy and second, we deal with domain shift that might result in completely new score-accuracy relationship.
Thus, as a potential remedy, we suggest. learning (x(x) PTD(c(á) + y se(x)) and using this conditional probability as a transformed score. This score represents the probability of an incorrect prediction given the original score. we define a competence region with an incompetence threshold of App we can expect an accuracy of least Anp on ID data. We hope that this relation also holds under domain shift. To predict Be(x)) = (ID(c(x) # y (x)) rely architecture that is constrained to be monotonic proposed in
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(Milee et al., 2021). Therefore, we do not change the order of the scores and equip the transformed score with an inductive bias that is consistent with Criterion 3.1. The transformed score also has predictable extrapolation behavior which is helpful when the distribution shifts. Note that since the transformation is monotonic, a threshold for the original score is also a valid threshold for the transformed score and vice versa. Therefore, we can also interpret this approach as estimating an incompetence threshold such that a certain accuracy is achievod.
Accordingly, Figure 7 depicts the ID-Gain and Coverage for ViM and Logit (transformed), if we select 1-Am as the incompetence threshold. The transformed ViM score suggests that achieve in most cases at least the ID accuracy, but at the cost of small coverage. The transformed Logit score has higher coverage, but it often fails to reproduce the ID accuracy (e.g., in the Terralncoguita data set). However, while we attain the ID accuracy for most cases, we still observe some failure cases, which makes the approach only tentative. Note, that these results also suggest. that the information contained the logits is not sufficient to give suitable competence regions in the sense of our question.
5 Conclusion
Accepting only predictions from the competence region of a classifier increases its accuracy dramatically under domain shift. Determining the fraction of samples where the classifier could be considered competent is question of how to approach the trade-off between accuracy and coverage. Addressing this trade-off via the and particularly challenging in the domain generalization
incompetence threshold is ooplication-dependdee (DG) setting where the test distribution differs from the training distribution per definition. Still, we showed that even in DG, it is possible to achieve higher than in-distribution accuracy under domain shift at the price of polentially diminished coverage (see Figure 2 or Appendix A.2).
Furthermore, we investigated a coverage-oriented threshold that would reject only pre-defined fraction (e.g., 5%) of all instances from the training distribution. In this case, achieved considerable improvement under distribution shift compared to naive application where no samples are rejected. However, at this particular threshold, we could recover the ID accuracy only in some scttings. Thus, we also studied whether we can learn an accuracy-oriented threshold where some predefined ID accuracy is guaranteed in the competence region. This approach was able to replicate the ID accuracy in the competence region for most investigated domain shifts. However, for a few domains, OOD accuracy drops significantly below the expected ID, calling for more detailed understanding of the behavior of incompetence scores in DG. Nevertheless, we observed that accuracy in the competence region behaves monotonically with the threshold (see Proposition 3.1 and Section 4.2).
Finally, we investigated differences between the closed and open world settings. We found that in the open world setting, feature-based methods, such as Deep-KNN (Sun et al., 2022) and ViM (Wang et al., 2022), elicit a particularly useful competence region. In a closed world DG setting, a clear winner does not emerge, but ViM and Deep-KNN seem to be competitive to logit-based approaches. We also analyzed whether we could find differences in the accuracy of the competence region with respect to different. classifiers. We could not find statistically significant effects on the accuracy in the competence region, leaving the benefit of robust algorithms for DG and different architectures for enlarged competence regions questionable.
All post-hoc methods investigated in this work are comparably fast to evaluate and therefore easily accessible for practitioners. However, the resolution of the trade-off between accuracy and coverage is not yet satisfactory in all cases, calling for more research on better competence scores. One interesting avenue concerns the use of multivoriate scores (i.e., a combination of multiple scores) with the potential elicit better competence regions.
Acknowledgments
JM and UK were supported by Informatics for Life funded by the Klaus Tschira Foundation. STR and FD were supported by the Deutsche Forschunggemeinschatt (DFG, German Research Foundation) under Ger- many's Excellence Strategy EXC-2181/1 390900948 (the Heidelberg STRUCTURES Cluster of Excellence). We thank the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for its facilities for high throughput calculations.
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References
Charu C Aggarwal and Charu C Aggarwal. An introduction to outlier analysis. Springer, 2017.
Charu C Aggarwal and Philip S Yu. Outlier detection for high dimensional data. In Proceedings of the 2001
ACM SIGMOD international conference on Management of dala, pp. 37 46, 2001.
Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization.
arXiv:1907.028: 09899, 2019.
Peter I. Bartlett and Marten H Wegkamp. Classification with reject option using hinge loss. Journal of Machine Learning Research, 9(8), 2008.
Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. In Proceedings of the European conference on computer vision (ECCV), pp. 456-473, 2018.
Shai Ben-David, John Blitzer, Koby Crammer,, and Fernando Pereira. Analysis of representations for domain adaptation. Advances in neural information processing systems, 19, 2006.
Liang Chen, Yong Zhang, Yibing Song, Ying Shan, and Lingqiao Liu. Improved test-time adaptation for domain generalization. arXiv preprint WrXi:2334404490 2023.
C Chow. On optimum recognition error and reject tradcoff. IEEE Transactions on information theory, 16 (1):41-46, 1970.
Chi-Keung Chow. An optimum character recognition system using decision functions. IRE Transactions on Electronic Computers, pp. 247-254, 1957.
Filipe Condessa, José Bioucas-Diaa Dias, and Jelena Kovačević. Performance measures for classification systems with rejection. Pattern Recognition, 63:437 450, 2017.
Steve Dias Da Cruz, Oliver Wasenmuller, Hans-Peter Beisc, Thomas Stifter, and Didicr Stricker. Sviro: Synthetic vehicle interior rear seat occupancy dataset and benchmark. In Proceedings of the IEEE/CVP Winter Conference on Applications of Computer Vision, pp. 973 982, 2020.
Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Un- terthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXic:2010. 2020.
Ran El- Yaniv et al. On the foundations noise- free selective classification. Journal of Machine Learning Research, 11(5), 2010.
Chen Fang, Ye Xu, and Daniel N Rockmore. Unbiased metric learning: On the utilization of multiple datascts and web images for softening bias. In Proceedings of the IEEE International Conference Computer Vision, pp. 1657-1664, 2013.
Tvan Flores. An optimum character recognition system using decision functions. IRE Transactions on Electronic Computers, pp. 180 180-180, 1958.
Giorgio Fumera and Fabio Roli. Support vector machines with embedded reject option. In Pattern Recog- vilion with Support Vector Machines: First International Workshop, SVM 2002 Niagora Falls, Canada, August 10, 2002 Proceedings, pp. 68 82. Springer, 2002.
Leon A Gatys, Alexander S Ecker, and Matthias Bethge. Image style transfer using convolutional neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2414- 2423, 2016.
Yonatan Geifman and Ran El-Yan.v. Selective classification for deep neural networks. Advances in neural information processing systems, 30, 2017.
14

---

Published in Transactions on Machine Learning Research (06/2023)
lan J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. 1xXiv1412.6577 2014.
Yves Grandvalet, Alain Rakolomamonjy, Joseph Keshet, and Stéphane Canu. Support vector machines with reject option. Advances in neural information processing systems, 21, 2008.
Ishaan Gulrajani and David Lopez-Paz. In search of lost domain generalization. orXiv:2007.01434, 2020.
Songqiao Han, Xiyang Hu, Hailiang Huang, Minggi Jiang, and Yue Zhao Adbench: Anomaly detection
benchmark. 2iv: 02222000602266 2022.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In
Proceedings of the IEFE conference on computer vision and pattern recognilion, pp- 770-778, 2016.
Martin E Hellman. The nearest neighbor classification rule with reject option. IEEE Transuctions Systems Science and Cyberneties, ((3):179-185, 1970.
Kilian Hendrickx, Lorenzo Perini, Dries Van der Plas, Wannes Meert, and Jesse Davis. Machine learning with reject option: A survey, 1ktt:2207.11277, 2021.
Dan Hendrycks and Kevin Gimpel. A baseline for detecting misclassified and distribiisti examples in neural networks. Xrxi::1600. 161.00216,,, 2016.
Dan Hendrycks, Steven Basart, Mantas Mazeika, Mohammadreza Mostajabi, Jacob Steinhardt, and Dawn Song. Scaling out-of-distribution detection for real-world scttings. arX1v:1911.11192, 2019.
Amita. Kamath, Robin Jia, and Percy Liang. arXiv:2006.09/4 09462, 2020.
Selective question answering under domain shift.
Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Balsubramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al. Wilds: A benchmark of in- the-wild distribution shifts. In International Conference on Machine Learning, pp. 5637-5664. PMLR, 2021.
Steven Korevaar, Ruwan Tennakoon, and Alireza Bab-Hadiashar. Failure to achieve domain invariance with domain generalization algorithms: An analysis in medical imaging. IEEE Access, 2023.
David Krueger, Ethan Caballero, Meenn-Herrik Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi Le Priol, and Aaron Courville. Out-of-distribution generalization via risk extrapolation (rex). In International Conference on Machine Learning, pp. 5815-5826. PMLR, 2021.
Kimin Lee, Kibok Lee, Honglak Lee, and Jinwoo Shin. A simple unified framework for detecting out-of- distribution samples and adversarial attacks. Advances in neural information processing systems, 31, 2018.
Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M Hospedales. Deeper, broader and artier domain gener- alization. In Proceedings of the IEEE international conference computer vision, pp. 5542-555 2017.
Pan Li, Da Li, Wei Li, Shaogang Gong, Yanwei Fu, and Timothy M Hospedales. A simple feature augmenta- tion for domain generalization. In Proceedings of the IEEF/CVF International Conference on Computer Vision, pp. 8886-8895, 2021.
Weitang Liu, Xiaoyun Wang, John Owens, and Yixuan Li. Energy-based out-of-distribution detection. Advances in neural information processing systems, 33:21464 21475, 2020.
Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. Swin transformer: Hicrarchical vision transformer using shifted windows. In Proceedings of the IEEE/CVF international conference on compuler vision, pp. 10012 10022-0002, 2021.
Diego PP Mesquita, Lincoln S Rocha, João Paulo P Gomes, and Ajalmar Rocha Neto. Classification with reject option for ware defect prediction. Applied Soft Computing, 49:1085-1088 1093, 2016.
15

---

Published in Transactions on Machine Learning Research (06/2023)
Krikamol Muandet, David Balduzzi, and Bernhard Schölkopf. Domain generalization via invariant feature representation. In International conference machine learning, pp. 10-18. PMT.R, 2013.
Jens Müller, Robert Schmier, Lynton Ardizzone, Carsten Rother, and Ullrich Köthe. Learning robust mod- els using the principle of independent causal mechanisms. In DAGM German Conference Pattern Recognition, pp. 79 110. Springer, 2021.
Malik Sajjad Ahmed Nadeem, Jean-Daniel Zucker, and Blaise Hanczar. Accuracy-rejcction curves (arcs) for comparing classification methods with reject option. In Machine Learning in Systems Biology, pp. 65-81. PMLR, 2009.
Eric Nalisnick, Akihiro Matsukawa, Yee Whye Teh, and Balaji Lakshminaraaanan. Detecting out-of- distribution imputs to deep generative models using typicality. arXiv:1906.0220 02994, 2019.
Hyeonseob Nam, HyunJae Lee, Jongchan Park, Wonjun Yoon, and Donggeun Yoo. Reducing domain gap by reducing style bias. In Proceedings of the IEEE/CVE Conference on Computer Vision and Pattern Recognition, pp. 8690-8699, 2021.
Anh Nguyen, Jason Yosinski, and Jeff Clune. Deep neural networks are easily fooled: High confidence prodictions for unrecognizable images. In Proceedings of the IEEE conference on computer vision and pallern recognilion,, pp. 427 0774 2015.
Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang, Kate Sacnko, and Bo Wang. Moment matching for multi-source domain adaptation. In Proceedings of the IEEE/CVF international conference on compuler vision, pp. 1406 1415, 2019.
Mohammad Pezeshki, Oumar Kaba, Yoshua Bengio, Aaron C Courville, Doina Precup, and Guillaume Lajoie. Gradient starvation: A learning proclivity in neural networks. Advances in Newral Information Processing Systems, 34:1256 1272, 2021.
Jie Ren, Jiaming Luo, Yao Zhao, Kundan Krishna, Mohammad Saleh, Balaji Lakshminarayanan, and Peler J Liu. Out-of-listribution delection and selective generation for conditional language models. arXiv:2209.155: 15558, 2022.
Lukas Ruff, Jacob R Kauffmann, Robert A Vandermeulen, Grégoire Montavon, Wojeiech Samek, Marius Kloft, Thomas G Dietterich, and Klaus- Robert Müller. A unifying review of deep and shallow anomaly detection. Proceedings of the IEEE, 109(5):756-755, 2021.
Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang. Distributionally robust neural net- works for group shifts: On the importance of regularization for worst-case generalization. brXiv:1911.08733 2019.
Antoine Sanner, Camila Gonzalez, and Anirban Mukhopadhyay. How reliable are distiiiiibttto general- ination methods for medical image segmentation? In DAGM German Conference Pattern Recognition, pp. 604 617. Springer, 2021.
Zheyan Shen, Jiashno Liu, Yue He, Xingxuan Zhang, Renzhe Xu, Han Yu, and Peng Cui. Towards out-of- distribution generalization: A survey. arXio:2108.1 136244 2021.
Yuge Shi, Jeffrey Sccly, Philip HS Torr, N Siddharth, Awni Hannun, Nicolas Usunier, and Gabriel Synnaeve. Gradient matching for domain generalization. urXiv:2104.09937 2021.
Raudenbush Stephen and Bryk Anthony. Hierarchical linear models. Sage Publications, Thousand Oaks, CA, 2002.
Yiyou Sun, Chuan Guo, and Yixuan Li. React: Out-of-distribution detection with rectified activations. Advances in Neural Information Processing Systems, 34:144-157, 2021.
Yiyou Sun, Yifei Ming, Xiaojin Zhu, and Yixuan Li. Oft-of-distribution detection with deep nearest neigh- bors. In International Conference Machine Learning, pp. 20627-20840. PMLR, 2022.
16

---

Published in Transactions on Machine Learning Research (06/2023)
Engkarat Techapamurak and Takayuki Okatani. Practical evaluation of out-of-distribution detection methods for image classification. arXiv:2101.02447, 2021.
Vladimir N Vapnik. An overview of statistical learning theory. IEEE transactions on neural networks, 10 (5):988-99) 999, 1999.
Neeraj Varshney, Swaroop Mishra, and Chitta Baral. Investigating selective prediction approaches across several tasks in iid, ood, and adversarial settings. arXii:2203.002: 2022.
Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. In Proceedings of the IEEE conference on computer vision and pollern recognilion, pp. 5018 5027, 2017.
Haoqi Wang, Zhizhong Li, Litong Feng, and Wayne Zhang. Vim: Out-of-distribution with virtual-logit matching. In Proceedings of the IEEE/CVF Conference Compuler Vision and Pullern Recognilion, pp. 4930, 2022.
Mei Wang and Weihong Deng. Deep visual domain adaptation: A survey, Neurocomputing, 312:135 153, 2018.
Marten Wegkamp and Ming Yuan. Support vector machines with a reject option. arXiv preprint arXiv:1201.1140, 2012.
Florian Wenzel, Andrea Dittadi, Peter Vincent Gchler, Carl-Johann Simon-Gabric,, Max Horn, Dominik Zietlow, David Kernert, Chris Russell, Thomas Brox, Berul Schiele, et al. Assaying out-of-lisuribution generalization in transfer learning. arXiv preprint arXiv:2207. 09239, 2022.
Yair Wiener and Ran El- Yaniv... Agnostic selective classification. Advances in neural information processing systems, 24, 2011.
Guoxuan Xia and Christos-Savvas Savvas Bouganis. Augmenting softmax information for selective classification with out-of-distribution data. In Proceedings of the Asian Conference Computer Vision, pp. 1995-2012, 2022.
Zhu Xiaojin. Semi-supervised learning literature survey. Computer Sciences TR, 1530, 2008.
Jigang Xie, Zhengding Qiu, and Jie Wu. Bootstrap methods for reject rules of fisher Ida. In 18th International Conference on Pattern Recognition (ICPR '66) volume 3, pp. 425-428. IEEE, 2006.
Shen Yan, Huan Song, Nanxiang Li, Lincan Zou, and Liu Ren. Improve unsupervisod domain adaptation with mixup training. vxXi::2011.0667, 2020.
Jingkang Yang, Kaiyang Zhou, Yixuan Li, and Ziwei Liu. Generalizod out-of-distribution detection: A survey.
arXiv:21 1222111.113301 11344, 2021.
Jingkang Yang, Pengyun Wang, Dejian Zou, Zitang Zhou, Kunyuan Ding, Wenxuan Peng, Haoqi Wang, Guangyao Chen, Bo Li, Yiyou Sun, et al. Openood: Benchmarking generalized out-of-distribution detec- tion. Xiw:2210.07242 2022.
Xu-Yao Zhang, Guo-Sen Xie, Xiuli Li, Tao Mei, and Cheng-Lin Liu. A survey on learning to reject. Pro- ceedings of the IEEE, 2023.
Kaiyang Zhou, Ziwei Luu, Yu Qiao, Tao Xiang, and Chen Change Loy. Domain generalization: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022.
17

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Published in Transactions on Machine Learning Research (06/2023)
A
Appendix
In this Appendix, we describe optimization procodures in detail, give additional detailed results and describe the open world data sets in detail. Furthermore, we give proof for Proposition 3.1.
A.1 Detailed Qualitative Results
Figure 8 and Figure 9 show for the PACS data set the three images attaining the highest and the lowest incompetence scores per class respectively. Images with lower scores achieve higher accuracy compared to the highest-scored images.
A.2 Detailed Quantitative Results- Dependence on Competence Threshold
We show the accuracy on OOD test data in dependence on the competence threshold for the DG data sets PACS, OfficeHome, VLCS, Terralncoguita, DomainNet and SVIRO in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 respectively. We only show results for Deep-KNN, Logit, ViM and GMM applied on the ERM classifier. For Deep-KNN, Logit, and Vim we see in almost all cases the monotonic behavior as predicted in Proposition 3.1. On the DG data sets VLCS, DomainNet and TerraIncognita the GMM score fails to show this monotonie behavior for some Lest domains. Therefore, GMM has not the monotonic behavior would expect from admissible incompetence score. For some domains, all scores do not behave as we aimed for. For instance, in the LabelMe test domain in VLCS (see Figure 12) we cannot achieve the ID accuracy for all thresholds a and all incompetence scores. While this behavior is extremely rare in our experiments (for the feature- and logit-based scores), it shows that the current competence scores can fail for some domain shifts.
In Figure 16 we also show results for different thresholds according to their percentile in the ID distribution. We can see that the relative performance of the scores stays considerably stable.
A.3 Detailed Quantitatively Results Extensive Study
In Table 2, Table 3 and Table 4 we give detailed results for all DG data sets considered in this work: PACS, VLCS, OfficeHome, TerraIncognita, DomainNet,, and SVIRO. We list the accuracics in the competence region where the incompetence threshold is chosen as the 95% percentile of the ID validation set. Here we show the median, the 5% and 95% percentiles over all test domains and classifiers. We can see that the deviations between different test domains are quite severe indicating different strengthus of domain shifts across the DG tasks. The main observations in Section 4.4 (e.g. the OOD-gain is quite significant and feature-based methods [ViM; Deep-KNN] are very successfull) hold across the different DG tasks.
A.4 Open World Setting
Open World Creation We use additional data to extend the closed world data sets to the open world setting. We use similar domains of other data sets with disjunct classes to generalize the DG data sets. The ID data sets and the open world extensions are listed in Table 5. We show examples of test data (closed world) and open world samples for all DG data sets. For PACS (in Figure 21) for VLCS (in Figure 22), for OfficeHome (in Figure 23) and for Terralncognita (in Figure 24)
Open World Results Figure 17 shows the ID-Gain and OOD-Gain for all incompetence scores consid- ered in this work depending on the fraction of open world samples. We see that ViM and Deep-KNN are particularly able to delincate unknown class instances from known class instances resulting in an improved ID- and OOD-Gain across all open world fractions.
In Figure 18 we investigate the behavior of different scores in detail. It shows the AUROC of delineating ID data vs. correctly classified samples of the test domain, ID data vs. wrongly classified samples of the test domain, and ID data vs. unknown class instances general. Here we consider an unknown test domain where 25% of all samples are open world outliers. We see interesting behavior here: ViM and Deep-KNN
18

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Published in Transactions on Machine Learning Research (06/2023)
(a) Lowest scores Art.
(b) Highest scores Art.
600
(c) Lowest scores Cartoon.
(d) Highest scores Cartoon.
Figure 8: Images with highest and lowest scores for different domains on PACS. Scores are computed with Decp-KNN on ERM.
19

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Published in Transactions on Machine Learning Research (06/2023)
(a) Lowest scores Photography.
(b) Highest scores Photography.
ERROR
(c) Lowest scores Sketch.
(d) Highest scores Sketch.
Figure 9: Images with highest and lowest scores for different domains on PACS. Scores are computed with Decp-KNN on ERM.
20

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Published in Transactions on Machine Learning Research (06/2023)
Logit
Deep-KNN
VIM
GMM
1.0
1.0
1.0
1.04
0.8
0.8
0.8
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0.6
0.6
0.8
0.6
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0.4
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5.0
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10
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11
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-2
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0.6
0.66
0.6
0.0
0.0
Percentile (Validation) Accuracy curve (Ag)
Coverage (OOD)
Accuracy (OOD)
(OOD) (ID)
Coverage (ID)
Accuracy (ID)
Figure 10: The accuracy of the ERM classifier OOD data (ooD(a) the competence region is enlarged by increasing the allowed incompetence Here we show the results for all DG tasks of the PACS data set.
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Published in Transactions on Machine Learning Research (06/2023)
Logit
Deep-KNN
VIM
GMM
1.0
1.0
1.0
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-2.
0.0
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Clipart
0.2
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Product
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-2
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World
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0.6
0.0
0.0
0.0
-2.5
Percentile (Validation) Accuracy curve (Ag)
Coverage (OOD)
Accuracy (OOD)
(OOD) (TD)
Coverage (TD)
Accuracy (TD)
Figure 11: The accuracy of the ERM classifier OOD data (a) as the competence region is enlarged by increasing the allowed incompetence Here we show the results for all DG tasks of the OfficeHome data set.
22

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Published in Transactions on Machine Learning Research (06/2023)
Logit:
Deep-KNN
VIM
GMM
1.0
0.8
0.8
0.8
0.8
0.6
0.6
0.6
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0.4
0.4
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0.4
1.0
1.0
1.0
1.0
LabelMe
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.0
7.5
0.8
0.8
0.6
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0.0
-2
-2.5
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2.5
5.0
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1.0
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VOC2007
0.8
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0.8
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0.6
0.8
0.6
0.6
0.2
0.0
0.0
0.0
0.0
95%-Percantie (Validation)
Coverage (OOD)
Accuracy (OOD)
Score (OOD)
Accuracy (A,)
Coverage (ID)
Accuracy
Score (ID)
Figure 12: The accuracy of the ERM classifier OOD data (ao as the competence region is enlarged by increasing the allowed incompetence Here we show the results for all DG tasks of the VLCS data set.
23

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Published in Transactions on Machine Learning Research (06/2023)
Logit
Deep-KNN
VIM
GMM
1,0
1.0
0.8
0.8
0.8
0.6
0.6
0.5
0.6
0.4
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0.2
0.2
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-2
10
0.0
2,5
5.0
10
1.0
1.0
1.04
1.0
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5.0
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0.6
0.6
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0.6
0.0
0.0
0.0
Percentile (Validation) Accuracy curve (Ag)
Coverage (OOD)
Accuracy (OOD)
(OOD)
Coverage (TD)
Accuracy (TD)
(TD)
Figure 13: The accuracy of the ERM classifier on OOD data (ao the competence region is enlarged by increasing the allowed incompetence a. Here show the results for all DG tasks of the Terralncoguita data set.
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Published in Transactions on Machine Learning Research (06/2023)
Logit
Deep-KNN
VIM
GMM
Clipart.
0.6
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02
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Infograph
0.8
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10
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0.8-
Painting
0.6
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1.0
1.0
1.0
Quickdraw
0.8
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0.0
0.0
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0
10
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-2
-2
20
30
1.0
1.0
1.0
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0.0
(Validation)
Coverage (OOD)
Accuracy (OOD)
(OOD) (TD)
Accuracy curve (A₂)
Coverage (ID)
Accuracy (TD)
Figure 14: The accuracy of the FRM classifier OOD data (ao the competence region is enlarged by increasing the allowed incompetence Here we show the results for all DG tasks of the NomainNet data set.
25

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Published in Transactions on Machine Learning Research (06/2023)
Logit
Deep-KNN
VIM
016-
0.4
0.5
02-
00
10
10
016
0.8
02.
18:
1.4
18.
0.4
16.
14
0.5
16:
1.0
90% Percenttle (Validatoon
Coverage (000)
Accuracy (COD) Accurnry (T))
Score (COD) Score (I))
Accuraty curee M
Covecage (TD)
Figure 15: The accuracy of the ERM classifier on OOD data (ooD(a) as the competence region is enlarged by increasing the allowed incompetence a. Here we show the results for all DG tasks of the SVIRO data set.
26

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Published in Transactions on Machine Learning Research (06/2023)
OOD-Gain
IID-Gain
Coverage
0.0
PACS
0.8
0.6
-0.1
0.0
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0.95
0.99
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VLCS
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DemainNet Terraincegita Officellome
0.10
-0.10
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0.4 0.90
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1.0 0.90
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-0.4
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1.0
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0.99
0.05
00.05
0.9
0.8
0.00
0.00
0.75
SVIRO
0.03
0.02
0..11
0.25
0.90
0.95
0.99
0.90
0.95
0.99
0.90
0.95 Percentile
0.99
Percentile
Percentile
Deep-KNN
Softmax
Logit
VIM
GMM
Figure 16: Median of accuracies in competence region for different thresholds (percentiles of ID distribution over all domain roles and classifiers.
27

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Published in Transactions on Machine Learning Research (06/2023)
tn Perecotoges (%)
Art
Carloon
Poleting
Skrbch
LEKGup
Lovenue
HLCGHPP
Curenuge
Conenuae
LESGap
Lovouue
Dxee-KNN KNN
12 [5-14]
2 211
54 [57-79
14 [1:11]]
-1 [6-]]
[63 [56-1]
211-3 [1-28
1/5-5
51 [53-66
13 [-17]
-0 F15-8
VIM
11 11-14
112-8
56 [51-71
17 [13-101
0 [4-31
59 [40.00)
[1-2
4.8-4
91 [86-6
3
-15 1:2 01
77 B1-50
Sabtmax
7 [[-10]
-4 1:511
56 (77-58
7 [G-10]
-10 1112
52 [78-90
13-21
97 13-06
12 17-18
-8 -10 11
58 151-50
Luge
01-11
1:4-
52 93-900
10 [--11
-9: 130-3
41
77 [72-64
11-4
(3-4)
56 [13-09
12 (4-16
-8 022
66 10-04
Rowrgy
[6-11]
144
|51-5
9 [--11
-9 [10-3
17 172-4
11:0
13-6
96 193.67
15-16
-0-11
66 190-0
Eurrgg-Boat Roart
[6-11]
[44
50 j33-66
9 [--11
-5 110-
17 (7--64
11-18
13-5
96 (22-07
15-16
-03-1
66 (99-0
Mohalonoots
[1400]
77
87 196-04
7
10
[50.04]
[0-11
87
91
GMIM
(1-100
56 (54-56
31
[50-67
10-11
13-4
83
PCA
-13
0--0
52
1:24
SUNI0
V0
GDp
Coverage
Cortrage
10:Caap
Civerage
DODD
100Cop
Coverage
EoNp-KNN
2 13-4
11:-3
[0-3]
[1-0
70 (E3-R
[G44
1--2
69--3
VIM
[0-3]
14 110:10
10-00
105-220
(87.00)
11:-4
-16 1/0-5
85 175-68
11:46
[0-11
154-7
Softmmo
15-1
[--14]
11:09
-26 (100-31)
(82-16)
(3-1
-1--1
80 (55-02
3 11--5
[11-11
90 103-00
Logit
13-5
13 [8-14
155-00
10-31
-26 1:2 221
96 (50-09)
12-4
-16 -10
57 [94-02
23 12:55
-6 -30 21
50 PL-04)
Energy
13-5
12 [6-14
155-00
10-11
-26 1.0 221
140 (50.00)
13-44
-15 .19 4]
80 (55-02
-11-1
58 175-4
Emsrgg Beart
1 10-2
12 [--44
10-31
-26 (126 22]
00 [0000))
13:05
-15 [00-4
50 |s5-42
11-5
-8 -11-4
58 (7--65
1 123
13 1--15
50 p3-00
10-01
-37 105-20
149 [S1-00]
-17 133-4
90 (87.00
11:00
1-9-6
76 P0-ed
GMM
1 1--
13 (1-15]
51 [19-56
10-01
-37 126-22
18 [ps-te]
10-19
-17 123-7
96 [77-06
0 11:04
-0 113--6
76 [53-M4
PCA
1 12.
13 15-16
52 (33000
10-41
-27
125 501
59) (56-00)
[0-11
-17 122 8
90 (85-07
11 1-3-8
-0 11:88.0
77 12-58
Art
Ciipect
Product
Hual Morks
10-Gap
Caserrage
113:3inp
Cieeragy
11-Coop
Cheeneoge
115-Go
Conerooon
Does- KNN
19
15 [21
11--11
14:
(65-79)
12
30
VIM
16-12)
-21
11 16-10
(86-80)
-12 (114-)
501
(-1--)
97-04
OllaaBonn
Leght
So8mex
[9-7]
-17 141-4
111-4
10
(65-91
[04-96
11..0
(15-5
(1--9)
57
11:0-11
-1-11
Eoergy
173-4
(e1-93)
11.77
(16-)
Enegg Heact
18-10
12
W5-50
17-10
.41
(68-82)
13-7
-12 (-1-2)
B84-0G
Malalanuiis
443-4
10-11
[-16-1
113-00
[-1--1
GMIM
-28 144-0
[-18-1
96 113-00
(-1--1
(97000
PCA
1
-18 (31-4
(77-00)
(1
10:00
-17 (11-1)
95 (34-06
10:09
(16-0)
L10
1.38
L-3
Cureeaae
Conernats
Dewg-KNN
34 (30-45
1 118.55
S1 [15-41]
311 (15.30
-16 (:36-0
55 [34.50]
333 (11:31)
-11 -00
48 131.5a
63 18:54
-5 [:00--1
14 100-00
VIM
34 (25-04
-6 (13--
32 [13-47
25 [16.3)
-154 (14
51 [35-cc]
17 (6-m9)
-16 [-55-17
47 [11-57
45 [20-05
-7 --31
13 (1:34
Temshooggit
Softmax
(2-12)
-31 1-4 18]
54 14450
(1-51
-13 0:53 54
53 [72.44
131-7
-25 1.54 211
91 180-0
4 [11-1)
-48 1::00 84)
78 153-000
Lauit
-90 1:
55 53-14
9 12-16
-39 354-20
76 [65:44]
21-46
-90 1-55-225
94 13-00
-47 [-1--)
56 [0-40]
Eaegy
[1-21)
-90 11-111
50 H0-09
10 [2-37]
-38 114-20
75 [32MM)
-31 (-5--25
94 (33-0
-6 (43-20
54 [41-44
Koerge Hoart
[[/91
10 17-17
[60-0]
110-0
-31 [-50-35]
94 [13
3
-10 (01--))))
119
Mahalsuokis
(:2:44
[57-04
13-4
11
15
GMM
10 1-3-77
425 148
13--7
[1-26
(45-90
5 0-12)
621
[3-44
14
PCA
(2211)
[46-30
10-10
[37-00)
[36-22)
50
(:6-20)
Table 2: Accuracy on competence region of OOD domain for different PACS, Office/lome, VLCS and Ter- raIncoynila domains and incompelence scores. As the threshold for the competence regions, we choose the 95% percentile of the ID validation set. For all metrics, higher value means better performance (t). All displayed values are medians over different domain roles and classifiers, brackets indicate 90% confidence interval.
are well-able filter out wrongly classified samples, but also filter out many correctly classified samples. The logit-based scores (Logit, Softmax, Energy, Energy-React) are less successful in filtering out wrongly classified samples, but also keep more correctly classified samples. In the optimal case, we would expect that the AUROC of ID vs. correct test data is S 0.5 and the AUROC of ID vs. false OOD data is 1. This would imply that we could successfully filter out wrongly predicted samples and keep a high coverage. Figure 18 shows that ViM and Deep-KNN are capable of filtering out new class instances across all DG data sets. For all other scores, we can find data sets where this behavior is not achieved. Consequently, ViM and Deep-KNN work best when unknown class instances occur.
A.5 Training Details and Classifiers
All classifiers are trained using the DomainBed repository? 2. We train three different noural network architoc- tures with Eaaprrical-Riss--Mminin
shortly ERM (Vapnik, 1999). Namely, a ResNet based architecture (He et al., 2016), a Vision Transformer (Dosovitskiy et al., 2020) and a Swin Transformer (Liu et al., 2021). If we just refer to ERM, we mcan the ResNet-basoo architecture. Furthermore, recent DG algorithms, namely Fish (Shi et al., 2021), GroupDRO (Sagawa al., 2019), SD (Pezeshki et al., 2021), SagNet (Nam et al., 2021), Mixup (Yan et al., 2020) and VREx (Krueger et al., 2021).
train classifiers with various
We use all the standard settings provided in the DomainBed repository and train all classifiers with hy- perparameters proposed in the repository. The Vision Transformer and SwinTransformer are trained with hyperparameters found useful on these data sets and architectures as (Wenzel et al., 2022). Each model is trained for 100 epochs on the smaller data sets (PACS, VLCS, TerraIncoguita and OfficeHome)) and for
28

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In Pereentages (%)
clip
info
point
DOD-GGin
TD-Gap T
Frac 1
OOD--Gai
ID-Gap 1
Frec
OOD
ID-Gap 1
Frac 7
Deep-KNN
4 [2-8]
2--7
86 (62-90
3 [0-6
-40 [-15-31
[71-89
[1-6]
-11-6
86 [84-122
VIM
4 [2-6]
[1-6]
90 [86-83
1 [--9
-42 [-44 28]
92 [66-97]
2 [1-8]
-10 [-13 3]
92 [80-316]
Softmmx
3 [1-4]
[1-5]
95 [94-97]
5 [1-7]
-40 [-42-29]
78 [74-88]
[1-4]
-10 [-12-8]
93 [91-96]
Logit
3 [1-4]
[2-5]
[50-96
[0-8]
[:44 30]
[12-96]
[1.4]
[-12 8]
93 (90-96]
Energy
3 [1-5]
[1-5]
91 [SS- 96
1 0-8)
-42 [-41-30
92 [73
[L-4
-10 [13-7]
93 [90-95]
Energy-Resse
3 [1-5]
[1-5]
92 [88-06)
1 [0-8
-42 [-44-30]
92 [72-98]
2 [1-4]]
-10 [-13-7]
93 |91-06]
Mahudonobis
[-3-3]
-2 [-3-6]
[84-93)
[-1-4]
[-45 32]
[19-97]
[11-]]
[-16 3]
95 [80-97]
GMM
-2 [-3-2]
-2 [-4-4]
87
[84-94
0 [-1-3]
-45 (46-332
97 [83-98]
-1 [11-0
-14 [-16-4]
95 [80-97]
PCA
-2 [-3-1]
-2 [-4-3]
8G [81-93
0 [-1-1]
-45
(-17-32)
96 [87-98)
-1 (-1-1)
-14
[-16-10)
96 [90-97]
quick
real
Aketch
OOD-Gain
ID-Gop 1
Coverage
OOD-Gain f
ID-Gap f
Coverage
OOD--Gan 1
ID-Gap 1
Coverage
Doep-KNN
3 [0-4
-48 [-52 27
78 [6G-91]
3 [1-3]
5 [2-90
92 [89-116]
6 [2-0]
-4 [-6-0]
83 [70-87]
VIM
2 [0:3]
-49 [-54 27]
80 (65-55)
2 [0-4
[2-11]
94 [88-97]
[0-6]
-7 [-9 1]
90 [86-96]
Softmax
0 [0:2]
-51 [34-27]
96-96
1-2)
[1-8]
(97-
[2-4]
-7 [-9-]
93 [92-9d
Logit
1 [0-2]
-50 1:4--27
80 [72-95]
2 1-2
3 [2-77
97 [16-118]
3 [1-6]
-7 [-8-1]
93 |01-94]
Energy
1 [0-2]
-50 [-54 27]
88 [7L-586
1 [0-2
3 [1-7]
98 [95-98]
3 [0-4]
-7 [-9 2
94 [91-96]
Energy-Ren React
1 [0:2]
-50 [-44-27]
[72-96
0-2]
[1-7]
(95 95
[0-4]
[-9-2]
[91-95]
Mahalonobia
0 -1-1]
-52 |35-27
94 [68- 100)
-1 1-355
1 [-2-11]
87 [81-91]
-1 [-1-6]
-10 [-13-2]
93 [86-95]
GMM
0 -1-0]
-52
[-55 27
95 [70-100]
(3-4
0 (3-100
87 [70-30]
[-2-0]
-11
[-14 2]
93 [89-95]
PCA
-1-0
-52 [-55-27]
(75-99)
-2 [-3-3]
(-2.9)
[19-90)
1-2-2
[-13-3]
[89-95]
Table 3: Accuracy on competence region of OOD domain for different DomainNet domains and incompetence scores. As the threshold for the competence regions, we choose the 95% percentile of the ID validation set. For all metrics, a higher value means better performance (t). All displayed values are medians over different domain roles and classifiers, brackets indicate 90% confidence interval.
Phecertagee (5)
ID-Gap
Covereg
Chenage
DCDX-eis
TD.Gup
Oweergg
ID-Cip
Cowwrgg
ID.Gap
Hap KNN
-11-1
01:11
54 (1:00)
11:1.3
311.3)
011
10
1120
15 (8.3)
[11.5
11
Sofhowe
11:1
07
1(20
Loght
19.0
11110 18
81 11100
13 30
11.5
0156
51 11:
121 11:0
51 05 mg
81 01
01 1130
09 06 80)
Bantrr
16-3
88 1144
72040
53.00
01 11068)
15 190-0
0166
13 (3:81
--30
55 33-75
3 11-1
01:34
09 DS40
16-3
7.040
19 105-00
0.66
55 35-75
114
0 01:44
(3400
GMDI
111-3
08:44
34-70
14-14
k-0-
19 (17-91
06-4
7 1106-34
014444
11-0
0 (1444)
51 11-47
01:-1
-15 23.01
53 997--8
1440
191 115.0
010-0
19-00
141 10: 100
11 (9-00
-3 4-11
161 (06-00
Highnt
80
HD4kapp
Conmrr
[1kiipp
ID-4Gap
Teep KNN
21:2 -01
01:11 11
5..10 11
01 1/1
13 190 mt
-7 (11.
30 10:33
0. 1-1
16 (11-1)
133 102
01110
13: (1)
VIM
km]
k1
18 110 k7
01.0
301 [14))]
(3:01)
0. 1.1
121
11:00
Sulasus
00-27
123-8
18 (3--6
311-4)
10-3
20 113-34
1344
42 [23-00
Battr
16-31
14544
32-74
13-4
0 1-2-
19 135.96
33-4
11-34
(11-3)
85 13-00
6:224
07 19477
4641--1
-2 14648
11-70
13-4
0 13-4
19 99c-1
13-4
24 19466
11-77
(17-1
55 13-00
07 (6-72
Makakmattle
7
14-0
35 82
0150
114-31
48 (1303-3
13: 50
P97A
22.00
50 (0-90
83 (1-00)
100
Table 4: Accuracy on competence region of OOD domain for different SVIRO domains and incompetence scores. As the threshold for the competence regions, we choose the 95% percentile of the ID validation set. For all metrics, a higher value means better performance (t). All displayed values are medians over different domain roles and classifiers, brackets indicate 90% confidence interval.
10 epochs on DomainNet and SVTRO. When no improvement in terms of accuracy on the validation set is achieved, we stop the training. The best model is chosen due to the accuracy on the ID distribution measured via the accuracy on the validation set.
Some scores are computed on the logits and some on the features. If computed on the features, we use the output of the penultimate layer of the model as input to the score function.
We distinguish between training, validation, and test set of the ID distribution. For the OOD distribution we only consider one data set provided by the DG task which is not seen during training, Score quantiles are always computed on the ID validation set. The ID accuracy is computed on the ID test data set. If score functions need optimization (as with G.M),, we train them on the ID training set. If a score function needs optimization, we restrict the training set to 50 000 samples. This only affects the DomainNet data set. We do only little to no optimization of the parameters of the score functions. We mainly stay in line with the standard settings found in the literature. For Deep-KNN we choose K because it shows slightly improved performance on the ID distribution (only inspected on PACS).
29

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Published in Transactions on Machine Learning Research (06/2023)
ID data set
Open world set
Test domain -> Open world domains
Open world classes
art + painting
alarm clock, ambulance, apple,
cartoon + clipart
PACS
DomainNet.
backpack, baseball, basketball,
photo photo
bat, bear,bed and bicyle
sketch sketch
VLCS
PACS
all environments - photo
elephant, giraffe and guitar
art > paint
clipart + clipart
OfficeHome
DomainNet
bread, butterfly, cake, carrot, cat
product - real
real world real
Terralncoguita
PACS
all enviroments - photo
elephant, giraffe and horse
Table 5: Open world extensions of different DG data sets and their test domains.
0.4
Softmax
Logit
Energy
Energy React
VIM
Deep KNN
Mahalanohis
GMM
PCA
0.2
ID-Gain
0.0
-0.2
-0.4
Fraction of open world samples
0.8
Logit
Esergy
Energy
Deep XNN
OOD-Gain
0.4
0.2
0.0
0.0
0.05
0.1
0.15
0.2
0.25
Fraction of open world samples
Figure 17: OOD-000Doaaaoee and ID-Ga Gain for different incompetence score for an increasing fraction of open world data (unknown classes) in the test domain (higher is better).
A.6 Trained Classifiers
Figure shows the accuracies of all different Classifiers on DG data sets for the data and the OOD data. Here we show the means and standard deviations over the different domains. All classifiers obtain a similar ID and OOD accuracy. One exception is VREx which did not converge for all domains DomainNet. In Figure 19 we show the accuracies for the different DG methods all data sets.
A.7 Proof of Proposition 3.1
Proof. (b) Take the limit (a(a) -000 R.D. Then there no restriction of the support of PooD, so the accuracy for large approaches the accuracy the entire OOD data set.
(a) By assumption PID and PooD share their support when restricted to the competence region Xc(a) when 5 a², Thus we can always assume that pooD(Xc(a)) > 0 for all milgesapp(PPn) sc(x) =; ag. which makes the accuracy well-defined for all relevant a 2 ag:
(ao = (oo)(X((a))c(() = Yire))
(3)
PooD(Xe(a))
Here, Yiree the correct label the input X.
For the remainder, we consider [ao. a²), so (ao An) Then, we have that Aood(a)
Anp (a) lim Ap(()) AID. The limit can be taken analogously the proof of (b) above.
30

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Published in Transactions on Machine Learning Research (06/2023)
Sonmax
Logat
Unweryy
Dosegy-laact
VIM
Deep-KNN
Mahalanoha
RMM
PCA
1.0
AUROC ID vs. OOD (correct)
0.6
344
0.2
IMCS
Oiticellome
Viucs
Terriincoonete
Dataset
Softmxx
Lagit
Energy
Eneegy Beart
VIM
Deep-KNM KNN
Mahalanohh
FRMM
PCA
AUROC vs. OOD (false)
3.9
0.7
0.5
IMCS
Officiltmm
Ternilnoognite
Dataset
Sotimax
Logat
Ensegy Heact
VIM
Deep KIN
Mahalanoba
LMM
PLA
AUROC vs. open world
8.8
0.4
IACS
Oficctteme
Vilcs
Dataset
Figure 18: Above: AUROC of delineating ID data vs. correctly classified samples the OOD data. Middle: AUROC of delineating ID data wrongly classified samples the OOD data. Below: AUROC of delin- eating ID data open world data in general. All test domains are enriched with 25% open world outliers.
31

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Published in Transactions on Machine Learning Research (06/2023)
1.0
0.8
Aood
0.6
0.4
0.2
ERM
Mixup
Fish
VREx
SD
SagNet
GroupDRO
0.0
PACS
VLCS
OfficeHome Terralncognita DomainNet
SVIRO
Data set
Figure 19: Accuracies for different classifiers OOD test data. The boxes show the quartiles and medians.
PACS
0.6-
0.2
OOD
ERM
sb
Mixup
Saphet
Fia
CrougORO
VREx
Swin Stutoforce
VLCS
0.4
ID Test
OOD
0.0
OfficeHome
0.2
ID Test
OOD
20M
sb
Mizup
Suphet
Fia
GroupORO
Vitts
Swis Dursdomme Yulnd/nassfoaat
Terraincognit
1.0
0.6
OOD
0.6
15AM
sb
Seghet
Grou@Oit
Viitt
Swin Towoedorme Vichn/nnssdtmaa
DomainNet
0.8
0.4
ID Test
OOD
Mixup
SayNet
Gioup-lil
Vilts
Uhuradormer Vasntimsssttmat
SVIRO
0.6
0.3
ID Test
OOD
IRM
sb
Misup
SaghOnt
Fich Classifier
GraugOR
VRit
Swis SwisTronssmmee VetucTrendomar
Figure 20: Accuracies for different classifiers on ID and OOD test data. We show the means and standard deviations over different DG tasks.
32

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Published in Transactions on Machine Learning Research (06/2023)
(a) Test data for domain Art
(b) Open world data for domain Art.
(c) Test data for domain Cartoon
(d) Open world data for domain Cartoon.
(e) Test data from domain Photo.
(I) Open world data for domain Photo.
(g) Test data from domain Sketch.
(h) Open world data for domain Sketch.
Figure 21: Test (left) and open world data (right) for the PACS data set.
33

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Published in Transactions on Machine Learning Research (06/2023)
(a) Test data from domain Ca1101.
(b) Open world data for domain Cal101.
(c) Test data from domain LabelMe.
(d) Open world data for domain LabelMe.
(c) Test data from domain SUN09.
(I) Open world data for domain SUN09.
(g) Test data from domain VOC2007.
(h) Open world data for domain VOC2007.
Figure 22: Test (left) and open world data (right) for the VLCS data set.
34

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Published in Transactions on Machine Learning Research (06/2023)
(a) Test data from domain Art.
(b) Open world data for domain Art.
T
(c) Test data from domain Clipart.
(d) Open world data for domain Clipart.
Coca-Cola
1000000
(c) Test data from domain Product.
(I) Open world data for domain Product.
(g) Test data from domain Real World.
(h) Open world data for domain Real World.
Figure 23: Test (left) and open world data (right) for the OfficeHome data set.
35

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Published in Transactions on Machine Learning Research (06/2023)
Test data from domain L100.
(b) Open world data for domain L100.
(c) Test data from domain L38.
(d) Open world data for domain L38.
(c) Test data from domain L43.
(I) Open world data for domain L43.
(g) Test data from domain L46.
(h) Open world data for domain L466,
Figure 24: Test (left) and open world data (right) for the Terralncognita data set.
36


Reviews:
[{"review_id": "mZxg0N3xWi", "reviewer": "Reviewer_zTpB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The motivation of this task is well written and the authors created an enjoyable story line throughout the paper, especially the Introduction.\n- The authors did a great job identifying different works to compute scores based on Feature-based, Density-based, Reconstruction-based,  and Logits-based methods\n- The authors compare these methods across different datasets and showcase their behavior which might be useful for practitioners who would like to use similar methods to maximize performance on similar datasets\n\n\nWeaknesses:\nThe Method Section needs more details to be understood. I was confused about how the \"Incompetence Score\" was computed until I started reading the Experiment Sections. I would encourage the authors to either reference how the scores are computed by adding a citation to the Experiment Section, or putting how the scores are computed in the Method section explicitly.\n\nFigure 3 is confusing, what is the x-axis value? it seems like it is alpha, but isn’t that the percentage of accepted samples? \n\nIt is very difficult to identify which Figure and Table support the statements in the Contributions listed at the end of the Introduction. For example, I still don’t know which Figure/Table shows that robust classifiers are worse than the naive baseline. I strongly suggest for the authors to add a citation of the Table and Figure in front of each of the 4 contributions listed. Otherwise I can't verify them.\n\nThe authors mentioned that their method also works with Regression but they do not explain how.\n\nThe contribution that there is a tradeoff between alpha and the classification performance is obvious. That  tradeoff is commonly observed with self-training methods in the semi-supervised learning literature [A] where picking samples using a certain confidence threshold such as   95% and above could lead to improved accuracy, but lowering that confidence threshold results in picking up bad samples which hurt performance. So this contribution that the authors list is very weak.\n\nThe authors should definitely include many of the semi-supervised literature where samples are selected based thresholds such as the ones here [A]  and the works that cite it.  The authors should also compare so some of these self-training methods especially the ones that pick examples using a confidence threshold.\n\nThe novelty in this work is very limited as the authors do not propose a new method but simply compare existing methods on few datasets.\n\nThe analysis made by the authors is very limited and difficult to draw conclusions as the datasets are small. Therefore the conclusions that Feature-based score is better than logits-based score is better for OOD cannot be verified unless more experiments are conducted on larger datasets like ImageNet. \n\nSpecifying the percentage as 95% is very specific and I doubt that it works well on different problem setups. Further, blindly doing a cut-off at 95% could lead to an imbalanced dataset in the case where one class is easy to classify and the other classes are more difficult leading to  selecting samples only from the easy classes leading to a biased dataset.\n\n[A] Zhu, X. 2006. Semi-Supervised Learning Literature Survey.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The task addressed in the article is Domain Generalization, where the distribution of the test data is different from that of the training data. The problem is to ensure that the trained model can perform well on the test data despite the domain shift, without requiring retraining on new data.\n\nThe authors attempt to address the task of domain generalization where the test data’s distribution differs from that of training. The authors propose a method that learns to reject samples based on an “incompetence threshold” empirically showing improved accuracy on the task of classification. The threshold is determined by the in-distribution data (seemingly coming from the training set) which is used to reject samples based on either logits-based incompetence score or feature-based incompetence score. The authors also claim to show experiments where robust classifiers failed in comparison to basic baselines. ", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I can't recommend an acceptance unless changes are made to address the Weaknesses mentioned in the previous text box.", "broader_impact_concerns": "I do not see Ethical Implications of this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "3tuicdh2Hr", "reviewer": "Reviewer_CkvU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "* The new problem setting is interesting and I think it is a sensible combination of existing settings.\n* The notion of an $\\alpha$ competence region seems quite useful, and the idea of constructing them via existing OOD detectors and some percentile of the in-domain data seems sound.\n* The experiments provide some useful insights. In particular: the demonstration of the trade-off between coverage and accuracy, the utility of different incompetence scores, and analysis in the open world setting.\n* Choose 5% as the acceptance threshold seems a bit arbitrary. It would have been interesting if the paper had investigated the sensitivity of this hyperparameter a bit more. I could see different tasks needing different thresholds here, and it would be nice if this was discussed a bit more.\n* The main weakness of the paper in my view is that the presentation could be significantly improved. I have provided several concrete suggestions below, but I suggest the authors make a pass over the paper and try to spell out some of the implications of their results a bit more clearly.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a novel problem setting that combines aspects of domain generalisation with the prediction with option to reject setting. The submission investigates how well existing scores for detecting out of distribution examples can be adapted to estimate the incompetence of a model on data that is guaranteed to be out of distribution, but possibly similar enough to still be correctly labelled by the model. It is found that accuracy of predictions where the classifier was estimated to be competent are much higher than those predictions where the classifier is presumed to be incompetent. As expected, there is a trade-off between accuracy and coverage. Other experiments investigate an open world variant of the new problem setting, and a heuristic for setting the threshold to be applied to the incompetence score.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "* The inroduction of the new problem setting could be made more explicit. The paper does not actually spell out the main differences between their proposed problem setting and the two most closely related previously studied settings: DG and prediction with option to reject.\n* There is too much going on in Figure 3. It Would be better to split this up into a few different figures, rather than trying to make several points with the same figure.\n* A more direct illustration of the trade-off between coverage and accuracy would be good. For example, plots with coverage on the x-axis and accuracy on the y axis would illustrate this. One could construct these in a similar way to precision--recall curves or ROC curves, by evaluating coverage/accuracy for a large number of different incompetence score thresholds. The area under this curve would also provide a single metric for doing evaluation in a threshold-free manner.", "broader_impact_concerns": "No significant broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "SeetMXpy7T", "reviewer": "Reviewer_hwUH", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I have several major concerns with this paper, most of which I believe are fixable, but probably not in this submission cycle. I encourage the authors to take the time to redesign the paper, reframe its contributions, and possibly resubmit, but I cannot recommend for acceptance currently, nor would I feel comfortable doing without major revisions.\n\n1. The earlier sections (even the abstract) belie the actual contribution of the work. The paper is framed as \"introducing\" this idea of abstention (\"learning to reject\") despite there being a long history of existing work on this (see point 2 for discussion on references). Simply giving a new name (\"competence score\") to an existing concept is not grounds for claiming to introduce a new framework, which is exactly what the first line of the abstract does. What this paper *does* contribute is an empirical analysis of several existing methods. The use of phrases with \"For comparability with prior work\" are misleading and suggest that this paper is presenting a new idea or method.\n\nThe line \"We present a comprehensive experimental evaluation of incompetence scores for classification and highlight the resulting trade-offs between rejection rate and accuracy gain\" is reasonably accurate, in that it begins to describe what it is that this paper actually does. But the first five pages of the paper discuss things that have already been done, and most of the experiments are relegated to the Appendix. Since this paper is an experimental study, why not spend more of the paper presenting and discussing actual findings?\n\n2.  References to prior work are *exceptionally* sparse and poorly researched. Several times the paper references long lines of work with a great deal of literature from as far back as half a century ago (e.g. Chow, 1970) by simply citing a single survey from the past few years. The Related Work section is a little better, but still leaves a lot to be desired. As an extreme example, I note that the first subsection which discusses OOD Detection and Generalization **does not cite a single source from before 2021**. Also, why is Gulrajani and Lopez-Paz (2020) given as the source for Domain Generalization in the intro?? This work did not introduce or popularize the problem, nor did it provide a meaningful survey. The paper is full of such questionable citations. The section on Selective Classification is better, but the only citations from before 2020 are a line of work by a single author, further indicating the lack of depth to this discussion.\n\n3. Definition 3.1 is unachievable. What is the value in a formal definition of this sort if the thing being defined does not exist? It also uses the phrase \"for any distribution of interest.\" What does this mean? Is this a formal definition or isn't it? Because that expression doesn't actually mean anything precise. The small amount of math in this paper strikes me as the classic example of math for math's sake---Proposition 3.1 doesn't seem to be saying anything that isn't just as clear in plain English. I think this paper would be much better served fully committing to the \"empirical evaluation\" angle.\n\n4. Finally, I question the meaningfulness of the experimental evaluation: I think such a direction *could* be quite interesting, but it doesn't feel that way in its current form. In particular, the fundamental problem of OOD generalization is that for any given task, one *does not know* the best competency score to use, nor the threshold to set. Rather than fixing $\\alpha=.95$, why not explore a range of values?\n\nWhile the number of experiments run is large, to me the actual value to be gained from these results feels quite minimal. I note that doing a \"post-hoc\" analysis is not very helpful, unless it shows that a particular method is universally superior, which is basically never going to be the case. I encourage the authors to consider the following alternative analysis which I think would be much more informative: given a set of training environments, do LOOCV to **select** the best competence score and/or threshold according to one's desired maximum rate of abstention. **Then**, use this to evaluate on the held-out domain. This study would give a meaningful demonstration of what to expect on a *future, real-world* distribution shift where we do not have access to the test domain and therefore cannot identify what is the correct competence score or threshold to be using. An important question to consider when framing the current results: what good does it do to tell us that the quality of the $\\alpha=0.95$ threshold depends on the competence score and the distribution? Is this not already obvious, given the difficulty and variety of distribution shift in general? Why is it helpful to show that for one specific benchmark, a specific pattern emerges, if this pattern won't be true for future shifts?\n\nAs I said before, all of these issues can be remedied---but they are large enough in the current submission that I can't recommend acceptance, even keeping in mind the TMLR guidelines.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper performs an empirical analysis of the performance of various existing methods for \"OOD detection\". This is in the context of using these methods to abstain on some fraction of the test points in the hope that average accuracy on the non-abstained points will increase.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "- The paper should be clearer about what it is *contributing as new* versus what it is *building upon*. The work does not present a new framework, nor does it suggest a new method to \"compare with prior work\". It is an empirical evaluation of existing approaches. This is a valuable contribution! But it must be honestly presented as such.\n\n- The references and related work need quite a bit of work to give credit appropriately. Simply citing one recent survey or famous paper from the past 3 years is not sufficient.\n\n- I think the work would be greatly improved by dropping Proposition 3.1, and presenting the definition as a desideratum (which is unachievable, but may be something to strive for). In general, the math here doesn't really add to the paper.\n\n- The idea of thresholding a \"competence score\" such as confidence or some other measure of \"OOD-ness\" already exists. Rather than spending so much time rehashing it (see the first point above), the paper would benefit from more time spent presenting experimental results, which are the meat of the contribution.\n\n- While the number of experiments run is large, the actual value to be gained from these results feels quite minimal. A more thorough evaluation which explores something more applicable to real-world use cases feels apropos and would strengthen the results.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Please see the summary of the strength and (addressed) weakness as above. Don't forget to consider reviewer hwUH's minor comments in the camera-ready version.
Decision: Accept (poster)