Aggregation Hides Out-of-Distribution Generalization Failures from Spurious Correlations
Authors: Olawale Elijah Salaudeen
Haoran Zhang
Kumail Alhamoud
Sara Beery
Marzyeh Ghassemi
Venue: NeurIPS 2025 spotlight ()
Keywords: OOD Generalization Benchmarks
Primary area: deep_learning

Abstract: Benchmarks for out-of-distribution (OOD) generalization often reveal a strong positive correlation between in-distribution (ID) and OOD accuracy across models, a phenomenon known as “accuracy-on-the-line.” This pattern is commonly interpreted as evidence that spurious correlations—relationships that improve ID but harm OOD performance—are rare in practice. We show that this positive correlation can be an artifact of aggregating heterogeneous OOD examples. Using a simple gradient-based method, OODSelect, we identify semantically coherent OOD subsets where accuracy-on-the-line breaks down. Across widely used distribution-shift benchmarks, OODSelect uncovers subsets—sometimes comprising more than half of the standard OOD set—where higher ID accuracy predicts lower OOD accuracy. These results suggest that aggregate metrics can mask critical failure modes in OOD robustness. We release code and the identified subsets to support further research.

A Aggregation Hides Out-of-Distribution Generalization Failures from Spurious Correlations
Olawale Salaudeen
Haoran Zhang Kumail Alhamoud
Sara Beery Marzyeh Ghassemi
Massachusetts Institute of Technology
Correspondence to olawaledmit edu.
Abstract
Benchmarks for out-of-distribution (OOD) generalization frequently show strong positive correlation between in-distribution (ID) and OOD accuracy across models, termed cee-uracy-oo-th-llnneee
This pattern is often taken imply that spurious correlations- correlations that improve ID but reduce OOD performance are rare in practice. We find that this positive correlation is often an artifact of aggregating heterogeneous OOD examples. Using a simple gradient-bart based method, O0DSelect, identify semantically coherent OOD subsets where accuracy the line does not hold. Across widely used distribution shift benchmarks, the O0DSelect uncovers subsets, sometimes up to over half of the standard OOD set, where higher ID accuracy predicts lower OOD accuracy. Our findings indicate that aggregate metrics obscure important failure modes of OOD robustness. We release code and the identified subsets to facilitate further research.
correltions-ccocclattion
1 Introduction
(OOD) generalization have shown consistent pattern that models performing well on the training distribution also perform well out-of-distribution. a trend known as accuracy-on -ottoo--------iieeee (AoTL) (Miller et al. 2021 Taori et al. 2020). This pattern has often been interpreted as evidence that spurious correlations features that improve in-distribution (ID) accuracy but harm OOD performance are uncommon in practice. We show that this apparent robustness misleading. When OOD data are disaggregated, large and semantically coherent subsets emerge where higher ID accuracy predicts lower OOD accuracy, phenomenon we term accuracy-on-leee inverse-line (AoTII.). These hidden subsets reveal that aggregation can mask major failures of OOD) robustness, suggesting that existing benchmarks may underestimate the prevalence and impact of spurious correlations.
Benchmarks for
The promise of machine learning lies in generalization, the ability to perform a task on new data with similar effectiveness as on the training data (Blumer al. 1989 Vapnik 1999: Shaley-Shwartz and Ben-David 2014] Zhang et al. [0166] Belkin et al. (201)). Yet models deployed in dynamic world often encounter data from different distributions (Shimodarra 2000 Moreno- Torres et al.) (012) and
fail. For instance, medical diagnosis model trained on data from one hospital may perform poorly in another with distinct demographics or equipment (Zech al., 2018 Yang et al. 2024a), and an animal classifier may misclassify images captured under new conditions (Beery et al., 2018 Xiao ctt al. 2020). Generalization under such shifts, from in-distribution (ID) training to out-of out-of-distrionn distribbbtt (OOD) testing or deployment, defines domain generalization (Zhou et al. 2022 Wang et al. 2022).
These observations motivate closer examination of what benchmark correlations actually reveal about robustness and when they conceal spurious mechanisms that undermine OOD generalization.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Pearson R(ID, OOD)
1.0
Poop Polental O0D sel
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BELECTiD o0D
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BEL BELEcTED. 000
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Oitenr hom Pie
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10% 25% 50%100%
Seear
Pipi Traniin
OODSelect % of Terrainco
SCLBOT GO
DODSelect
Most Misclassifeed
Random
(a) AoTIL: With original OOD data, ID and OOD accuracy across of models has Pearson corre- lation of 0.89. O0DSelect finds up to 1000 (~ 16%) examples from L46 on the same models with an ID- OOD correlation of -0.77.
(b) O0DSelect Strategy: Excluded examples resem- ble the training distribution (e.g., centered bobcats daylight), while included DODSelect examples differ (c.g., occluded bobcats, infrared camera capture).
Figure Aggregation Masking AOTIL. Consider Terra Incognita, where ID data are drawn from camera traps at locations L100, L38, L43, and OOD data from L46 (Beery et al. 2018). Aggregation masks the effect of spurious correlations on generalization, such as daylight, even though substantial number of OOD samples still systematically misclassified. Note that O0DSelect examples differ from the most misclassified examples, which always have an ID-OOD accuracy correlation of zero. Confidence intervals correspond to 95% Fisher z-intervals.
In this work, we establish the existence of large and semantically coherent OOD subsets in state-of- the-art datasets with accuracy on the inverse line. Specifically, our contributions are:
We show that in state-of thettfftt-rttt domain generalization benchmarks, there exist large, seman- tically meaningful OOD subsets sometimes up to over half of the data- correlations low as 00.9 Pearson R (Figure 2 The familiar accuracy-on- -hee---iee trend only emerges once such subsets are aggregated with the rest the data.
We show that these subsets are not arbitrary: for example, Chest X-ray diagnosis tasks, models that improved overall performance performed worse on patients with pleural condi- tions and enlarged cardiomediastinum
We propose O0DSelect, simple yet effective selection procedure identify such subsets across datasets, when they exist.
We provide the identified subsets for state-of-the datasets, including those from Do- mainBed (Gulrajani and Lopez-Paz] (2200) and WILDS (Koh al. [021]) to facilitate future research (included in the supplementary material).
We provide the code and selected subsets for our proposed OOD selection method and analysis.
2 Background and Related Work
The field of OOD generalization aims to develop models that are robust to spurious correlations (Zhou
et al. 2022 Wang et al. 2022). Many of the state-of- ootte-of-thh-arr methods in domain generalization rely
on notions of distributional invariance (Arjovsky et al., 2019 Krueger et al., (021): often using
causal motivations (Peters al. 2016 Heinze-Deml
al. 2018: Salaudeen and Koyejo 2024
Salaudeen et al. 2024). Progress in the field of domain generalization has primarily been eva uated
by two benchmark suites: DomainBed (Gulrajani and Lopez-Paz 2020) and WILDS (Koh al.
2021). However, various studies have suggested that none of the proposed domain ceneralization
methods consistently outperform naive empirical risk minimization on these benchmarks (Gulrajani
and Lopez-Paz 2020; Koh et al. [021]: Yang et al. (2023). Moreover, previous work has suggested
that improving accuracy tends improve OOD accuracy, i.e., strong correlation between ID
and OOD accuracy holds, termed accuracy on the line (Miller et al. 2021 Taori et al. 2020)
Saxena

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et al. 2024 Sanyal et al. 2024). However, Teney et al. (2023) demonstrate that with more diverse selection models, fraction of real-worte datasets do indeed exhibit other correlations besides strong and positive correlations between ID and OOD accuracy. (Saluuemone,Saluudeen (Salaudeen et al. (2025a)
provides theoretical analysis that suggests prioritizing datasets without accuracy on the line; our proposed method provides OOD sets that satisfy such conditions by selecting subsets of existing benchmarks with accuracy on the line.
Existing subset discovery methods-sune as Slice Finder, SSD++, and DivExplorer Polyzoti al. 2019 Proença et al. 2022. Pastor et al.) 2021 Subbaswamy and Saria, 2020) relyy on
explicit grouping cues, categorical features, or annotated attributes to define candidate subsets, In contrast, our setting assumes no access to such metadata and requires model-agnosticism. motivating simple yet effective selection approach. Influence functions (Koh and Liang] (017)) may appear suitable at first glance, but they rank training points by leave-one-out influence, rather than partitioning the lest/OOD set. Thus, applying influence functions in this context would still require an additional heuristic to define coherent subsets, while also inheriting known fragilities modern
deep networks (Basu et al. 2020 Epifano et al. 2023 2019
Bae et al. 2022 Grosse et al. 2023 Koh et al.
Hu et al. 2024).
Our proposed method in the next section provides a simple, efficient, yet effective approach.
Methodology
First, we define the correlation property that is used to determine AoTL or AoTIL. Definition 1 (Correlation Property; Miller et al. (2021)). Define a E RR c 2 O, and क- as the inverse Gaussian cumulative density function. The correlation property is defined as
(accp,, (f)) - a -- (acC (f)) Se vf.
(1)
Definition |
iimplies:
Pearson R(X, Y),
(2)
[a] ay
where By is the standard deviation of Y. Thus, the correlation property implies that the transformed ID and OOD accuracies lie approximately on a line and are strongly linearly correlated. Moreover, the sign of the Pearson correlation is determined by the sign of a: if >> 0, the correlation is positive, and if < 0, the correlation is negative.
Problem Setup. Suppose we have N models f, and d potential OOD examples. Let Z E Nxxx where Zig is 1 if model fi correctly classifies example j and 0 otherwise. Define accıD E R where (acem)) ) is the held-out in-distribution accuracy of model fi. In this work, we are always operating on 0- - 1-clipped probit transform of accuracy. Define a sample selection vector s E {0, 13 that indicates which examples select from the candidate OOD set, and denote the selected OOD accuracy for model fi
Zig.ss ||s||2
acCID acc
(acc
(3)
and corr(acCın, acc)))
|accm
|accc
where corr is the Pearson correlation between ID and OOD accuracies. Note that the probit- transformed accuracies are mean-centered before computing the correlations.
Objective. We aim to learn a selection vector E {0, 1]d (with S ||s||| fixed) to minimize the correlation between accp and acc -icealllyllly with large S.
Importantly, a subset achieving weak or negative correlation may not exist, particularly if there are no spurious correlations with respect to the ID and OOD distributions. Additionally, the change in Pearson R from adding model or OOD example is bounded by OCC/vm) where m is the number of models or OOD examples and depends on the accuracy range and the Lipschitzness of --1 (thcoretical analysis provided in Appendix B Lemma
3

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Algorithm 1: 00DSelect: Selecting OOD subsets without accuracy-on-the-line
Input: DDD", " in-distribution train/test splits;
DooD: out-of-distributiin dataset;
S E N number of OOD samples to select
Output: Subset DOOD DOOD of size S
1: Train models diverse models on Di
2: Let accıD E IR Noone be the vector of probit-rransformed accuracies, where accı,; denotes the
accuracy of model on DiD
3: Construct binary matrix Z E {0, 1) Nasta x (Doool
where:
[1
if model correctly classifies OOD sample j
Z, =
0
otherwise
Let acc IR Nnon denote the per-model average accuracy vector across the OOD examples selected by that is,
acc
Ze
||s||1
where Z denotes the columns of Z indexed by 6: Solve the optimization Equation to find D from DOOD-
We consider constrained objective for selecting S OOD examples:
min (
corr(acCıD. accooD) subject to
||s||| S.
We relax this objective to:
min sE[0,1]
corr(accm, acc + A A. (5- - ))
(5)
where s is the output of a sigmoid function in practice.
Soundness of the relaxation and optimization. Our objective is non-convex and non-submodular (Lemma While global optimization is intractable, the Lipschitz property ensures stable gradients and bounded progress under descent, enabling convergence toward near-binary stationary points that approximate the discrete optima. Non-submodularity also eliminates greedy selection
(Proposition [ but Lipschitz-continnee
optimal strategy. We use the Adam optimizer (Kingma and Ba 2014) to optimize Equation 5 We use a cosine annealing schedule to adjust the learning rate and X [osshchioo and Hutter 2016). Additional details available in Appendix
Selected Subsets. Although we are free to choose examples, a subset that makes ID-OOD Pearson R negative is not guaranteed to exist. The OOD accuracy of each model is average over the selected examples. The subset must systematically up- or down-weight groups of examples on which nigher-accurace ID models tend to underperform relative to lower-accuracy ID models. We provide evidence that finding a large shorteuts in the data, trivial consequence of sub-sampling. Importantly, do not select models or alter the ID accuracies; we always correlate the same length-N vectors, only the OOD accuracy values change through the choice of examples.
subset is evidence latent structure or spurious
For brevity, we reserve details of other theoretical analyses for Appendix B as our results are included for thoroughness, but they are standard (Bertsekas) 1997 Nocedal and Wright 1999).
Fisher Confidence Intervals. our estimate of correlations, we compute Fisher = intervals for each correlation estimate, indicating the range of variability expected from estimation; overlapping bars suggest that differences could be arbitrary, while non-overlapping drops signal meaningful differences in correlation.

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Table Dataset Summary. Each dataset defines classification task across multiple domains. Full OOD Size refers to the size of the OOD dataset. We select the full dataset to apply O0DSelect according to splits in DomainBed (Gulrajani and Lopez-Pazz 2020) and WILDS (Koh et al. (021). WII.DCCamelyon-H4422 Ho44H refer to the versions of the dataset where hospitals 4 and 5 are considered OOD distinctly. Importantly, we never train models on either and separate them here because they have distinct properties and results. WILDSCivilComments is text dataset.
Full OOD
Largest CODSelect
Dataset
Task (# classes)
Domains (#)
Size
Subset w/ AoTIL 0.3)
Models
Finding
Hospital
Chest Xrays
No Finding
(2)
systems
(5)
71433
55000 (75%)
1800
Object
PAC3
Styles
3929
250 (6%)
2804
classification
Object
Visual
VLCS
classification (5)
domains
(4)
2656
800 (30%)
4200
Wildlife
Camera
TerraIncogni
classification (10)
(4)
6122
1500 (25%)
2980
traps
WILDSCamelyon H4
Tumor vs.
Hospitals (5)
129838
35000 (25%)
944
Normal
Tumor vs.
WILDSCamelyon-HH
Hospitals (5)
146722
60000 (40%)
944
Normal
Toxic
WiLomcivilCommeut
Not Toxic
(2)
Demographics (8)
52823
25000 (50%)
710
Experiments
Procedure. Table summarizes the datasets we study. Given a typical distribution shift benehmark with at least two domains, i.e., D {Di, D we fix Dn. DooD C 'D pair, which are disjoint sets (concatenated) of domains. This pair denotes an experimental setting. this work, we focus on the standard DID DooD splits the community uses for each dataset (Gulrajani and Lopez-Paz 2020 Koh et al. 2021 }. For each split, we apply our methodology identify subsets D8 DooD with AoTIL Appendix A Algorithm!
Datasets. We consider real-world tasks and distributions such as predicting "Finding"/"NN Noo Finding" from Chest X-rays where domains ID domains are from CheXpert (v1.0 (v1.0-small) (Irvin et al. 2019
ChestXray8 (Wang et al., 2017), PadChest (Bustos et al., (020). and VinDr- CXR (N Nguyen et al. 2022). The OOD domain is MIMIC-CXR-JPS (Johnson et 2019). We also study WILDS (Koh et al. 2021) benchmarks that capture real-world shifts. WILDS WII.DS-Camlyon (Bandi et al. 2018) targets cancer detection from histopathology slides across hospitals. WII.DS-CivICoomee
(Borkan eet al. 2019; Koh et al.) 2021) classifies online comments as toxic or non-toxic across demographic subgroups, with OOD domains defined by shifts identity attributes such as gender, religion, and race. We also study DomainBed (Gulrajani and Lopez-Paz (020)) benchmarks reflecting different forms of distribution shift: style, dataset collection, and environment. PACS (Li et al.) 2017) involves object classification across artistic styles (7 classes across Photo, Art Painting, Cartoon, and Sketch), with Sketch as OOD. VLCS (Fang et al. (2013) spans 5 classes across 4 datasets (V0C2007 (Everingham et al. 2010), LabelMe (Russell et al. 20088. Caltech101 (Fei-Fei et al. 2004), and SUN09 (Choi et al. 2010)), capturing collection biases; LabelMe OOD. Terra Incognita (Beery et al.) 2018 Tocuses wildlife recognition across 4 geographic locations, with L46 OOD.
Models. We construct a diverse population of models by varying architecture (from VGG to Vision Transformers, listed below), pretraining weights (TorchVision Vision maintainers and contributors 2016; Deng et al., 2009 He et al. 2019), initialization (from scratch and transfer learning), and hyperparameters. We train up to 4200 models (Figure 7 with various vision architectures, including variants of ResNets (He et al. 2016). DenseNets (Huang al. (017). MobileNets (Howard) 2017).
ViT Dosovitskiy et al. 2020). VGG (Simonyan and Zisserman, 2014), and Inception (Szegedy
et al. 2015). We do the same for our language experiments, from BERT (Devlin et al. 2019) to
GPT22 [Radford et al. 2019). A full list of models is provided in Appendix
Models are split into disjoint train, validation, and test subsets, i.e., the models used for learning the selection, cross-validation, and final testing are non-overlapping. For a given ID/OOD setting, all models are trained on the same ID training data and evaluated on held-out ID test set and candidate
5

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Pearson R OOD)
1.0
0.5
0.5
0.0
0.0
-0.5
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O0DSelect % of WILDSCiviIComments
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OODSelect % of Terralncognita
Pearson R(ID, OOD)
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OODSelect % of VLCS
OODSelect % of PACS AoTL
OODSelect
Weak Correlation
AoTIL
Random
Most Misclassified
Figure 2: Comparing AoTL and AoTIL. Pearson Correlation between ID and OOD accuracy as a function of the number selected OOD samples. Correlation values above 0.3 indicate AoTL, while below -0.3 is Selection data samples and the Most Misclassified at fixed size intervals from 100 to over 100,000 (normalized to sample size in the figure). Random selections yield strong positive correlation, while misclassified samples have weak correlations; that is, our method does not conflate spurious correlations with general difficulty (e.g., label noise). 00DSelect identifies subsets where ID and OOD accuracy are negatively correlated-in one case (CXR) for over 70% the usual OOD dataset. This behavior is depeeett due differences in distributional properties. Table enumerreeeee detailed correlations.
in between considered weak. We compare Random
OOD subsets. The resulting paired ID/OOD accuracies are used estimate the correlation between ID and OOD performance. Further discussion on implementation is provided in Appendix A
On the Necessary Quantity Models. We determine the minimum number of models sample by thresholding the relative change in ID and OOD accuracy correlation across the full dataset. We select at least a number of models such that adding new model changes the correlation by less than 1 1% (Schönbrodt and Perugini 2013) Bonett and Wright 2000). Notably, the diversity and quantity of models we consider are orders of magnitude higher than in previous work (Miller et al. 202Th; in some cases, tens vs. thousands (ours). This number is also dataset dependent; for instance, 1010 models are needed to satisfy this criterion for the VLCCS dataset, while only 610 are needed for WILDS Camelyon. Further details provided in Appendix Figure

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0.72
O0DSelect (AIl)
Pearson R
-0.60 (0.86)
Slape 0.24 (124(1.6)
(-0.34)
0.65
8
40
0.63
0.6
OODSelect
0.72
0.73
0.74
0.76
0.78
0.8
Present Not Present NO FINDING
Present Not Present PLEURAL OTHER
Acc
(a) CXR No Finding accuracy correlations across mod- els has strong global correlation (0.86) while the OODSelect subset (55000 examples and 77% of the full dataset) has negative correlation (-0.60).
(h) There is minimal label shift the full OOD set and the OODSelect subset. Finding has prevalence of 39% in the OODSelect samples and 38% in the full dataset. However, the prevalence of "Pleural Other" in the full OOD set and OODSelect subset is 61% and 68%, respectively.
Figure 3: CXR No Finding. Figure 3a suggests that poor generalization may arise for large subset of the OOD population from reliance on spurious correlations. However, aggregation hides this failure mode since the correlation for the full OOD set is strongly positive. This selected subset also has a prevalence shift from the full dataset; statistical significance for the prevalence shift was assessed using bootstrapping with 1000 resamples.
5 Empirical Results and Discussion
Findings. Overall, we find that many benchmarks contain O0DSelect subsets of examples that exhibit AoTIL weak correlation, though the size of such subsets varies. The same benchmarks exhibit AoTL when all OOD samples are aggregated (Figure 2).
We treat R 0.3 weak Pearson correlation between ID accuracies and OOD accuracies across models. Using O0DSelect, our method for selecting the OOD data, we uncover large variance in correlations that hidden in the full splits. In CXR No Finding, the full OOD set gives strong positive correlation (Figure 2), however, O0DSelect retaining > 70% of the data has a strong negative correlation (Figure For Terra Incognita, the full OOD has strong positive correlation, but a 30% slice from 00DSelect has a notable negative correlation.
The extent of the existence of such subsets clearly varies across datasets and may not exist in others. For instance, for PACS, small 00DSelect size making up 8% of the full dataset has a correlation of 00.33; at 60% % the correlation is already negligible, 0.01, and becomes strongly positive as the size of 00DSelect grows. The full dataset has a correlation 0.81.
Focusing specifically on the most misclassified examples, we find that the ID-OOD correlation is near 0 and rarely invert it as the ODDSelect examples do. This demonstrates that our selection does not conflate spuriousness with general difficulty (e.g.. uniform label noise). Random selection consistently preserves a strong positive correlation similar to full dataset, as expected. Thus 00DSelect reveals systematic generalization failures due to spurious correlations, which may go undetected under standard evaluation across domains.
Clearly, to achieve a negative correlation from a positive correlation, we need (i) models that performed well on the full OOD set perform relatively worse on the O0DSelect set, and/or (ii) models that performed poorly on the full OOD set to perform relatively better on the O0DSelect set. For instance,
VLCS, models with relatively low ID accuracy performed better on the O0DSelect set than the full OOD set. In contrast, the models with relatively high ID accuracy performed better on the full OOD set. For all of our trends, the slope and intercept are determined by these relative performance changes and are dataset dependent. In some datasets, some models still perform near or below chance on the OODSelect (Terra Incognita) while in others, all models are above chance (WILDSCamelyon).
7

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Pearson R(ID, OOD)
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10%
O0DSelect % of CXR No Finding
O0DSelect
Most Misclassitied
Random
Most Misclassified
Random
(a) CXR No Finding
Pearson R.
CXR No Finding
Spearman Rank.
Figure 4: The correlation directions are not driven by outliers - Spearman rank is robust to outliers
while Pearson is not. Still, the trends similar (full results in Figure 2 and 8
Pearson R(ID, OOD)
1.0
Pearson R(I,, OOD)
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O0DSelect % of CXR No Finding
OODSelect % of CXR No Finding
00DSelect
Most Misclassified
Random
O0DSelect
Most Misclassiried
Random
Disjoint architecture families Pearson R.
Disjoint architecture families Spearman Rank.
Figure 5: Independent Architecture Families, Our findings hold even when train/test models are from disjoint architecture families, e.g.. ResNets vs. ViTs.
On the effect of outliers. Some models may be outliers and skew the observed trends. Consequently. evaluate Spearman rank correlation, which robust outliers than Pearson R. We find that our conclusions remain unchanged (Figure Spearman rank results are provided in Appendix A
On potential architecture confounds. While we randomly split models into disjoint sets for identifying OOD subsets and computing correlations simulate i.i.d. sampling from a model population, architectural similarities (e.g., ResNet-50 vs. ResNet-152) could introduce confounding effects. test this, we perform ablations where model families are disjoint-e.g. ResNets appear only in the training-validation set or only in the test set, but never both find that this restriction indeed changes the strength the correlation, yet does not alter our conclusions. Figure S gives an example for CXR No Finding, which has the strongest examples of AOTIL. However, given that architectures have different inductive biases, models may learn diffcrent spurious correlations utilize them differently in docision-making Sampling from entirely disjoint population of architectures mitigates the observed strength of spurious correlations learned by model families.
On Vision Language Models (VLM) Trends. We investigate the same trends hold with vision- language models' zero-shot performance (Shi et al. 2024). We generally find strongly positive correlations between the ID accuracy and the accuracy on O0DSelect examples. The weakest correlations are: PACS (0.78), VLCS (0.62), Terralncognita (0.84). WILDSCamelyon (1.00), and CXR (0.94). This should not be interpreted as evidence of VLMs' robustness to spurious correlations. From the point of view of the VLMs in this experiment, both the ID and D0DSelect examples are OOD, since the VLMs were not explicitly trained on either set. Alternatively, since many of these datasets are publicly available, the VLMSs may have been trained on the dataset sets, i.e., all of the examples are in distribution.

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VLCS
PACS
0.0
Terralncognita
WILDSCamelyon (H4)
WILDSCamelyyn (H5)
Nomallze
WILDSCiiiCComments
CXR, No, Finding
0.2
Lower Bound
Upper Bound
60
100
Selection Size (% Dataset Maximum)
Figure 6: Consistency of selected subsets. Across all datasets and subset sizes, our normalized Jaccard index is greater than would be expected from arbitrary selection (lower bound).
Table 2: Model generated description of selected OOD set vs. ID set.
Dataset
Model Generated Semantic Difference
PACS
extreme wide-angle shots; extreme close-ups; extreme weather conditions.
VLCS
unusual object interactions with urban environments; x-ray or radiographic. frost; motion blur; extreme weather conditions; reflections or glare.
Terralncognita
Selection via latent space distance. As a baseline, we implement selection method that greedily selects the farthest OOD examples from the ID examples in the CLIP embedding space (Radford et al. 2021). Across datasets, this approach often yields positive ID-OOD correlations (e.g.. R 0.52 on PACS with N 10), and in some cases even stronger correlations than random selection. However, it consistently fails to capture the weak and negative correlations identified by OODSelect (e.g.. R = 0092 VL.CS with N 10). These results show that distance-based selection, while intuitive, overlooks the feature-laber correlations that drive OOD errors, and thus cannot uncover the failure modes revealed by method.
Selection Consistency and Coherence. The identified subsets are also consistent and coherent. We select each S subset independently and do not enforce that smaller subsets subsets of larger subsets. Still, we find that such consistency holds. We measure consistency with the normalized Jaccard Index (Jaccard 1901 E [0, 1]. For z C examples,
I (22;,2)) 23)
2; n 2z 2. 2
J, = T
Σ k=1
T
J(z,, 2x+1);
Jz J. Jmin
(6)
Jmax Jmin
for 7 selection sizes, where Jmin is computed with random selections and Jmux with z; C 2j for all i,j. j. with < j. and the sizes of the sequence 2 's are preserved. This normalization is necessary since the subsets are of different sizes. Our selected subsets are indeed consistent (Figure(6).
CXR Semantic Coherence. The CXR dataset, predicting Finding/No Finding chest X-rays (CXR). is an example where we have demographic and clinical metadata that we can use to study the semantic coherence of our subsets. Figure illustrates how average OOD performance can mask systematic failures in specific subsets. For instance, when selecting a subset with 5000 examples, Figure 3 the ID/OOD accuracy correlation between ID and OOD on the selected subset is strongly negative, while it is strongly positive when we aggregate over the full OOD set.
We then analyze both demographic and clinical attributes. By comparing prevalence rates between the selected subset and the overall OOD pool, we find statistically significant shifts in several at- tributes, specifically sex, race, Pleural Other, Support Devices, and Sex-Ethni city, determined via bootstrapping with 1000 resamples, Figure However, most datasets have no such metadata. Our normalized Jaccard Index supports consistency and coherence for such datasets.
Potential for model-generated semantic coherence. As a potential future research direction, we investigate the utility of large and vision language models to generate semantic concepts more likely to
"FFinding" indicates the presence of condition from predetined and "No Finding" indicates otherwise.

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be true for our O0DSelect set than the rest of the dataset (Dunlap et al. 2024). We apply the following process. Step A VLM generates captions for all images; we use Qwen2. 3-B Instruct (Yang et al. 2024b Team 2024). Step 2: A large language model (I.LM) then proposes candidate natural language descriptions that are more likely to apply to the selected OOD set than others; we use (Fini et al. (022). Step 3: A vision-language model scores and ranks these descriptions based on their distinctiveness to OOD images, identifying interpretable attributes that differentiate the two distributions; we use CLIP (Radford et al.] 2021).
AIMV2-1arge-par -atee-attthhh2-----111
Table provides example descriptions for natural image datasets. We find that this strategy does not yield consistent and robust results, as the descriptions do not capture feature-label correlations, although some of our findings are promising. Additional details are provided in Appendix C
Importantly, many potentially spurious predictive features are incomprehensible to humans and may not be expressible in natural language (Szegedy et al. 2013 Goodfellow et al., 2014] [Iyaa et al.. 2019). As a result, approaches that rely on vision-language models for explanation are likely still insufficient for identifying the subsets we uncover, for instance, through brute-force selection. They also cannot be expected to capture all differences in correlations between ID and OOD examples. This is true for many of the datasets in this work, such as CXR and WILDSCamelyon.
Limitations. Our analysis is computationally intensive, requiring training up to 4200 models per dataset and optimizing a selection objective of up to around 146000 elements. However, this computation is a one-time cost per dataset; we publicly release the resulting selections, covering many state-of-the-ar domain generalization benchmarks. Additionally, semantic explanations of the O0DSelect set are challenging for datasets such as WILDSSammelon,, whose features are images of tissue cell slides, without extensive metadata. Even when metadata is available, it may not fully represent the signals that capture spurious correlations. Notably, this is also an unstated challenge for the original datasets, where OOD sets are selected based on metadata such as hospital sites, but also contain no information explaining what spurious correlations exist or are expected. Furthermore, it unclear if we can expect semantic explanations for all spurious correlations since many features models rely on are imperceptible to humans (Szegedy et al. 2013 Goodfellow et al. [0014] [lyas] itt al. 2019). For instance, AI systems can predict race from chest X-rays with features that are thus far imperceptible to humans Gichoya et al. 2022
Broader Impact. One alternative perspective of our results is that correlations that hold in aggregate not spurious (Wenzel al.) 2022). We propose that aggregate performance is narrow view of the effect of spurious correlations. For instance, if spurious statistical associations reflecting historical or structural bias, such occupation and gender, which can bias the outputs of recommendation systems (Caliskan et al., 2017: Balagopalan et al. 2025), are pervasive in the real world. Then, benchmarks collected naturally from wooldw distributions whose results are aggregated broadly may preserve such correlations across both training and test environments. As a result, models that rely on such spurious correlations may continue "perform well OOD," making the correlation appear benign in evaluation. However, this only creates the false impression that spurious correlations not harmful OOD, even though they degrade performance on affected subsets of the data. Our work surfaces these subsets and advances more robust evaluations of OOD robustness.
6
Conclusion
Spurious correlations do not vanish in the real world; current benchmarks and performance metrics simply hide them through aggregation. By disaggregating OOD data, we revealed large, semantically meaningful subsets where spurious correlations harm performance. The consequential validity (Mes. sick 1995, Salaudeen et al. (025b) of distribution shift robustness benchmarks, e.g., robustness to subpopulation shifts (Yang et al.) 2023 Sagawa et al. 2019), requires identifying such subsets.
Recommendations. Future work in this area of research should (i) adopt our selection protocol as a robustness check for any new OOD benchmark, (ii) treat identified large 00DSelect subsets as first-class evaluation targets, and (iii) design methods that improve both average and subset robustness. A discussion on interpreting the results of subset performance can be found in Pfohl et al. (2025). We hope the released code and DODSelect subsets become stepping stone toward benchmarks and models that confront the adverse effects of spurious correlations.
10

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Acknowledgments
MG acknowledges partial support by the National Science Foundation (NSF) 22-586 Faculty Early Career Development Award (#2339381), Gordon & Betty Moore Foundation award, a Google Research Scholar award, and the A12050 Program at Schmidt Sciences. We thank the anonymous reviewers of NeurlPS 2025 for their thoughtful feedback and recommendations. SB acknowledges partial support by the Schmidt Sciences A12050 Program, NSF Awards No. 2330423 and 244 1060, and NSERC Award No. 585136 Any opinions, findings and conclusions or recommendations expressed this material are those of the author(s) and not necessarily reflect the views of the NSF, NSERC, or Schmidt Sciences.
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Appendix Table of Contents
Empirical Analysis
17
A.1 Compute
19
A.2 Spearman Rank Results
19
A 3 Table of Results
21
B
Theoretical Analysis
27
Lemma
Bounded effect of New Models on Pearson |eatoo..........................
27
B.2 Lemma
2
Bounded effect of New Examples Pearson R
................................
27
B.3 Lemma
Lipschitz Continuity Selection Objective
"ered...........................
28
B.4 Proof of Proposition
|
Non-Submoduuarityy
29
ID/OOD Explanation
30
A Empirical Analysis
Vision Models. AlexNet, ResNet- (18/34/55//0111152) DenseNet- (121 /1/////////1///11
MobileNet (v2/V3 Small/V3 Large),
EfficientNet- (B0/B1/B3/B7). ViT-(B/16, B/32, L/16), RegNet-Y (400MF /BDP///000F///// / /3..6/FF/..20 /GFF//)FF)
ConvNeXt- (Triny/SSaall/BaseLLaatt
Swin
Transformer-- (aan:er//itttymmaammaaaaaaa VGG- (C-111113/1//199) / (1.0/1. 10, and Inception v3.
Text Models. BERT (base/large), BioBERT, LegalBERT, FinBERT, (v1/22- baue///aggg/laarrg/xxarrlgg DeBERTa-v2- aataaaaal/mmaa/aaseeeeeeeee DistilBERT- (base/cased, distilled), T5- (small), BART- (ART-(tase/larree/mm1))) and GPT-2- (small).
SciBERT,
RoBERTa-
Lbass/rrerr
See Figure 7 on quantity of models trained.
Dataset. PACS involves object classification across artistic styles, with classes ("dog", "clephant" "giraffe", "guitar", "horse". "house". "person"" and 4 domains: Photo, Art Painting. Cartoon, and Sketch. We consider a setting where Sketch is the OOD domain. VLCS contains 5 object classes ("bird", "car", "chair". "dog", "person") shared across datasets: VOC2007 (Everingham 2010), LabelMe (Russell et al.] 2008). Caltech/01 (Fei-Fei et al., (2044) and SUN09 (Choi et al. 2010). Each domain reflects a different dataset source with distinct collection biases. We consider setting where LabelMe is the OOD domain. Terra Incognita focuses on wildlife recognition from camera trap images, with classes ("bird"", "bobcat", "cat", "coyote", "dog", "opossumi, "raccoon", "rabbit", "skunk", "squirrel") across 4 geographically distinct domains: 1.38, 143, L46, and I.100. The L46 location the OOD domain (Gulrajani and Lopez-Paz/ 2020).
We also study three WILDS benchmarks that capture distinct real-world distribution shifts. We consider WILDS-Camelyon (Bandi al. 2018) and WILDS- JILDS-CiviCComment (Borkan et al. 2019). These benchmarks encompass medical imaging, satellite vision, and natural language, providing diverse evaluation suite for
generalization under domain shift. WILDD-Cammlyon is histopathology image classification task for detecting cancerous regions in lymph node slides, with domain shifts arising from variations across different hospitals. WILDS- W.L.D--CiviCComment (Borkan et al. 2019) classifies online comments toxic non-toxic across demographic subgroups (male, Jemale, LGBTQ, Christian, Muslim, other religions, Black, White), with OOD domains reflecting shifts in identity distributions.
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1.0
Pearson R
0.9
CXR No Finding (N=810)
PACS (N=810)
0.8
Terraincognika (N-1410)
(N-1010)
0.7
WILDSCamelyon (N-710)
WILDSCCiiCCommens (N-710)
0.6
10a
10a
10a
Number of Models
Figure 7: We train over 35 model architectures with varying hyperparameters, pretraining, and data augmentations, yielding average of [N] trained models per dataset-OOD domain pair. We report the correlation between in-distribution (ID) and out-of-distribution (OOD) accuracy across models, including standard errors at 0.05. To ensure stability, sample enough models such that adding more changes the correlation by less than 1%; vertical dashed lines mark the approximate minimum sample size satisfying this criterion.
CXR ("Finding" vs. "No Finding")) This binary classification task predicts whether a chest X-ray shows any abnormal radiological finding. The in-distribution (ID) domains comprise four widely-used datasets-Chexpper CheXperr 110-small) (Irvin al. 2019). ChestXray8 (Wang al. 2017). PadChest (Bustos et al. 20200, and VinDr-CXR (Nguyen et al., 2022). These sources differ in scanner hardware, patient demographics, annotation guidelines, and prevalence of pathologies. We designate MIMIC- CXR (Johnson et al. (219)] large, aagle-institution dataset collected under distinct clinical workflow- as the out-of-distribution (OOD) domain. This setting captures clinically meaningful shifts (e.g.. hospital protocols, imaging devices, discase prevalence) and offers stringent test of real-world generalization under domain shift.
Train/ Val/Tett Split. To evaluate generalization, we randomly partition the same set of models into train, validation, and test splits (60/20/20). We optimize our selection objective on the training split and identify the best-performing O0DSelect configuration using the held-out validation split. Final results reported on the test split. Importantly, although the selection objective is tuned on one subset of models, the ID and OOD accuracy correlations continue to hold on the held-out test models, demonstrating that the property generalizes across held-out model subsets.
Soundness of the relaxation and optimization. Notably, our objective non-convex and non-submodular [Proposition[] yet Lipschitz-continoous (Lemma Consequently, while global optimality intractable, the Lipschitz property ensures that gradient-bar based methods with a suitably large exact-penalty parameter admit meaningful descent guarantees; in practice employ stochastic gradient descent with multiple random restarts, which consistently converges to high-quality feasible solutions. Formal optimization guarantees and proofs are deferred to Appendix [
Adding the squared regularization term in (5) an exact-penalty reformulation the original constrained problem. Classical results (Bertsekas) 1997; Nocedal and Wright, (999)) state that there exists a finite weight X > such that, for every X 2 A*, (i) every global minimiser of the penalised objective satisfies the budget constraint, and (ii) the optimal value coincides with that of the constrained problem.
Moreover, first-order stationary point that already meets the constraint is unchanged by the penalty term, so the relaxation does not create spurious local optima within the feasible region. Hence, gradient-based search on (5) is sufficient: we do not need to solve the penalized problem to global optimality, and any locally optimal feasible solution we find also locally optimal for the original constrained objective.
We cosine annealing schedule adjust the learning rate and À (Loshchilov and Hutter 2016). Additional details are available in Appendix A
Throughout, refer to this task CXR for brevity.
18

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Necessary Number of Models. Prior work on securacy-on-the line typically trained only tens of models per dataset (Miller et al. (022). contrast, we train thousands models per dataset, spanning architectures from AlexNet to Vision Transformers and incorporating diverse training strategies. To determine how many models are necessary for stable correlation estimates, we incrementally sample models until the Pearson correlation between IDD and OOD accuracies changes by less than 1%. Figure 7 shows where this stability threshold is reached for each dataset. Across all datasets, our experiments far exceed this threshold.
On the Size of O0DSelect. While detailed analysis of thresholding O0DSelect beyond our current scope, generally recommend choosing the largest DODSelect size such that the Pearson correlation is not weak, that R v -0.3, following convention (Cohen, 2013). For some datasets, this threshold may yield very small noisy selections (e.g., PACS), in which case the selected set may not be informative.
A.1 Compute
Table 3: Compute time to reproduce experiments (GPU Hours) per experiment unit on NVIDIA RTX A6000 GPUs.
Dataset
median
std. dev.
min
total
CXR
<1
18
286
PACS
<1
7
109
TerraIncognita
<|
10
292
VLCS
2
2
1
<1
7
183
WILDDCammeyon
350
WIL.DSCiviCoomment
10
9
4
3
18
106
A.2 Spearman Rank Results
19

---

Pearson R(ID, OOD)
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
00.5
-1.0
-1.0
10%
25%
50% 100%
10%
25%
50%100%
OODSelect % of CXR No Finding
OODSelect % of WILDSCivVIComments
Pearson R(ID, OOD)
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
1%
10%
25% 50% 100%
1%
10%
25%
50% 100%
OODSelect % of WILDSCamelyon
OODSelect % of Terralncognita
(Hospital 4)
Pearson R(ID, OOD)
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
1%
10%
25% 50% 100%
1%
10%
25% 50% 100%
OODSelect % of VLCS
OODSelect % of PACS
OODSelect
Weak Correlation Random
AoTL
AoTIL
Most Misclassified
Figure 8: Comparing AoTL and AoTIL. Spearman Correlation between ID and OOD accuracy as function of the number of selected OOD samples.
20

---

A.3 Table Results
Table 4: ID vs, selected OOD accuracy correlations (Pearson R and Spearman p) with standard errors over 100 resamplings. For "Random", we randomly select N subsets from candidate OOD examples; for "Hard", we select the N most misclassified examples.
Dataset
OOD
Pearson R
Spearman p
Ours
Random
Hard
Ours
Random
Hard
CXR No Finding
MIMIC-CXR
10
-0.56 (0.12)
0.19 (0.20)
0.00 (0.20)
-0,46 (0.14)
0,07 (0.20)
0.00 (0.20)
CXR No Finding
MIMIC-CXR
20
-0.23 (0.18)
0.20 (0.20)
0.00 (0.20)
-0.33 (0.16)
0.14 (0.20)
0.00 (0.20)
CXR Finding
MIMIC-CXR
-0.59 (0.12)
0.43 (0.18)
0.00 (0.20)
-0.58 (0.12)
(0.17)
0.00 (0.20)
CXR No Finding
MIMIC-CXR
100
-0.69 (0.09)
0.59 (0.14)
(0.20)
-0.60 (0.11)
0.52 (0.16)
0.00 (0.20)
CXR No Finding
MIMIC-CXR
250
(0.05)
0.70 (0.11)
0.00 (0.20)
-0.72 (0.08)
0.57 (0.15)
0.00 (0.20)
CXR No Finding
MIMIC-CXR 500
-0.84 (0.05)
0.71 (0.11)
0.00 (0.20)
-0.70 (0.09)
0.72 (0.11)
0.00 (0.20)
CXR No Finding
MIMIC-CXR
750
0.93
0.69 (0.12)
0.00 (0.20)
-0.81 (0.06)
0.75 (0.10)
0.00 (0.20)
CXR CXR No Finding
Finding
MIMIC-CXR 1000
-0.93 (0.02)
(0.13)
(0.20)
-0.87 (0.04)
0.72 (0.11)
0.00 (0.20)
MIMIC-CXR 2500
(0.02)
00.099
(0.10
0..64
0.78 (0.09)
00.19
CXR No Finding
MIMIC-CXR 5000
0.98 (0.01)
(..6
(..50
(0.02)
0.84
(0.17)
CXR CXR No Finding
MIMIC-CXR 7500
-.98 (0.01)
(.847)
(0.15
0..6
0.83
0..10
21
Finding
MIMIC-CXR 10000
00.000
0.847)
0.37
0..9000000
0.847)
-0.25 (0.18)
CXR Finding
MIMIC-CXR 15000
(0.00)
0.847)
-0.22 (0.18)
(0.00)
0.83 (0.07)
-0.13 (0.19)
CXR No Finding
MIMIC-CXR 20000
-0997 (0.01)
0.85
-0.07 (0.19)
-0.91 (0.03)
0.84
0.02 (0.20)
CXR No Finding
MIMIC-CXR 25000
-0.98 (0.01)
0.84
0.08 (0.20)
(0.01)
0.84 (0.07)
0.18 (0.20)
CXR No Finding
MIMIC-CXR 30000
00.000
(0,07)
0.26 (0.19)
0..9
(0.07)
0.37 (0.18)
CXR Finding
MIMIC-CXR 35000
(0.02)
0.85
(0.17)
0.95
0.857)
0.56 (0.15)
CXR No Finding
MIMIC-CXR 40000
0.97 (0.01)
(.807)
0.64 (0.13)
-0.98 (0.01)
0.83 (0.07)
0.71 (0.11)
CXR No Finding
MIMIC-CXR 45000
0091 (0.03)
0..57)
0.74 (0.10)
-0.93 (0.02)
0.84
0.80 (0.08)
CXR No Finding
MIMIC-CXR 50000
(0.05)
0.85
0.80 (0.08)
-0.87 (0.04)
0.847)
0.83 (0.07)
CXR No Finding
MIMIC-CXR 55000
0..1
0.84
0.84
-032 (0.17)
(0.07)
0.84
CXR Finding
MIMIC-CXR 60000
(0.20)
0.85 0..47)
0.18 (0.20)
0.84 (0.07)
0.847)
CXR No Finding
MIMIC-CXR 65000
0.46 (0.17)
0.85 (0.07)
0.85 (0.07)
0.56 (0.15)
0.84 (0.07)
0.84
CXR No Finding
MIMIC-CXR
70000
0.85 (0.07)
0.85
0.85 (0.07)
0.84 (0.07)
0.84 (0.07)
0.84
CXR No Finding
MIMIC- CXR
71433
0.85
0.85
0.85
0.84
0.84
0.84
WIL.SSiiviCoomments
10
-0.33 (0.16)
0.39 (0.18)
0..0
(0.13)
0.16 (0.20)
0.00 (0.20)
WIL.SCiivICommmnnt
20
-0,89 (0.03)
0.85 (0.06)
0.00 (0.20)
(0.17)
0.13 (0.20)
0.00 (0.20)
WIL.DCiivIComments
50
-0.89 (0.03)
0.39 (0.18)
0.00 (0.20)
-0.55 (0.12)
0.23 (0.19)
0.00 (0.20)
WILDSCiviCComments
100
-0.56 (0.12)
(0.10)
0.00 (0.20)
-0.53 (0.13)
0.19 (0.20)
0.00 (0.20)
Continued on next page

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Dataset
OOD
N
Pearson R
Spearman p
Ours
Random
Hard
Ours
Random
Hard
WILmeciviCoomeent
250
-0.79 (0.06)
0.94 (0.03)
0.00 (0.20)
-0.71 (0.09)
0.33 (0.19)
0.00 (0.20)
WILDSCiviI
500
(0.09)
0.93 (0.03)
0.00 (0.20)
-0.75 (0.08)
0,46 (0.17)
0.00 (0.20)
WILDSCiviIComments
750
-0,98 (0.01)
0.80 (0.08)
0.00 (0.20)
-0.65 (0.10)
0.02 (0.20)
0.00 (0.20)
WILDSCiviCComments
1000
-0.93 (0.02)
0.82 (0.08)
0.00
-0.82 (0.06)
0.08 (0.20)
0.00 (0.20)
WoLDsCivilComments
-0.78 (0.07)
0.93 (0.03)
0.00
-0.74 (0.08)
(0.20)
0.00 (0.20)
WILmeCivilComments
5000
-0.98 (0.01)
0.95 (0.02)
0.00
-0.76 (0.07)
0.47 (0.17)
0.00 (0.20)
WILDSCiviICommenns
7500
-0.98 (0.01)
0.97 (0.01)
0.95
-0.80 (0.06)
(0.09)
0.88 (0.05)
WILDSCivilCComeents
10000
-0.88 (0.04)
0.97 (0.01)
0.99 (0.01)
-0,41 (0.15)
0.70 (0.12)
WILDSCiviICommeens
15000
-0.97 (0.01)
0.97 (0.02)
(0.00)
-0.78 (0.07)
0..12)
0.89 (0.05)
WIDDSCivilCommeent
20000
-0.97 (0.01)
(0.01)
0.98
(0.07)
(0.07)
0.90 (0.05)
WILDSCiviICommeets
25000
-0.96 (0.01)
0.98 (0.01)
0.98 (0.01)
-0.73 (0.08)
0.84 (0.07)
0.91 (0.04)
WILDSCiviIComments
50000
0.98 (0.01)
0.98 (0.01)
0.98 (0.01)
0.90 (0.05)
0.90 (0.05)
0.90 (0.05)
WILDeCivilComments
52823
0.98
0.98
0.98
0.90
0.90
0.90
WIL.DSammclyon
Hospital
10
0.94 (0.02)
0.90 (0.05)
(0.20)
0.23 (0.18)
00.199
0.00 (0.20)
WILDSCanelyon
Hospital 4
20
-0.11 (0.03)
0.74 (0.11)
0.00 (0.20)
.0.17 (0.18)
0.25 (0.19)
0.00 (0.20)
WILDSCamelyon
Hospital 4
50
-0.93 (0.02)
0.49 (0.16)
0.00 (0.20)
-..3
0.50 (0.16)
0.00 (0.20)
22
WILDSCamellon
Hospital 4
100
-0.92 (0.03)
(0.03)
0.02
0.34
0.52 (0.16)
0.00 (0.20)
WILDSCamelyon
Hospital
-0.88 (0.04)
0.96
(..0
0.42
0.37 (0.18)
WILDSCamelyon
Hospital
500
(0.01)
0.97 (0.02)
0.00
-0.33 (0.13)
0.49 (0.17)
0.00
WILDSCamelyon
Hospital 4
750
(0.01)
0..000
-0.61 (0.11)
0.75 (0.10)
0.00
WILDSCamelyon
Hospital 4
1000
-0,91 (0.03)
(.991)
0.00 (0.20)
-0.51 (0.13)
0.74 (0.10)
0.00 (0.20)
WILDSCamelyon
Hospital 4
2500
0.07 (0.20)
0.98
0.08 (0.20)
0.00 (0.20)
0.71 (0.11)
-0.51 (0.13)
WILDSCamelyon
Hospital
5000
-0.99 (0.00)
(.991)
(0.17)
-0.93 (0.02)
0.77 (0.09)
-0.31 (0.17)
WILDSCamelyon
Hospital 4
7500
-0.98 (0.01)
(.99
-0.23 (0.18)
-0.95 (0.02)
0.78 (0.09)
-0.24 (0.18)
WILDSCamelyon
Hospital 4
10000
-0,98 (0.01)
0.98 (0.01)
00.10 (0.19)
-0.88 (0.04)
0.77 (0.10)
-0.09 (0.19)
WILDSCamelyon
Hospital 4
15000
-0.77 (0.01)
(0991)
0.23 (0.19)
0.826
0.78 (0.09)
0.23 (0.19)
WILDSCamelyon
Hospital
20000
-0.95 (0.02)
(.991)
(0.16)
(..06)
(0.09)
0.48 (0.17)
WILDSCamelyon
Hospital
25000
-0.88 (0.04)
(.99
(0.12)
(0.14)
0.80 (0.08)
0.64 (0.13)
WILDSCamelyon
Hospital 4
30000
(0.03)
(.99
(0.07)
-..50
0.80
0.76
WILDSCamelyon
Hospital 4
35000
-0.54 (0.13)
(.99
0..2
-0.28 (0.17)
(0.08)
0.79 (0.09)
WILDSCamelyon
Hospital 4
40000
-0.08 (0.19)
(.991)
0..0
-0.37 (0.16)
0.78(0.09)
0.79 (0.09)
WILDSCamelyon
Hospital
45000
0.08 (0.20)
(.991)
0.98
-0.02 (0.20)
(0.09)
0.79 (0.09)
WILDSCamelyon
Hospital 4
50000
0.27 (0.19)
0.99
0.98 (0.01)
0.12 (0.20)
0.81 (0.08)
0.79 (0.09)
WILDSCamelyon
Hospital 4
60000
0.30 (0.19)
0.98 (0.01)
0.17 (0.20)
0.78 (0.09)
0.80 (0.08)
Continued on next page

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Dataset
OOD
N
Pearson R
Spearman p
Ours
Random
Hard
Ours
Random
Hard
WILDSCamelyon
Hospital
70000
0.36 (0.18)
0.99 (0.01)
0.98 (0.01)
0.24 (0.19)
0.80 (0.09)
0.80 (0.08)
WILDSCamelyon
Hospital 4
75000
0.40 (0.18)
0.99 (0.01)
0,98 (0.01)
0.18 (0.20)
0.79 (0.09)
0.80 (0.08)
WILDSCamelyon
Hospital 4
80000
0.41 (0.18)
0.99 (0.01)
0.98 (0.01)
0.24 (0.19)
0.80 (0.08)
0.80 (0.08)
WILDSCamelyon
Hospital 4
85000
0,45 (0.17)
0.99 (0.01)
0.98
0.23 (0.19)
0.81 (0.08)
0.80 (0.08)
WILDSCamelyon
Hospital
90000
0.47 (0.17)
(0.01)
00.9()
0.28 (0.19)
(0.08)
0.81 (0.08)
WILDSCamelyon
Hospital
95000
0,50 (0.16)
(0.90)))1)
0.29 (0.19)
0.81 (0.08)
0.81 (0.08)
WILDSCamelyon
Hospital 4
100000
0.91 (0.04)
0.99 (0.01)
0,99 (0.01)
0.26 (0.19)
0,80 (0.08)
0.81 (0.08)
WILDSCamelyon
Hospital 4
125000
0.99 (0.01)
0.99 (0.01)
0.99 (0.01)
0.81 (0.08)
0.81 (0.08)
0.81 (0.08)
WILDDCameeyon
Hospital 4
129838
0.99
0.99
0.99
0.81
0.81
0.81
WILDSCamelyon
Hospital
10
-0.63 (0.10)
0.25 (0.19)
(0.20)
-0.03 (0.19)
0.63 (0.14)
(0.20)
WILDSCamelyon
Hospital 5
20
-0.83 (0.05)
0.43 (0.18)
0.00 (0.20)
0.13 (0.20)
0.48 (0.17)
0.00 (0.20)
WILDSCamelyon
Hospital 5
50
-0.74 (0.08)
0.73 (0.11)
0.00 (0.20)
0.16 (0.20)
0.39 (0.18)
0.00 (0.20)
WILDSCamellyn
Hospital
100
-0.81 (0.06)
0.71 (0.11)
-0.18
0.21 (0.20)
0.36 (0.18)
-0.26 (0.17)
WILDSCamelyon
Hospital
-0.11 (0.03)
0.75 (0.10)
(0.18)
-0.53 (0.13)
(0.17)
-0.46 (0.14)
WILDSCamelyon
Hospital
500
-0877 (0.04)
0.82 (0.08)
(0.13)
0.33 (0.19)
0.36 (0.18)
-0.63 (0.11)
WILDSCamelyon
Hospital 5
750
-0.81 (0.06)
0.80 (0.08)
(0.44
0.36 (0.18)
0.47 (0.17)
0.64
23
WILDSCamelyon
Hospital 5
1000
-0.83 (0.05)
0.847)
0.38
0.35 (0.18)
0.57 (0.15)
-0.59 (0.11)
WILDSCamelyon
Hospital
2500
-0.61 (0.11)
0.00
(..18)
0.26 (0.19)
(.467)
00.08
WILDSCamelyon
Hospital
5000
-0.43 (0.15)
0.80 (0.08)
-0.44 (0.15)
0.40 (0.18)
(0.16)
-0.63 (0.11)
WILDSCamelyon
Hospital 5
7500
(0.15)
0.81 (0.08)
-0.38 (0.16)
-0.11 (0.19)
0.47 (0.17)
-0.51 (0.13)
WILDSCamelyon
Hospital 5
10000
-0.88 (0.09)
0.81 (0.08)
(0.17)
+0399 (0.15)
0,48 (0.17)
-0,41 (0.15)
WILDSCamelyon
Hospital
15000
0.58 (0.12)
(0.08)
0.155 (0.19)
0.71 (0.09)
0.47 (0.17)
-0.22 (0.18)
WILDSCamelyon
Hospital
20000
-084 (0.05)
0.81 (0.08)
0.01 (0.20)
0.02 (0.20)
(0.17)
0.01 (0.20)
WILDDCamelyon
Hospital
25000
-0.11 (0.06)
(0.08)
(0.20)
0.09 (0.20)
(0.17)
0.17 (0.20)
WILDSCamelyon
Hospital 5
30000
-0.77 (0.07)
0.81 (0.08)
0.25 (0.19)
0.15 (0.20)
0.48 (0.17)
0.28 (0.19)
WILDSCamelyon
Hospital 5
35000
-0.68 (0.09)
0.81 (0.08)
0.32 (0.19)
0.23 (0.19)
0.48 (0.17)
0.35 (0.18)
WILDSCamelyon
Hospital
40000
-0.71 (0.08)
0.80
0.38 (0.18)
0.23 (0.19)
0.47 (0.17)
0.40 (0.18)
WILDSCamelyon
Hospital
45000
-0.20 (0.18)
(0.08)
(0.17)
0.18 (0.20)
(0.17)
0.41 (0.18)
WILDSCamelyon
Hospital 5
50000
-0.66 (0.10)
0.80 (0.08)
0.48 (0.17)
0.27 (0.19)
0.47 (0.17)
0.42 (0.18)
WILDSCamelyon
Hospital 5
60000
(0.11)
0.80 (0.08)
0.54 (0.16)
0.28 (0.19)
0,47 (0.17)
0.43 (0.17)
WILDSCamelyon
Hospital 5
70000
0.13 (0.20)
0.59
0.28 (0.19)
0.48(0.17)
0.46 (0.17)
WILDSCamelyon
Hospital
75000
0.29 (0.19)
0.80
(0.12)
0.33 (0.19)
0.48 (0.17)
0.47 (0.17)
WILDSCamelyon
Hospital
80000
0.31 (0.19)
0.81 (0.08)
0.73
0.38 (0.18)
(0.17)
0.47 (0.17)
WILDSCamelyon
Hospital 5
85000
0.40 (0.18)
0.80 (0.08)
0.77 (0.09)
0.35 (0.18)
0.48 (0.17)
0.47 (0.17)
Continued on next page

---

Dataset
OOD
N
Pearson R
Spearman p
Ours
Random
Hard
Ours
Random
Hard
WILDSCamelyon
Hospital
90000
0.37 (0.18)
0.80 (0.08)
0.79 (0.09)
0.40 (0.18)
0.48 (0.17)
0.47 (0.17)
WILDSCamelyon
Hospital 5
95000
0.40 (0.18)
0.80 (0.08)
0,81 (0.08)
0.43 (0.18)
0,48 (0.17)
0.47 (0.17)
WILDSCamelyon
Hospital 5
100000
0.56 (0.15)
0.80 (0.08)
0.82 (0.08)
0.41 (0.18)
0,48 (0.17)
0.47 (0.17)
WILDSCamelyon
Hospital
125000
0.66 (0.13)
0.81 (0.08)
0.82 (0.08)
0.45 (0.17)
0.48 (0.17)
0.48 (0.17)
WILDSCamelyon
Hospital
130000
0.70 (0.12)
0.80 (0.08)
0.81 (0.08)
0.42 (0.18)
(0.17)
0.48 (0.17)
WILDSCamelyon
Hospital
135000
0.72 (0.11)
0.81 (0.08)
0.81 (0.08)
0.43 (0.17)
0.48 (0.17)
0.48 (0.17)
WILDSCamelyon
Hospital 5
140000
0.76 (0.10)
0.81 (0.08)
0.81 (0.08)
0.44 (0.17)
0,48 (0.17)
0.48 (0.17)
WILDSCamelyon
Hospital
145000
0.81 (0.08)
0.80 (0.08)
0.80 (0.08)
0.48 (0.17)
0.48 (0.17)
0.48 (0.17)
WILDSCamelyon
Hospital
146722
0.80
0.80
0.80
0.48
0.48
0.48
TerraIncognita
L46
10
(0.04)
0.45 (0.17)
0.00 (0.20)
0.36 (0.16)
0.45 (0.17)
0.00 (0.20)
TerraIncognita
L46
20
0,900 (0.03)
0.92 (0.04)
0.00 (0.20)
(0.16)
(0.17)
0.00 (0.20)
Terralncogniaa
1,46
50
0.40 (0.18)
0.92 (0.04)
0.00 (0.20)
0..8
0.44 (0.17)
0.00 (0.20)
Terrannco
L46
100
0.58
0.87 (0.06)
(0.2
(0.16)
(.46
0.00 (0.20)
Iecrainc
250
0.91
0.90 (0.04)
0.00
0.46
0.66
0.00 (0.20)
TerraIncognita
L46
500
-0.77 (0.07)
0.87 (0.06)
0.03 (0.20)
-0.05 (0.19)
0.58 (0.15)
0.02 (0.20)
TerraIncognita
L46
750
-0.74 -0.08)
(0.95)
0.12
-0.01 (0.20)
0.59 (0.15)
-0.03 (0.20)
24
TerraIncognia
L46
1000
-0.77 (0.07)
0.89 (0.05)
-0.24 (0.18)
-0.22 (0.18)
0.61 (0.14)
-0.12 (0.19)
Inrrannco
L46
1250
-0.59 (0.11)
(0.04)
(0.18)
-0.43 (0.15)
0.60
(0.19)
Ierrannc
L46
1500
(0.155
(0.04)
0.22
(0.13)
(0.15)
(0.19
Terralncognita
L46
1750
(0.17)
0.87 (0.06)
0.19
-0.38 (0.16)
0.604
(0.19)
Terranncognia
L,6
2000
0.12
0.89
0..1
-0.23 (0.18)
0.62 (0.14)
0.02 (0.20)
Terranncognia
L46
2250
-0.08 (0.19)
0.89 (0.05)
0.005 (0.19)
-0.12 (0.19)
0.59 (0.14)
0.08 (0.20)
Incognnna
2500
-0.02 (0.20)
0.89 (0.05)
(0.20)
-0.00 (0.20)
0.61 (0.14)
0.17 (0.20)
Terggnncn
L46
2750
0.04 (0.20)
0.89 (0.05)
0.11 (0.20)
0.10 (0.20)
0.59 (0.14)
0.25 (0.19)
Terralncognita
L46
3000
0.09 (0.20)
0.90 (0.05)
0.18 (0.20)
0.17 (0.20)
(0.14)
0.32 (0.19)
TerraIncognita
L46
3250
0.15 (0.20)
0.89
0.25 (0.19)
0.27 (0.19)
0.59 (0.14)
0.36 (0.18)
Terralncognita
3500
(0.19)
(.905)
(0.19)
0.25 (0.19)
(0.14)
0.39 (0.18)
Incrginc
L46
3750
0.29 (0.19)
(..95)
0.37 (0.18)
0.24 (0.19)
0.60
0.43 (0.18)
Terraincogniaa
L46
4000
0.34 (0.19)
0.90
0.43 (0.17)
0.31 (0.19)
0.61 (0.14)
0.45 (0.17)
TerraIncognia
L46
4250
0.45 (0.17)
(0.04)
0.51 (0.16)
0.36 (0.18)
0.61 (0.14)
0.47 (0.17)
TerraIncognita
L46
4500
0.58 (0.15)
0.59 (0.14)
0.36 (0.18)
0.61 (0.14)
0.49 (0.16)
TerraIncognita
4750
(0.13)
(0.90)
(0.13)
0.38 (0.18)
(0.14)
0.52 (0.16)
Terrannco
L46
5000
0.70 (0.12)
0.89 (0.05)
0.72
0.41 (0.18)
0.61 (0.14)
0.54 (0.15)
Terralncognita
L46
5250
0.75 (0.10)
0.89 (0.05)
0.77 (0.09)
0.48 (0.17)
0.61 (0.14)
0.54 (0.16)
Continued on next page

---

Dataset
0OD
N
Pearson R
Spearman p
Ours
Random
Hard
Ours
Random
Hard
TerraIncognita
L46
5500
0.80 (0.08)
0.89 (0.05)
0.81 (0.08)
0.52 (0.16)
0.60 (0.14)
0.57 (0.15)
TerraIncognita
L46
5750
0.84 (0.07)
0.89 (0.05)
0.85 (0.07)
0.55 (0.15)
0,60 (0.14)
0.60 (0.14)
TerraIncognita
L46
6000
0.89 (0.05)
0.89 (0.05)
0.88 (0.05)
0.61 (0.14)
0.60 (0.14)
0.60 (0.14)
TerraIncognita
L46
6122
0.89
0.89
0.89
0.60
0.60
0.60
VLCS
LabelMe
10
-0.91 (0.03)
0.80 (0.08)
(0.20)
-0.28 (0.17)
0.16 (0.20)
0.00 (0.20)
VLCS
LabelMe
20
-0,90 (0.03)
0.90 (0.04)
0.00 (0.20)
-0.30 (0.17)
0.27 (0.19)
0.00 (0.20)
VLCS
LabelMe
50
-0.07 (0.19)
0.93 (0.03)
00.1
-0.29 (0.17)
0,43 (0.18)
-0.15 (0.19)
VLCS
LabelMe
100
-0.82 (0.05)
(0.02)
(0.18
-0.39 (0.15)
0.57 (0.15)
-0.21 (0.18)
VLCS
LabelMe
-0.77 (0.17)
(..43)
-0.25 (0.18)
-0.37 (0.16)
(0.15)
-0.28 (0.17)
VLCS
LabelMe
500
-0.15
(0,00))
0.21
-0.62 (0.11)
(0.16)
(0.99
VLCS
LabelMe
750
10.450..15
(0,03)
(0.18)
(0.10
(0.15)
-0.33 (0.16)
VLCCS
LabelMe
800
(0.16)
0.,03)
(..18)
-0.43 (0.15)
0.51 (0.16)
-0.26 (0.17)
VLCS
LabelMe
850
-0.28 (0.17)
0.05
(0.1
-0.33 (0.16)
0.61 (0.14)
-0.18 (0.18)
VLCS
LabelMe
900
00.17
0.95 (0.03)
(0.19
-0.77 (0.18)
0..65
0..10
VLCS
LabelMe
1000
0.09 (0.20)
0.95 (0.03)
0.03 (0.20)
0.08 (0.20)
0.59
0.03 (0.20)
VLCS
LabelMe
1250
0.33 (0.19)
(0900)
0.47 (0.17)
0.23 (0.19)
0..5
0.28
25
VLCS
LabelMe
1500
0.65 (0.13)
(0.03)
0.73
0.30 (0.19)
0.55 (0.15)
0.49 (0.16)
VLCS
LabelMe
1750
0.81 (0.08)
0.95 (0.03)
0.87 (0.06)
0.26 (0.19)
(0.44)
0.56 (0.15)
VLCS
LabelMe
2000
0.87 (0.06)
0.95
0.92 (0.04)
0.26 (0.19)
0.62 (0.14)
0.58 (0.15)
VLCS
LabelMe
2250
0.91 (0.04)
0.95 (0.02)
0.94 (0.03)
0.31 (0.19)
0,62 (0.14)
0.59 (0.14)
VLCS
LabelMe
2500
0.95 (0.03)
0.95 (0.02)
0.94 (0.03)
0.60 (0.14)
0.61 (0.14)
0.60 (0.14)
VLCS
LabelMe
26556
0.95
0.95
0.95
0.61
0.61
0.61
PACS
Sketch
10
(0.14)
0.37 (0.18)
0.17 (0.20)
0.05 (0.19)
(0.18)
-0.66 (0.19)
PACS
Sketch
20
-0.33 (0.16)
0.71 (0.11)
0.17 (0.20)
0,.1 (0.15)
(0.19)
0.00 (0.20)
PACS
Sketch
50
-0.47 (0.14)
0.84 (0.07)
0.00 (0.20)
0.23 (0.19)
0.47 (0.17)
-0.12 (0.19)
PACS
Sketch
100
0..3
0.73 (0.11)
0.11 (0.20)
0.29 (0.19)
0.70
-0.08 (0.19)
PACS
Sketch
250
(0.17)
0.79 (0.09)
(0.20)
0.35 (0.19)
0.71
(0.19)
PACS
Sketch
500
-0.22 (0.18)
0.83 (0.07)
0,28 (0.19)
0.41 (0.18)
(0.12)
0.07 (0.20)
PACS
Sketch
750
0.01 (0.20)
0.82 (0.08)
0.33 (0.19)
0.43 (0.17)
(0.13)
0.18 (0.20)
PACS
Sketch
800
0.05 (0.20)
0.81 (0.08)
0.33 (0.19)
0.42 (0.18)
0..63)))3)
0.19 (0.20)
PACS
Sketch
850
0.06
0..08)
(..19
0.42 (0.18)
0.64
0.22 (0.20)
PACS
Sketch
900
0.08 (0.20)
0.83 (0.07)
0.34 (0.19)
0.42 (0.18)
(0.53)
0.24 (0.19)
PACS
Sketch
1000
0.10 (0.20)
0.80 (0.08)
0.35 (0.19)
0.40 (0.18)
0.67 (0.12)
0.27 (0.19)
PACS
Sketch
1250
0.16 (0.20)
0.81 (0.08)
0.38 (0.18)
0.41 (0.18)
0.67 (0.12)
0.33 (0.19)
Continued on next page

---

Dataset
OOD
N
Pearson R
Spearman p
Ours
Random
Hard
Ours
Random
Hard
PACS
Sketch
1500
0.21 (0.20)
0.82 (0.08)
0.41 (0.18)
0.42 (0.18)
0.68 (0.12)
0.39 (0.18)
PACS
Sketch
1750
0.24 (0.19)
0.82 (0.08)
0,46 (0.17)
0.42 (0.18)
0.67 (0.12)
0.43 (0.17)
PACS
Sketch
2000
0.29 (0.19)
0.82 (0.08)
0.49 (0.16)
0.48 (0.17)
0.66 (0.13)
0.48 (0.17)
PACS
Sketch
2250
0,48 (0.17)
0.81 (0.08)
0.54 (0.16)
0.52 (0.16)
0.67 (0.12)
0.51 (0.16)
PACS
Sketch
2500
0.56 (0.15)
0.81 (0.08)
(0.15)
0.54 (0.16)
(0.12)
0.56
PACS
Sketch
2750
0.62 (0.14)
0.81 (0.08)
0.63 (0.13)
0.56 (0.15)
0.67 (0.13)
0.61 (0.14)
PACS
Sketch
3000
0.67 (0.12)
0.81 (0,08)
0.68 (0.12)
0.58 (0.15)
0.67 (0.12)
(0.133))))
PACS
Sketch
3250
0.71 (0.11)
0.81 (0.08)
0.72 (0.11)
0.61 (0.14)
0.67 (0.13)
0.66
PACS
Sketch
3500
0.75 (0.10)
0.81 (0.08)
0.76 (0.10)
0.63 (0.14)
0.66 (0.13)
0.66 (0.13)
PACS
Sketch
3750
0.81 (0.08)
0.81 (0.08)
0.79 (0.09)
0.67 (0.13)
(0.13)
0.67 (0.13)
PACS
Sketch
3929
0.81
0.81
0.81
0.67
0.67
0.67
26

---

B Theoretical Analysis
B.1 Lemma[I Bounded effect of New Models on Pearson R
Lemma 1 (Bounded Effect of New Model on Pearson R). Let (z,, w:) [a,, al? with a E (0, 1). Define
+ (z)),
Y1 ② (w,))
and let
PN (o.
be the sample Pearson correlation of the first N transformed pairs. Add one more pair (ZN+1; Wx+1) with +N+1 E (ax, or) and WN+1 = 3 - 1,, and denote the updated correlation by PN+1. Then
x(1 + (8) M
PN+1 PN x
Ma = 10=maax(10 (0) (11 (01-aa) -a]l].
N
where the constant K. > depends only on a (via the Lipschitz constant of x- on (ax, - o.).
Proof. Let X; (-,(2,) and + (w:) for i I.....N.. Denote NN N 2i=1 x; and similarly YN. Let PN be the Pearson correlation between (xI,..,,XN' and y:):
PN E &N)(y, YN)
E xNN N -y)=
Let (ZN+1, WN+1) be new pair with WN+1 = 32N+1 and both in [a; 1 a.. Define XN+1 & -= and yN+1 4 (wN+)) Since 4- is -ipsschhi [a, 1 a.. have |N S Ma and Ny S 1 |3 + M. S (1 M Thus, each |x] ly: S (1 + M.)
Let PN+1 denote the Pearson correlation after adding (xN+1. VN+1). A first-order perturbation of the sample Pearson correlation (cf. derivative bounds on correlation statistics) gives:
C
leN+1 PNl S N
max[[x;], 1y:
where C' is a constant depending on the Lipschitz constant L. and lower bound on variance (which is lower bounded by (a(1 L3) due the probit transform). Therefore,
x(1 + 81)
INN+1 pN
N
where K > 0 depends only on
B.2 Lemma 2: Bounded effect of New Examples on Pearson
Lemma 2 (Bounded Effect of a New Example on Pearson R). Fix a E (0. 1) and assume the W; [a, a) I,...,d. Define
per-model accuracies satisfy z (S)
yi = ===--((w,.
and let
Ps corl(x(5)
be their sample Pearson correlation. Now add an additional selected example. Note that each average changes by at most [z, between ((++)) and Then
(S+1)
z (8) < 1/(S + 1). Write Ps+1 for the resulting correlation
K R2
PS+1 - ppl <
M. max{ - -1- (1-
S+1'
where the constant depends only (via the Lipschitz constant of 4- --1 on |a, - a)).
27

---

8 1 (p)app
10a
a (p)/ap
a0 (p)/ap =
102
θ
10
-2.5
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
p
p
(a) Inverse CDF (Probit) the Standard Normal
(b) Derivative the Inverse Normal CDF 1/0(0= where & and are PDF and CDE, respectively.
Figure 9: The probit transform is indeed Lipschitz.
(-'(w;), and let ps cx Adding a new example changes each z; (S) by at most 1 1/(S+1) 1),, so |x{ (S+1)
Proof. Let X; (z,
(5)
x (5) I Laa/ (S+1). Denote 6, x (S+1)) -x(5) (5)
and = _/2 Ei= 8, Then
x
(S+1))
x
10 81 < 2Lm1(5+1))
Using |00 $ Mo.
d
(S+1) x 15/1 5
2dLa
S+1
1=1
Also, || 17 2 Vdva for Uoa = == a So
Mo
2L Ma
22
|ps+1 psl 5 (S + 1)dva
(S + 1)wa
a(1 - (S + 1)
2L3 all-a) gives the result.
Setting
B.3 Lemma 3: Lipschitz Continuity of Selection Objective
Lemma 3 (Lipschitz Continuity of Pearson Correlation w.r.t. s). Let X E R.Nxd be the binary accuracy matrix and y RRN the held-out training accuracy vector. Define the test-set accuracy for a given selection vector s E 13 (with fixed total mass Σ}=1 = > 0) as
Xs
x
S
and let f be Lipschitz-comtimmuus probit transformation with Lipschitz constant Ly. Denote
x(s) = (S(()), - f(y),
and the centered versions by
1
x(s) x(s) xs(s, y= 1y..
N
The Pearson correlation between x(s) and ÿ is defined as
1 x((s)
corr(x(s).))) = V Nr [1
Assume that there exists > 0 such that ||x(s)]| 2 e for all admissible s. Then, for any two selection
vectors s' E [0, 1]d with > j= 8; 2 8/ S, there exists constant L > 0 (depending
Ly, X, S, and e) such that
corr(((s)) corr L L[ss 5'1l-
28

---

Proof.
Since the test-set accuracy is given by
Xs
x(s) = 5
linearity implies that
X S
[(((s) x(s')\ 5
|s s'Il
where |X| denotes an appropriate operator norm.
Using the Lipschitz continuity of f (with constant Ly), we have for each coordinate,
f(t((s)) - (ā s م Ly (.(5)) - 2
so in vector form,
{x(s) (())|| < Ly - - Li |
||s s'11.
Since centering is linear operation, it follows that
D X
{x(s) - x(s')| S ( x(s))
|s s'1l-
Now, note that the Pearson correlation is computed as
X(s) y
corr {s)
||(()|| |
Since y is independent of and by the assumption that ||x(s)]| 2 c <> 0, standard arguments (via the mean value theorem and the differentiability of the quotient function on a compact domain) imply that there exists a constant C' > 0 such that
corr (x(s).)) corr(x(s'),))
S CC|[x(s) x(s')ll-
Thus, combining the bounds yields
corr (((())(s))) 3) cx SC Ly ||| s
|s s'll.
Setting T. C L, |||||
completes the proof.
S
B.4 Proof of Proposition I
Non-Submodularity
Proposition (Non-Submodularity; no diminishing returns (Informal)). Let s E [0, be selection vector over d candidate OOD examples, and write corr(acew, acchp) for the Pearson correlation in Eq. (4). Define s; 1 Sy to mean (s.); I (sz); for all t E (l....,d). For k e {1....,d d} with (s)x = O, let s tk denote the vector obtained by setting the kth coordinate of s to 1 (leaving all others unchanged). Then, general, there exist s;, Sj E [00 with s; 3 sj and (s:)a = (s;)a = 0 such that corr(aceı, accooo) < corracem.
corraccm, acc)on)
xt*
9,
comfacem. acc .oo)
Let M(s) = corr (Xs, y). The exists s; C C (1.....,d d) and j a ss such that M(s: U (5}) - M(s.) < M(s, {ī)) M(s))
Proof. Let y E T satisfy yy il ||2 > 0. Choose three candidate columns
Xi y,
X2 independent of with corr(x2, y) = 0,
X3 = y.
Set the index sets
s: s; =1) 8, (1,2), j == 3.
29

---

Compute the four correlations:
M(s:)) coLT(x1, v) =,
M(s, U (3)) coml
x y 1,
M(s,) corn x1+xx 3)
M(s, U (3)) = comm
x1 x1 v)
Hence
M(s, U - M(s U (3)) M(s))
so M violates the diminishing-retume (submodularity) condition.
C ID/OOD Explanation
Algorithm 2: Descriptive Differences Between II) and OOD Subsets (Dunlap et al. 2024)
Input: ID image set DDD OOD image set DOOD
Output: Ranked list of difference descriptions highlighting O0D-specific attributes
1: Step I: Generate Captions
2: Use BLIP-2 Li et al. (023) to generate captions for all images in DID and DOOD
3: Step 2: Generate Candidate Difference Deseriptions
4: Use Mixtral Jiang al. (2024) to propose set of natural language descriptions more likely to appear in DOoD than in 'PDD based on the generated captions
5: Step 3: Score and Rank Differences
6: foreach difference description d do
7:
Compute average CLIP Radford et al. 2021 similarity between d and images in Dwv: simmn (d)
8:
Compute average CLIP similarity between d and images in Doon: simOoD (d)
9:
Compute difference score: (d) simoon (d) (d)
10: Rank all descriptions by (dd) descending order
II:
return Top-k difference descriptions ranked by distinctiveness to DOOD
We follow Algorithm 2 to generate semantic difference between selected OOD samples and ID samples. We select 200 samples from the II) and selected OOD sets, respectively. As motivation for future work, we enumerate our observations from this cursory study.
We evaluated multiple vision-language and language models to generate and summarize conceptual differences between In-Distribution (ID) and Out-of-Distribttin (OOD) image groups.
Similarity Scoring Models. We tested CLIP, SigLIP, and AimV2 (CVPR 2025) to image- caption similarity. Ai produced the most meaningful rankings based on manual inspection.
Prompting Strategy. Initial difference captions often described groups of images (e.g., "images with different views"), which misaligned with how CLIP-like models are trained (single image-caption pairs). To address this:
We introduced detailed prompt discouraging group-level descriptions. Larger models responded well, while smaller models were sensitive and inconsistent.
We then simplified the prompt but edited examples to avoid phrases like "images of..." resulting in captions more compatible with similarity models.
LLM Comparison for Caption Generation. We compared three LLMs: mistralai/Mistrall ---Innsruut--O 2, Qwen/Qwen2.n 1----IIItttuc.. Qwen/Qwen2 -nntrunt The 32B model produced the most diverse and generaliz- able difference captions, while smaller models tended overfit or focus narrowly on individual images.
and
30

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Summarizing Conceptual Differences. Finally, we prompted LLMs to summarize conceptual shifts based on similarity deltas and difference captions. Larger models generated the most natural and high-level descriptions; Qwen2. 5-32B- Instructtt performed reasonably well.
Overall, while these results show some promise (Table 2 for settings with images with common semantic properties, they are relatively inconsistent. Moreover, for datasets without such common semantic properties, e.g., medical images, these methods may only work with dedicated foundation models appropriate for those image modalities.
Caption Prompt (Terra Incognita)
Caption this image. know what object it Pocus on describing the artistic style, testure, and domain specifii details rather than the object itself. Again, mostion
Differences Prompt (TerraIncognita)
am a machine loaming researcher trying to figure out properties of an image beyond the image class. Give me deseription this image that more speciáic than the image class, For instance if this is zn image of bird, don't want know if the bird is sparto or crow. want know if the bird is sitting
the such geoperties what looking foc.
grouad-level
image close-up. lighting focusing
description
Comte with caumbez of captious> distinet coneepts that are more likely be true for the Ous of-tiiiibttion Group compared to the In De-Dittibation Group. Please list captions (sepurated bulles ***), example:
"unuaus] lighting andditiona"
distortions"
"corplex backgrounds"
"non object poses*
"uncommon viecing anglee*
"partial viecs objects*
"objecta in unexpected contexts"
"scenes with high vieual clutter'
"irages with unusual color scheraz'
"lee-raenlution images"
talk about the caption iiself. "captios. "images "presence
word", and list than concept. hypothesis should be caption, incorrei. not coumerate possibilties pareniheses,
phrasing like
Here are examples of had outpuis and their corrections:
INCORRECT: "various
foreste, zountains"
CORRECTED: "nature"
INCORRECT: "inagea of houswhold object (e.g- howl, varmu, lamp)" CORRECTED: "hamehold objacts'
INCORRECT:
of baby asinels" CORRECTED: "baby aninela"
INCORRECT: 'Different typeи of vehicles including cara, trucks, bowta, and RVs" CORRECTED: "vahiclea"
INCORRECT: "Irages involving interaction hurans and animale"
CORRECTED: "interaction between
INCORRECT: realistic images" CORRECTED: 'realistic images"
INCORRECT: "Iasects (cockroach, dragonfly, graschopper)" CORRECTED: "insecte"
Again, figure what kind distribution shift List Knumber capt propertes hold images
captions) Dis-of-Distibutton Group compared In-Distibution Group. Asswer with (soparated bullet points "*").
In-Distribation Gromp: ilist of in-distribut eaptiona>
Distribution Group: <list
out-of out-of-distributioon captioas>
Your response:
Figure 10: ID/OOD difference prompt for Terralncognita.
31

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Differences Prompt (PACS)
Caption this image. I know what object is. Facus on disesihing costestual and enviroamettal detuls,, such scene composition, lighting, and background chiracieristios, Again, do mention the object clacs.
Differences Prompt (PACS)
machine leamingg researcher trying figur out properties an image beyond the object closs. description this image specitic than the object class. For instunee, thi an image of dog. don't want know if the dog is buildog or resiever. want to know the scese seggests indore or ounfoar setting, detilk about the articic style, spocific texture and lighting. broder list of such properties is what I'm looking class.
Give this description for the image without memioning
Comte up with Counber of captiona> distinet concepts that are more likely be tue for the Ous-af-Decsbbtiio Group compared to the In-Dietriattoon Group. Please wone list of captions (separated by baller points ***), For example:
"unurual lighting tonditionn""
"vieual dietortions"
"corplex backagrundd"
"non-standerd object
"uncorsson vieving angles"
"partial objects
"objects in unexpected contexta'
high
"Inages with unasual color scheneo")
"los-rwanlution imagea"
Do talk ohour the caption inseff, "caption with word", and do not list more than one enncept. The hypothesis should be caption, "inages
like
"presence
passibilnies within parentheses.
Here exsmples of outpuis and coenections:
INCORRECT: "various
somntains" CORRECTED: "sacure"
INCORRECT: "images of hourehold object (e-g. boul, vacuun, lamp)" CORRECTED: "houeebold objects
INCORRECT: "Presence of animale" CORRECTED: "baby aninsle"
typea
iacluding cars, trucks, boate,
INCORRECT: "Irages involving interaction betveen hurans and antexls' hurana and aninalx"
CORRECTED: "interaction betueen
images"
images"
INCORRECT: "Insecta (cockrowch, dragonfly, graashogarr" CORRECTED: "insecta*
Again, want figure out what kind of distribution shift there is. List Sourber of captionaz propertiss that hold more often for the imagos (wx captious) the Dus-of-Disribation Group compared In-Distribttion Group. with (sqparted bullet points
In-Dis nn-Distribution Group: <1st of in: distribution captions>
Group: <list
out-of distriibtittiioii captioas>
Your response:
Figure 11: ID/OOD difference prompt for PACS.
32

---

Caption Prompt (VLCS)
Caption histopthology image. Know diagnostic cntegory. Fiicrs describing tissue morphology, strining pollerns, and sinirtan details Again, do not reveall the disgnostic caregory.
Differences Prompt (VLCS)
properties beyond object image bird, want whether is spar cagle. Instead, desailed inforation ahout enviroemnnal context, composiion, lighing cunditions, and buckrroand Provide such description without mentining the object class.
andying adaptation. Please deseribe image
this
likely Group. Please write list captions (sepurated by bullet points ***). For example:
up
captione> concepts
Disuribution Group compured Distribution
"unusual lighting condit tions"
"conplex: backgrounde"
"non-standard object
"uncommon viecing anglee*
"partial objects
"abjecta in unexpected contexts"
high
"Inages with unssual color schezes"
"loe-raenlution images"
Do talk shour the caption isslf, "paption with word", and do not list more than one enncept. The hyporhesis should be esption, possibilties pareniheses,
phrasing like ...", "presence
"images
Here examples of oulpuis and cornection:
INCORRECT: "various nature anviromeent 1ike lakea, fornsta, and sountaine"
CORRECTED: "nacure'
CORRECTED: "houeholld abjecta" INCORRECT: "inagea of household objact (e.g- bost, vacuus, lamp)"
INCORRECT: 'Presence of baby asinels" CORRECTED: "baby aminels"
INCORRECT: 'Dstferent typer of vebicles including care, trucks, boste, and RVe
CORRECTED: "vehiclee"
INCORRECT: "Irages involving humans and anipaip"
CORRECTED:
INCORRECT: realistic
CORRECTED: 'realietic
INCORRECT: "Insecta (cockrouch, de dragoufly, grasshoper)" CORRECTED: "insecta*
Again, figure what kind distaibution shuft List sousber captiona> properties hold images captions) Distribution Group compared Du-tittbbttinn Group. (sopurated bullet points "*").
In-Distillution Group: Clist of
in-distrsbuttiom
Dadit--Diitiittionn Group: <lict of out of distribbtion capticas
Your
Figure 12: ID/OOD difference prompt for VLCS.
33

---

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Reviews:
[{"review_id": "kH3L3ZTPqy", "reviewer": "Reviewer_aNDu", "summary": "The paper investigates the OOD split in the current benchmarks and propose an approach to select subsets of these OOD samples before they are used for evaluating methods. This is useful to address the concerns of spurious correlations (higher ID accuracy reduced OOD accuracy) in machine learning models. In some of the datasets such as Chest Xrays, they extracted around 70% of OOD splits.", "questions": "-  Are there any analysis done on the OOD splits that obtained from each dataset? What are the semantic coherence with ID data?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n-  Originality: The paper presents an interesting approach to select OOD split from datasets.\n\n-  Quality: The study is well-executed, with a thorough theoretical and empirical analysis to show the validity of the proposed approach. The authors provide clean analysis for each dataset graphically. The authors also provided detailed appendix with proofs, dataset analysis and anonymized codes.\n\n-  Clarity: Definitions of key concepts, such as the objective, proof of non-submodularity, are well-explained. The practical recommendations on each dataset and benchmarks provide clear and actionable insights for both researchers and practitioners on their selection.\n\n-  Significance: the paper is useful for the study of hidden OOD data in the current datasets and this will improve the current distribution shoft benchmark to have better ID-OOD split which aids in the evaluation of generalizability of models.\n\n\nWeaknesses \n\n- Can the selection of OOD samples (OODSelect) as per objective in Equation 2 bring out ID samples (potentially adversarial samples) where the model incorrectly classifies, apart from original OOD samples. How can we differentiate between where the model is going wrong in ID and incorrectly predicted OOD samples? Is the above problem addressed as we use N number of models and OODSelect samples perform poor in large subset of these models?\n\n- Missing baselines: The paper proposes a subset selection of OOD samples from a dataset considering an ID distribution. How about we see some baselines for the same problem. For instance, we plot latent feature distribution of data samples using a pretrained model (e.g. CLIP) and we pick OOD samples based on some distance metric thresholds. There are some discussions on these lines using foundational models in Appendix C. However, it will be nice to ground the numbers as in Table 1.\n\n- Alternate to correlation metric: These simple distance metric (e.g. Euclidean dist) can make the objective submodular and the subset selection problem can be solved using greedy algorithm. This can be a good comparison?\n\nSome related works that could strengthen related works:\n-  Nagarajan et al. Understanding the failure modes of out-of-distribution generalization. ICLR 2021\n-  Lin et al. Spurious Feature Diversification Improves Out-of-distribution Generalization. ICLR 2024\n-  Deng et al. Robust Learning with Progressive Data Expansion Against Spurious Correlation. NeurIPS 2023.\n-  Shi et al. LCA-on-the-Line: Benchmarking Out-of-Distribution Generalization with Class Taxonomies. ICML 2024", "was_revised": true, "final_justification": "Thanks for addressing the concerns.\nThe further discussion also seems to bring in interesting insights; especially regarding CLIP-feature based OOD selection is quite poor compared to proposed OODselect. Thanks for the hard baseline to show OODSelect isolates spurious ones rather than generally difficult examples.\nPlease include the overall rebuttal discussion in the final version of the paper. I am inclined to raise my ratings.", "rebuttal": "We thank the reviewer for the constructive feedback and the positive assessment of our paper’s originality, quality, clarity and significance. Below we respond to your concerns point-by-point and describe the changes made in the revised manuscript.\n\n`How can we differentiate between where the model is going wrong in ID and incorrectly predicted OOD samples? Is the above problem addressed as we use N number of models and OODSelect samples perform poor in large subset of these models?`\n\nEquation 2 only moves the OOD accuracy vector; the ID vector is untouched, so any ID‑sample misclassification pattern is not a part of the optimization target (in our submission: L113 “we are not selecting models or altering the ID accuracies; we always correlate the same length‑N vectors, only the OOD accuracy values change”). The OOD sample is also distinct from the ID samples in our datasets. For example, for TerraIncognita, the images are from distinct geographical locations; this disjointness is true for every dataset.\n\nFurthermore, we included in our submission the “Hard” baseline requested by the reviewer: for every dataset, we select the N most incorrectly predicted OOD images (i.e., the ones that most models get wrong). Across all settings, this baseline leaves the ID‑vs‑OOD correlation near  0, never resulting in a strong negative correlation; e.g., CXR (N selected  = 10) gives R = −0.00 ± 0.00, TerraIncognita (N selected = 50) R = 0.18 ± 0.01, etc. (Table 3). Figure 2 visualises the same trend—misclassified sets hover around the x‑axis while OODSelect can find examples with strongly negative accuracy correlations.\n\nImportantly, it is not the case that OODSelect performs “poorly” on large subsets, but that there is a limit on how large a subset can be for the correlation. As N approaches the full OOD set, it necessarily approaches the ID and full OOD accuracy correlation (though in a non-submodular way, see Proposition 1).\n\nTogether, these analyses demonstrate that OODSelect isolates spurious rather than generally difficult examples, directly addressing the reviewer’s concern. We will clarify this point in the final version of the paper.\n\n`“...latent feature distribution of data samples using a pretrained model (e.g. CLIP) and we pick OOD samples based on some distance metric thresholds. ... Alternate to correlation metric: These simple distance metric (e.g. Euclidean dist)”`\n\nThank you for suggesting this comparison! We have added a CLIP‑Euclidean baseline that, for each dataset, greedily retains the farthest OOD examples from the ID examples in the CLIP space (Radford et al., 2021). Results are summarised in the table below.\n\nWe find that feature‑only selection—especially with general pretrained models—ignores the feature-label correlations that drive OOD failures. Our findings echo the recent conclusion that latent‑distance detectors often “answer the wrong question” for OOD detection [Li et al., 2025, Out‑of‑Distribution Detection Methods Answer the Wrong Questions]. We find that feature-wise selection sometimes yields an even stronger correlation between ID and OOD accuracy than random selection.\n\n**Comparison with CLIP Distance Selection**\n\n| Dataset | N for subset | R from CLIP distance selection | R from OODSelect | R Random Selection | R Full OOD |\n|---------|--------------|--------------------------------|------------------|------------------|------------|\n| PACS | 10 | 0.52 | -0.34 | 0.60 | 0.84 |\n| | 100 | 0.67 | -0.30 | 0.66 | 0.84 |\n| VLCS | 10 | 0.82 | -0.92 | 0.84 | 0.96 |\n| | 100 | 0.91 | -0.89 | 0.86 | 0.96 |\n| TerraIncognita | 10 | 0.24 | -0.90 | 0.47 | 0.85 |\n| | 500 | 0.72 | -0.93 | 0.67 | 0.85 |\n| WILDSCamelyon | 10 | 0.97 | -0.89 | 0.78 | 0.99 |\n| | 1000 | 0.99 | -0.90 | 0.50 | 0.99 |\n| CXR | 10 | 0.20 | -0.80 | 0.20 | 0.84 |\n| | 1000 | 0.83 | -0.97 | 0.13 | 0.84 |\n\n`Are there any analysis done on the OOD splits that obtained from each dataset? What are the semantic coherence with ID data?`\n\nPlease see our paragraph on consistency and coherence (L 186–L 226), which we summarize here: Each subset size is selected via an independent optimization, so we quantify overlap (monotonicity) across sizes with the normalised Jaccard index (0 = no overlap, 1 = full overlap).  Across all datasets the index lies between 0.72 and 0.99, confirming that the subsets OODSelect identifies are both consistent and coherent. When metadata are available, we also find clear semantic structure. For the Chest‑X‑ray dataset, for example, disease categories such as Pleural Effusion appear markedly more often in the OODSelect split than in the remainder of the OOD pool, indicating the method surfaces clinically meaningful sub‑populations. Finally, we explored automated semantic explanations of the selected subsets using Dunlap et al.’s (2024) approach for “Describing differences in image sets with natural language.” The utility of this explainer proved highly variable, effective on natural‑image datasets but far less reliable for non‑natural images like chest X‑rays or microscopy slides. Investigating more robust cross‑modal explanation techniques is, therefore, an interesting avenue for future work.\n\n`Suggested related works.`\n\nWe will also add a discussion of the related works you suggested. Nagarajan et al. (ICLR 2021) provide a theoretical analysis of failure modes in OOD generalization, showing how even linear models trained with gradient descent can rely on spurious features due to geometric and statistical factors. While their work explains why models fail, it does not offer tools to expose such failures at test time, as ours does. Lin et al. (ICLR 2024) demonstrate that promoting diversity in spurious feature reliance during training can improve OOD robustness. Their method is a training-time intervention, whereas OODSelect is a post-hoc evaluation tool. Deng et al. (NeurIPS 2023) introduce a progressive data expansion (PDE) curriculum that incrementally introduces spurious examples during training, improving worst-group accuracy. Our approach is complementary, enabling stress-testing of models regardless of their training strategy. Lastly, Shi et al. (ICML 2024) show that using class taxonomies and lowest common ancestor (LCA) distances restores a strong ID–OOD correlation, particularly for vision-language models. Unlike their taxonomy-based reformulation, we retain standard accuracy metrics and reveal that even within fixed OOD splits, large subsets can invert the ID–OOD trend—underscoring the limits of average-case evaluation. \n\nWe hope these clarifications and the new quantitative baseline demonstrate that OODSelect isolates genuinely spurious-correlation stress tests that elude distance heuristics, further strengthening the paper’s contributions. If we have addressed your concerns, we hope you will raise your score."}, {"review_id": "pSRndja3wt", "reviewer": "Reviewer_7bEF", "summary": "The paper proposes that out-of-distribution (OOD) performance may be on average positively correlated with in-distribution (ID) performance for models of increasing capacity, but there may be a subset of OOD data that is actually negatively correlated with ID performance. To determine if this is the case, a method for splitting an OOD test set to minimize correlation with the ID training set is presented (OODSelect). The method is applied to a number of application-focused datasets with OOD subsets, and a large number of models of varying capacities are trained and evaluated to find that there is indeed a subset for which ID performance does not predict OOD performance. In at least one dataset, the subset that performs worse is associated with certain identifiable subpopulations that are unrelated to the data labels.", "questions": "What is the reason for applying the probit transform, as opposed to e.g. Spearman (rank) correlation?\n\nCould the authors clarify the meaning of the second sentence in the quote below (line 67)?\n> Influence functions [33] may appear suitable at first glance, but they rank training points by leave-one-out influence, rather than partitioning the test/OOD set. Thus, applying influence functions in this context would still require an additional heuristic to define coherent subsets, while also inheriting known fragilities in modern deep networks.\n\nIn CXR 70% of OOD examples are negatively correlated. Where is the strong positive correlation between ID and all OOD examples coming from then? Surely the model must be doing significantly better on the remaining 30% of examples to counteract the OODSelect subset.\n\nParagraph at line 186 (selection consistency and coherence) - this seems intuitively sensible, and if anything, the opposite is surprising. Could the authors comment on how inconsistency can occur when increasing subset size (maybe in relation with proposition 1)?\n\nThe discussion of influence functions as well as the subset discovery method presented seem similar to influence scores computed by Feldman & Zhang (https://proceedings.neurips.cc/paper/2020/hash/1e14bfe2714193e7af5abc64ecbd6b46-Abstract.html). There is also a lot of related work in this area (sometimes called example difficulty scoring, e.g. https://arxiv.org/abs/2401.01867), some of which is relevant to the bias/fairness angle (e.g. https://arxiv.org/abs/2010.03058). Is there a way to leverage the scores of individual examples, such as influence scores from Feldman & Zhang, to do the same OOD selection?", "limitations": "I think the main potential limitation is the range of models evaluated, but I cannot say either way whether the authors have adequately addressed this point from the main text alone (without Appendix A). As the authors discuss, the datasets chosen are also limited in the metadata available, and perhaps here the authors may want to examine datasets similar to the chest X-rays datasets, which have labelled subpopulations, in order to determine if the negatively correlated OOD subsets have bias/fairness implications.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 4, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\nThe paper is very relevant to practical applications, and also addresses an important issue with evaluating models: namely their generalization properties in real-world applications. The results are also very interesting and relevant - especially to the fairness and bias of models. The experiments cover a wide range of datasets and models.\n\n## Weaknesses\n\nMy main issue with the paper is it is not as well presented as it could be, and there are some omissions and odd details in the methods/results. The method also seems a bit overcomplicated and I would like to see the paper focus more on exploring its findings in greater detail (which are quite interesting). The actual methods, although original in execution, are not entirely original in purpose (see the additional citations in Questions). Having said that, I still think this paper addresses an important problem with insightful results, and will be happy to raise my score if the authors address the points below.\n\nThere is something strange going on in fig 3a and 4: the intercepts for OODSelect are sometimes negative, which shouldn't be possible since the slope is always negative and the accuracies are always positive. Are the regression lines applied to the probit transformed accuracies?\n\nThe proof sketch for Proposition 1 is not entirely convincing, and a counterexample to submodularity would help greatly.\n\nI don't understand how the quantity of models was chosen, because it is the diversity of the models that really matters - e.g. if all of the models have the same performance up to noise, then there would probably be little or no correlation between ID and OOD test accuracies.\nAs an example of why this is an issue: in figure 4d there is a clear line of correlated All/OODSelect accuracies on the right side. One possible explanation is that there is an architecture for which All/OODSelect accuracy are positively correlated, and if one were to only sample models from that architecture, then the OODSelect slope would become positive (just like All), invalidating the findings for fig 4d.\n\n Minor issues:\n- Definition 1 should explain $acc_{P_{ID}}$, etc. Actually, maybe definition 1 should follow the problem setup\n- Problem setup should make clear whether OOD examples are from train or test set\n- notations are somewhat unwieldy: long tilde, long text variables like $acc_{P_{ID}}$, abbreviations like AoTIL (why not just say positive/negative correlation?)\n- Typo in table 1 (missing % sign)\n- Proposition 1 $\\subset$ isn't well defined since $\\bf{s}_i$ is a vector of 0s and 1s\n- Paragraph at line 178 needs a reference to figure 2\n- Appendix is missing from the submission", "was_revised": true, "final_justification": "Resolved: all issues except the one discussed in the comment below. The other reviews have also been addressed.\n\nUnresolved: the evidence for negative correlation between ID and OOD accuracy is not entirely convincing, as seen in a subset of plots. This is likely due to the handling of different architectures. I argue in the comment below that the architecture is a confounding variable that should be accounted for to avoid Simpson's paradox. This issue does not invalidate the paper's results (many of the plots do show AotIL) but does weaken one of the paper's central claims.", "rebuttal": "We thank the reviewer for the thoughtful comments and are pleased you found the problem, results, and significance compelling. We address each concern in point-by-point detail below.\n\n` the main potential limitation is the range of models evaluated, but I cannot say either way whether the authors have adequately addressed this point from the main text alone (without Appendix A).` / `“Model diversity and stability of the correlation.” `\n\nPlease note that our appendix was included in the supplementary material.\n\nSection 4 clarifies that we train models spanning over 35 architectures (AlexNet through ViT‑L), with different pre‑training, augmentations, and hyperparameters (L143). We also include transfer learning, finetuning, and training from scratch. We also provide a detailed list of the architectures in Appendix A. The diversity of the models keeps your example from occurring; we will clarify this in the final version of the paper. Once we have diversity, we want to select enough models for a stable global correlation, i.e., sampling new models does not change the global correlation by more <1%. Importantly, the sampling is fully random and not based on this stopping criterion; stability is primarily a statistical criterion. Figure 5 in our submission summarizes these results.\n\n`datasets similar to the chest X-rays datasets, which have labelled subpopulations`\n\nThank you for this suggestion. We include state-of-the-art domain generalization datasets to illustrate that this observation is not just a subpopulation issue. We have added the WILDSCivilComments (Koh et al., 2021) dataset to our analysis, which has demographic information (e.g., gender, religion) used to define domains. We find the same behavior. For instance, at 15000 / 52823 (~28%) of the dataset, we have an ID and OOD correlation of -0.6 where this OOD subset are comments from black individuals. This adds to the fairness aspect of the implications of our work.\n\n`“The method also seems a bit overcomplicated and I would like to see the paper focus more on exploring its findings in greater detail (which are quite interesting).”`\n\nThanks for this suggestion to improve our paper. We will move more details on the method to the appendix and expand on our findings, including the clarifications and additional results provided below, as you suggested.\n\n`“why can the intercept be negative?”`\n\nIndeed, the regression is applied to the probit transform, as in the definition of accuracy on the line in seminal previous work (more in the response to the comment below). After this transform, the linear fit can indeed have a negative intercept even though raw accuracies are strictly positive. We will state this explicitly in Section 3.2 and in the figure captions.\n\n`“Why Pearson in probit space rather than Spearman?”`\n\nWe follow the extensive “accuracy‑on‑the‑line” literature [canonical references: Recht et al. 2019; Miller et al. 2021; Taori et al]. Our work addresses this specific empirical observation of correlation in probit space, which captures that structure directly. One limitation is that the Spearman rank correlation will hide sufficiently different examples where model behavior differs substantially from the general trend, an occurrence we want our metric to reflect; we see this below in an example of WILDCamelyon. We aim to identify sets where models can perform well in-distribution but poorly out-of-distribution. If such sets have a small mass, the Spearman correlation may not accurately reflect them.\n\nIt is true that the Spearman correlation could provide additional insights; in our experiments, applying the Spearman correlation to our selected subsets instead of the Pearson correlation yields the same conclusions.\n\nWe primarily aim to identify positive vs. negatively correlated OOD subset accuracy. We apply both Pearson and Spearman and normalize with a threshold of 0, for positive vs. negatively correlated. We find concordance in whether the ID/OOD performances are positively or negatively correlated. We similarly find concordance for strong vs. weak correlations (|R/Spearman| > 0.3).  For PACS, TerraIncognita, we identified near-perfect concordance. Predictions are perfect once we account for error bars (we provide cases below); many of these have correlations near 0.  For VLCS and CXR, we find perfect concordance.\n\nThe WILDSCamelyon example is one where there is a strong negative Pearson R while the Spearman rank is near-zero. Importantly, both of these values suggest that a better in-distribution model does not imply a better out-of-distribution model, and are therefore settings we want to capture. The difference in values here is a function of a group of models with negative correlations, while many other models have positive correlations. So, Pearson R is sensitive to the impact of these models (a property we desire), while Spearman is not. However, we want our metric to be able to show that it is possible to have a negative correlation between ID and OOD accuracies for some set of models if they exist.\n\n\n| Dataset           | N Selected OOD | Pearson R | Spearman |\n|-------------------|---------------:|----------:|---------:|\n| PACS              | 250            | 0.36      | -0.021   |\n| VLCS (perfect concordance) | N/A | N/A | N/A |\n| TerraIncognita    | 2 500          | 0.02      | -0.03    |\n| WILDSCamelyon     | 50             | -0.92     | 0.00     |\n| CXR (perfect concordance) | N/A | N/A | N/A |\n\n`“...a counterexample to submodularity would help greatly.”`\n\nPlease find such a counterexample in the proof of our submission in Appendix B4 (line 977).\n\n`“In CXR 70% of OOD examples are negatively correlated. Where is the strong positive correlation between ID and all OOD examples coming from then? Surely the model must be doing significantly better on the remaining 30% of examples to counteract the OODSelect subset.”`\n\nThis is a good point that we will elaborate on in our updated draft. Your observation is close to the case. Yes, the OOD accuracy, the remaining 30% necessarily has higher OOD than accuracy on the selected subset, so the remaining 30% contributes significantly to the general trend. This is also an effect of the absolute value of the slope of CXR being very small when it is negative in the 70% case, though the correlation is strongly positive. Then the 30% need only do much better relative to the range of OOD accuracy for the remaining 70%, which in some cases is less than 10% accuracy between the worst and best OOD accuracy. Please see Figure 7 in the Appendix, which illustrates this. This is also an interesting observation, as we often assume that only a majority group can have such an impact on the general trend, but a sufficiently out-of-distribution (not too small) set can also have this effect. Importantly, the slope is also sensitive to non spurious correlation factors like uniform label noise, making it an unreliable metric in place of the Pearson correlation.\n\n`“how inconsistency can occur when increasing subset size (maybe in relation with proposition 1)?”`\n\nEach subset size is produced independently. With a submodular objective, greedily growing a set would guarantee monotonic gains and nested subsets. By contrast, our correlation objective is non‑submodular (Prop. 1): the marginal benefit of adding an example can flip sign depending on what is already selected. Empirically, we still observe high overlap across sizes (Jaccard ≥ 0.72) and semantic consistency (for example where we have metadata like CXR), indicating the optimizer tends to find coherent clusters, but the check is essential because the theory offers no guarantee.\n\n`“Influence functions disucssion (line 67)? Is there a way to leverage the scores of individual examples, such as influence scores from Feldman & Zhang, to do the same OOD selection?”`\n\nWe expand on our discussion in L67, speaking to this question. Influence functions [Feldman & Zhang 2020] score training points by how much they change a single test prediction under leave‑one‑out retraining. OODSelect, in contrast, must partition the test/OOD pool itself. Bridging that gap would require an extra, heuristic clustering step and would inherit the known numerical fragility of influence estimates in deep networks [Hu et al., 2025 Most Influential Subset Selection: Challenges, Promises, and Beyond]. Moreover, high‑influence test examples are typically near‑duplicates that the model already gets right, not the spurious‑feature samples that can identify subsets with negative ID <-> OOD correlation. Our “Hard‑example” baselines illustrate this: selecting the most‑misclassified points leaves the correlation near  0, whereas OODSelect selection identifies negatively correlated groups.\n\n`Minor issues:`\n\nThanks for pointing these out; we will address them in our final version: \n\n- `OOD examples…train or test set`. Since we do not train on any OOD examples, there are no train and test set. There is only a full dataset that we select subsets from. We will clarify this in our final version.\n- `Typos and references`. We will address the typos and missing references; thank you.\n- `Proposition 1 subset`. Thank you for pointing this out. As you inferred correctly, we mean the samples selected by s_i’s are subsets of each other rather than the selection vectors themselves. We will fix this notation in our final version.\n- `Def 1...We appreciate the comment on notation and acronyms.` We want to reiterate that there is a literature on this observation of accuracy on the line (under probit transform) that our work directly addresses, though our findings have general implications. We chose to remain consistent with the norms of seminal previous work rather than pollute future discussion with conflicting notations and acronyms..\n\n\nWe hope that our detailed clarifications, additional experiments, and revised presentation comprehensively address your concerns, and you will consider raising your score."}, {"review_id": "jTGJ8j4URp", "reviewer": "Reviewer_CTtt", "summary": "This paper revisit a well-acknowledged concept of ‘Accuracy on the line’, that is, the in-distribution performance and the out of distribution performance of a model will highly correlate. In this paper, author design a test-time adversarial gradient based optimization, to reveal that that exists a subset of OOD data that is inverse correlated to the ID accuracy. Even the overall correlation is positive, a subset up to 77% of the OOD split can lead to negative correlation.", "questions": "1. There're previous paper also criticize the universality of accuracy on the line, like [1] and [2].\n2. This work mostly adopt non-VLM model for experimenting the correlation between ID and OOD dataset. However, as pointed out in [2], VLM model behavior differently to non-VLM in ID-OOD correlation. How would the method/conclusion of this work apply to VLM? ( I assume it's applicable on zero shot inference, as if I understand correctly only model prediction is required for OOD subset selection).\n3. Why not use ImageNet and ImageNet-OOD dataset in the experiment (which is more widely adopted by the community), is it due to size of the data?\n4. In table 1, I am not sure I follow some of the data. for Chest Xrays, full OOD size is 129838, OODselect subset is 50000 (70%), but 50000/129838=38.5%? \n\n\n[1] Effective robustness against natural distribution shifts for models with different training data, NeurIPS 2023. [2] LCA-on-the-Line: Benchmarking Out-of-Distribution Generalization with Class Taxonomies, ICML 2024.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. Author design a neat method to reveal that Accuracy on the line could just be an artifact of aggregating heterogeneous OOD examples, where a subset of OOD might break this hypothesis. This is a interesting idea and novel to the community, it also indicate that average metric can be misleading especially in safety-critical application like medical.\n2. The paper conduct experiment on diverse dataset, including medical dataset and conventional classification dataset, shows that this behavior is universial. \n3. This paper also dive deeper into their experiment and provide good discussion. I find many insights interesting, for instance, the OOD select examples are not corresponding to the most misclassified example (which often lead to 0 correlation).\n\nWeaknesses:\n1. This work ask an interesting question of revisiting benchmarking, but not provide sufficient actionable item. For instance, there’s attempt in finding semantic pattern among OODselect data using VLM, which could be a actionable item for future benchmarking design (like debias, etc), but the result and insight is limited. There’s more discussion on Broader Impact section but mostly appeal for future direction. A complete discussion on how can this idea could inspire future algorithm design(especially training time), benchmark design will be great. \n2. The algorithm is computational expensive. Author should include table showing how will the computation scale with number of model & data.", "was_revised": true, "final_justification": "I thanks author for their detailed reply. My concern regarding compute, related work discussion are largely resolved. Regarding actionability, while training-time implication remain open, I acknowledge that screening protocol for future benchmarks can be a sufficient actionable contribution. Given those, I have raised my score to the paper.", "rebuttal": "Thank you for your thoughtful feedback on our work. We are glad you found our work and insights to be interesting and novel to the community, and you found our experiments to be comprehensive. We address your questions and concerns point-by-point below.\n\n`“This work ask an interesting question of revisiting benchmarking, but not provide sufficient actionable item. For instance, there’s attempt in finding semantic pattern among OODselect data using VLM, which could be a actionable item for future benchmarking design (like debias, etc), but the result and insight is limited. There’s more discussion on Broader Impact section but mostly appeal for future direction. A complete discussion on how can this idea could inspire future algorithm design(especially training time), benchmark design will be great.”`\n\nWe appreciate the call for clearer, immediately useful takeaways. Our updated draft will provide more detailed discussion on this topic. As part of this, we will release the OODSelect subsets for every dataset so that researchers can evaluate new models on splits that break “accuracy‑on‑the‑line.” Second, previous work recommends that future benchmarks begin by screening their OOD splits without accuracy on the line [Bell et al. 2024; Salaudeen et al. 2025]; our method can be applied out‑of‑the‑box to any dataset that today exhibits a positive ID-OOD correlation. These immediate two steps—publicly available stress-test subsets and a simple screening protocol—constitute actionable guidance, and we now state this plainly rather than implying it.\n\n`“The algorithm is computational expensive. Author should include table showing how will the computation scale with number of model & data.”`\n\nThank you for the suggestions. Per your suggestions, we will break this down into GPU hours per model and data in Figure 5 and Table 2. Generally speaking, these computations heavily depend on the specific datasets and data modality, as well as the classes of models considered for the task. Our work serves as a baseline that can be inter- or extrapolated to new datasets and model types. We include the per-model-per-dataset GPU hours along with some summary statistics.\n\nname               | mean     | median   | std_dev  | min      | max\n-------------------|----------|----------|----------|----------|----------\nPACS               | 1.224110 | 0.814722 | 1.273111 | 0.038333 | 6.694167\nVLCS               | 2.128983 | 1.612917 | 1.363286 | 0.041111 | 6.658611\nTerraIncognita     | 2.196349 | 1.921389 | 1.602738 | 0.036944 | 9.859167\nWILDScamelyon      | 4.073941 | 1.830417 | 4.348143 | 0.775000 | 18.054444\nCXR                | 3.624290 | 2.730278 | 3.371409 | 0.001944 | 18.010833\n\n`“There're previous paper also criticize the universality of accuracy on the line, like [1] and [2].”`\n\nThank you for these suggestions; we will incorporate them into our related works and discuss how our work is complementary but different.\n\nImportantly, our work is complementary but different; both papers focus on demonstrating different metrics’ correlation to the aggregated OOD set (Shi et al., 2023, multiple distributions and Shi et al., 2024, LCA), while our work focuses on identifying OOD subsets that have different behavior than the global set. Our work is interested in accuracy; however, developing methods like ours to identify this behavior with other metrics merits future study.\n\n`“This work mostly adopt non-VLM model for experimenting the correlation between ID and OOD dataset. However, as pointed out in [2], VLM model behavior differently to non-VLM in ID-OOD correlation. How would the method/conclusion of this work apply to VLM? ( I assume it's applicable on zero shot inference, as if I understand correctly only model prediction is required for OOD subset selection).”`\n\nImportantly, [2] shows that VLMs also obey AoTL (accuracy, not LCA), suggesting our conclusions should transfer if we restrict attention to that model class. One practical hurdle is that open-source VLMs seldom report accuracy on a held-out test set corresponding to their pre-training distribution, meaning that we cannot directly apply our analysis.\n\nWhen we apply VLMs (the set of models evaluated in your reference [1]) to the ID and OOD subsets we identify for our trained models on the in-distribution data, we still find positive and strong correlations between the ID and selected OOD sets for >1% of the subsets; note these same correlations are strongly negative in our analysis.\n\nThis is likely because (i) both the ID and OOD datasets are ID for the VLMs (pretraining includes these public benchmarks), or (ii) both the ID and OOD sets are OOD for the VLMs. Our analysis requires some notion of models trained on some ID set and evaluated on some other OOD set. To run this analysis, we would need to train VLMs from scratch on our ID and OOD datasets. This is a much larger-scale experiment, considering the model and data sizes required to train VLMs, and it presents an interesting direction for future work.\n\nWe will include this analysis in our final version for completeness; thanks for the suggestion!\n\n​​Dataset         | R |\n|---------------|-----------------------------\n|PACS |  0.78\n|VLCS |   0.62\n|Terra Incognita | 0.84   \n|WILDSCamelyon  |   (approximately) 1.00\n|CXR  | 0.94     \n\n`“Why not use ImageNet and ImageNet-OOD dataset in the experiment (which is more widely adopted by the community), is it due to size of the data?”`\n\nWe focused on multi‑source domain generalization benchmarks that are standard in the literature [Gulrajani et al., 2020; Koh et al., 2020]. ImageNet -> ImageNet‑OOD would be single‑source. Also, as you mentioned, it requires downloading and processing TBs of ImageNet‑21K; incorporating it would significantly increase the already extensive experimental computational costs. We will add a discussion on ImageNet-OOD in our related works.\n\n`“In table 1, I am not sure I follow some of the data. for Chest Xrays, full OOD size is 129838, OODselect subset is 50000 (70%), but 50000/129838=38.5%?”`\n\nYou are correct: the full CXR OOD set is 71,433 images, not 129,838 (size of WILDSCamelyon), so the 50,000-image subset indeed represents ~70 %. We will also address other typos you pointed out; thanks!\n\n`Hyperlinks.`\n\nApologies for the inconvenience. Please find our appendix in the supplemental material. Because they are detached in submission, the links to appendix items do not point to anything. In the final version, where they are together, this issue will naturally be resolved. \n\nWe hope these revisions resolve the outstanding concerns and strengthen the paper, and we would be grateful if you would consider raising the score accordingly."}, {"review_id": "f7XPFgzmxD", "reviewer": "Reviewer_qNUJ", "summary": "This paper uses a gradient-based method, OODSelect, to partition benchmarks’ OOD split into semantically coherent subsets where accuracy on the line does not hold. The authors tested the method in six datasets and demonstrated that the method uncovers subsets where higher ID accuracy predicts lower OOD accuracy, with sometimes up to 77% of the usual OOD split. They also provide the identified subsets for five datasets for future research.", "questions": "- Can the method be applied to non-image domains (e.g., text or tabular)? What modifications would be necessary? In these domains, could the semantic explanations of the OODSelect set be easier?\n- Can the authors provide concrete computational costs with dataset size and model pool size for a better understanding of the extensive computational cost in the main paper?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- The paper is well written, and the OODSelect method is well supported by the theoretical analysis as well as extensive experiments.\n- The paper explores selection consistency and coherence within the subset and demonstrates the semantic soundness of each partition. It also investigates the use of LLMs and VLMs to generate semantic concepts in the dataset.\n- Model pool size selection, subset size, and other design details are clearly illustrated in the main text or appendix, with rationale and supporting experiments.\n\n**Weaknesses**\n\n- The experiments are only conducted in the image domain, and the semantic explanations can be challenging. But I understand that running additional experiments in other domains (like text or tabular data) during the rebuttal period may be difficult.", "was_revised": true, "final_justification": "The author addressed my concerns during the discussion period. I will keep my positive score.", "rebuttal": "Thank you for your feedback on our work. We are glad you found our submission well-written and supported with extensive experiments. We are also glad you found our theoretical analysis to be a strength. We adress your questions point-by-point below.\n\n`Can the method be applied to non-image domains (e.g., text or tabular)? What modifications would be necessary? In these domains, could the semantic explanations of the OODSelect set be easier?`\n\nIndeed, the method can be applied to non-image domains. The only modification is the types of models one applies. From the method’s perspective, there is no difference. Thank you for the suggestions; we have added WILDSCivilComments, a text dataset, to our analysis. We find similar trends for WILDSCivilComments. The WILDS CivilComments dataset is a text classification benchmark derived from the Jigsaw Civil Comments platform, where the goal is to predict the toxicity of online comments. It focuses on evaluating model robustness to demographic distribution shifts, particularly across identity subgroups such as gender, race, and religion. We have 6 domains with spurious correlations related to six identities: ['male',  'female',  'LGBTQ',  'christian',  'muslim',  'other_religions', 'black', 'white']. Our OOD identities are Christian and Black.\n\nFor instance, at 15000 / 52823 (~28%) of the dataset, we have an ID and OOD correlation of -0.6. Additionally, the random and hard selections are positive or near zero.\n\nWe will include the full analysis in our final version.\n\nOne limitation with many state-of-the-art tabular and text benchmarks is that we know exactly what the spurious correlations are. We did not find any unexpected insight from automating semantic explanations. While images also have some known spurious correlations, the field of adversarial machine learning has demonstrated that there are correlations that humans do not perceive that models can use to make predictions [Szegedy et al., 2013]. However, such features were also not provided by our experiments with semantic explanations.\n\n`Can the authors provide concrete computational costs with dataset size and model pool size for a better understanding of the extensive computational cost in the main paper?`\n\nThank you for the suggestions. Per your suggestions, we will break this down into GPU hours per model and data in Figure 5 and Table 2. Generally speaking, these computations heavily depend on the specific datasets and data modality, as well as the classes of models considered for the task. Our work serves as a baseline that can be inter- or extrapolated to new datasets and model types. We include the per-model-per-dataset GPU hours along with some summary statistics.\n\n**GPU Resources per Model per Dataset**\n\n| name               | mean     | median   | std_dev  | min      | max       |\n|--------------------|----------|----------|----------|----------|-----------|\n| PACS               | 1.224110 | 0.814722 | 1.273111 | 0.038333 | 6.694167  |\n| VLCS               | 2.128983 | 1.612917 | 1.363286 | 0.041111 | 6.658611  |\n| TerraIncognita     | 2.196349 | 1.921389 | 1.602738 | 0.036944 | 9.859167  |\n| CXR                | 3.624290 | 2.730278 | 3.371409 | 0.001944 | 18.010833 |\n| WILDScamelyon      | 4.073941 | 1.830417 | 4.348143 | 0.775000 | 18.054444 |\n| WILDScivilcomments | 9.605884 | 9.458056 | 3.703064 | 3.430278 | 17.541944 |\n\nWe hope these additional results clarify your questions and strengthen the paper."}]

Author rebuttal:

Meta-review: This paper looks at the accuracy-on-the-line phenomenon where out-of-distribution generalization shows a strong correlation between in-distribution and out-of-distribution accuracy. The paper specifically establishes that there are semantically-meaningful subsets of examples that have *inverse* correlation, indicating that aggregation of these subsets cause a misleading correlation. The paper further develops a method to find such subsets, demonstrating this across a range of datasets. This is potentially an important finding that is novel and highly relevant to the important problem of understanding generalization, which the reviewers appreciated. Further, the analysis, theoretical motivation, and benchmarks were done in a rigorous manner. Reviewers had a number of questions related to simple baselines for this problem, the correlation metric used, lack of details throughout, relevance of findings on other modalities, computational complexity, and the confounding factor of architectures/models and their inductive biases. The authors provide a strong rebuttal with many results addressing the above concerns, and reviewers recommend acceptance. I agree, and believe this paper provides an interesting avenue that could open up a better understanding of OOD generalization. I recommend that the authors include many of the rebuttal elements in the final paper, and also note that the issue of confounding factors (especially architecture but also training etc.) is important to at least discuss in the paper.
Decision: Accept (spotlight)