Are Domain Generalization Benchmarks with Accuracy on the Line Misspecified?
Authors: Olawale Elijah Salaudeen
Nicole Chiou
Shiny Weng
Sanmi Koyejo
Venue: Accepted by TMLR ()

Abstract: Spurious correlations, unstable statistical shortcuts a model can exploit, are expected to degrade performance out-of-distribution (OOD). However, across many popular OOD generalization benchmarks, vanilla empirical risk minimization (ERM) often achieves the highest OOD accuracy. Moreover, gains in in-distribution accuracy generally improve OOD accuracy, a phenomenon termed accuracy on the line, which contradicts the expected harm of spurious correlations. We show that these observations are an artifact of misspecified OOD datasets that do not include shifts in spurious correlations that harm OOD generalization, the setting they are meant to evaluate. Consequently, current practice evaluates "robustness" without truly stressing the spurious signals we seek to eliminate; our work pinpoints when that happens and how to fix it. Contributions. (i) We derive necessary and sufficient conditions for a distribution shift to reveal a model's reliance on spurious features; when these conditions hold, "accuracy on the line" disappears. (ii) We audit leading OOD datasets and find that most still display accuracy on the line, suggesting they are misspecified for evaluating robustness to spurious correlations. (iii) We catalog the few well-specified datasets and summarize generalizable design principles, such as identifying datasets of natural interventions (e.g., a pandemic), to guide future well-specified benchmarks.

Published in Transactions on Machine Learning Research (08/2025)
Are Domain Generalization Benchmarks with Accuracy on the Line Misspecified?
Olawale Salaudeen*
olowale@mit.ccm edu
Massachusetts Institute of Technology
Nicole Chiou'
nicchsou@@stanford.cco edu
Stanford University
Shiny Weng Stanford University
shimgur@sttaforddcomm edu
Sanmi Koyejo Stanford University
sammi@cs..ttnforde edu
Reviewed on OpenReview: https: // openreview. net/ forum? id- PiyyuRygPQo
Abstract
Spurious correlations unstable statistical shortcuts model can exploit are to degrade performance out-of-listribution (OOD). However, across many popular OOD generalization benchmarks, vanilla empirical risk minimization (ERM) often achieves the highest OOD accuracy. Moreover, gains in-distribution accuracy generally improve OOD accuracy phenomenon termed accuracy on the line-winn which contradicts the expected harm of spurious correlations. We show that these observations dalasels that do not include shifts in spurious correlations that harm OOD generalization; the setting they are meant to evaluate. Consequently, current practice evaluates "robustness" without truly stressing the spurious signals we seek to eliminate; our work pinpoints when that happens and how to fix it. Contributions. (i) We derive necessory and sufficient conditions for a distribution shift. to reveal model's reliance on spurious features; when these conditions hold, "accuracy on the line" disappears. (ii) We audit leading OOD datasets and find that most still display accuracy the line, suggesting they misspecified for evaluating robustness to spurious correlations. (iii) We catalog the few well-specified datasets and summarize generalizable design principles, such as identifying datasets of natural interventions such as pandemic, to guide future well-specified benchmarks.
are expected
artifact of misspecified OOD
1 Introduction
Domain gencralization aims to develop classifiers that generalize to new and potentially worst-case unobscrvod distributions (Arjovsky et al. 2019 Rosenfeld et al., 2020). Spurious correlations, also referred to shortculs, are coincidental statistical associations between features and labels in training data that fail generalize
beyond the training distribution, hindering domain generalization (Nagarajan et al. 2020 Geirhos et al.
2020 Makar et al.) 2022). Thus, many amss for domain generalization have locused on learning Jassifiers that ignore these unreliable paiterns- e.g., invariaace or fealure disenlanglement methods (Arjovsky
et al. 2019 Wang et al. 2019 Parascandolo et al. 2020 Salaudeen Koyejo 2022 Creager et al. 2021
Krueger al. 2021 Ahuja et al. 2021 Shi et al. 2021 Zhou al. 2022 Wang et al. 2022 ct al. 2022
Salaudeen & Koyejo 2024).
Zhou al. (2022) Wang al. (0022) provide a comprehensive survey on domain
*Correspondence to olawale@mitt.dde Work partly done while PhD student the University of Illinois at Urhana- Champaigu and visiting PhD student at Stanford University. Equal contribution; listed alphabetical order.

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Published in Transactions on Machine Learning Research (08/2025)
(b) Out-of-Distribution (misspecified)
(c) Out-of-Distribution
Legend
(a) In-Distribution
(well-specified)
Pasture
Cow
Desert
Camel
Domain
Spurious
General
Figure 1: Simplified illustration of misspecified vs. well-specified domain generalization benchmarks. Consider classifiation task of predicting cows camels. In (a) (in-distribution) most cows appear green pasture backgrounds and most camels on yellow desert backgrounds, the background is spuriously correlated with the label. In (b) (OOD, misspecified), the dataset differs from (a) only in image rotation while preserving the pasture->cow and cesert->camel association. A classifier that relies on background therefore still performs well. (c) (OOD, well-specified), the background-las labell pairing is reversed (camels on pasture, cows on desert). background- classifier now misclassifies most examples, so models that depend on the spurious correlations underperform those that use only domain-general (animal) features.
generalization. Notably, domain generalization is distinct from domain adaptation, where one aims to adapt.
unstable correlations rather than remove them (Saenko et al., 2010 Long et al. 2018 Ganin et al. 2016
Wilson & Cook 2020: Alabdulmohsin et al. 2023 Gupta et al.
2023 Tsai et al. 2024
To ground this problem setting, there are many real-world setlings ings where spurious correlations lead to adverse real-world behavior. For instance, when predicting medical diagnoses from chest X-rays, the relationship between physiological features and diagnoses expected to be stable from site to site (domain-general) while the relationship between site-specific markings on X-rays and diagnoses is unstable (spurious). In practice, classifiers learn to rely site-specific markings and fail out-of-distribution (Zech al. 2018). Furthermore, clinical classifiers trained data from hospital may be less effective in diffcrent hospital with distinct patient demographics, equipment, or clinical practices, inducing spurious correlations (Yang et al. 2024b). Similarly, an animal detection classifier trained on data from one environment and sensor type may be less effective classifying animals in images captured under a different environment and sensor confiurrationinndue spurious correlations (Beery et al. 2018 Xiao el al. 2020
Consequently, the standard approach benchmarking domain generalization algorithms involves training classifiers on set of in-distribution data and subsoquently evaluating their performance distinct out- of of-distribution datasets (Gulrajani & Lopez-Pazi 2020 Koh al. 2021 The underlying hypothesis is that classifiers that rely on spurious correlations will exhibit poorer transfer performance when evaluated on new domains, whereas classifiers that avoid these correlations will generalize better. Consequently, effective algorithms for mitigating or eliminating spurious correlations during training should give classifiers with improved transferability and higher out-of-distribution accuracy. Importantly, standard empirical risk
minimizers (ERM) leverage spurious correlations if doing so minimizes empirical risk (Xiao al. 2020
Vapnik 1991 1999). In some settings, they may even prefer the spurious correlations (Jain et al. 2024
Yang
et al. 2024a
Still, empirical evidence suggests that standard empirical risk minimizers often achieve the highest out-of-
distribution (OOD) accuracy on widely used domain generalization benchmarks (Gulrajani & Lopez- 2020:
Yao et al. 2022 Gagnon-Aree Audet et al. 2022 Yang et al. 2023). Additionally, Taori et al. (2020); Miller et al.
(2020] 2021 Wenzel al. (2022); Back al. (2022 Saxena et al. (2024): Nastl & Hardt (2024) demonstrate
strong correlation between and out-of of--iittibtttin accuracy across several taate-of-thoarr domain generalization benchmarks, suggesting that better in-distribution (TD) performance generally predicts better
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out-of-distribution performance. On popular tabular distribution shift datasets (Gardner et al. 2023 Nastl & Hardt (2024) further show that a classifier that uses all available, and potentially spurious, features generally results in better OOD performance than a classifier that uses a select subset, assumed to be causal and stable (Peters et al. 2016 Heinze-Deml et al. 2018 Schölkopf et al., 2012). These observations scomingly
contradict the thesis that there can be spurious stalistical associalions Training data that can improve ID
performance but worsen OOD performance (Geirhos et al., 2020 Makar et al., 2022
These findings raise critical question: Does better in-dissrrbbttin classifier imply better out-of-distribution classifier, challenging the necessity of targeted algorithms for domain generalization?
We propose reconciling perspective and provide supporting evidence. Drawing on the concept of underspec- ification in modern machine learning pipelines (D' [D'Amour et al., 2022), we propose that the in-distribution empirical risk minimizer's apparent superiority domain generalization benchmarks for asscssing robustness to spurious corrclations.
stems from the misspecification of popular
The core of this work studies benchmark's ability distinguish between two lypes of classifiers: one that ignores spurious correlations Glomain-generss correlations (including spurious) that maximize in-distribution accuracy (domain-speciffc classifiers), Figure I Specifically, we consider the setting where correlations between features and labels exist in training data and a subset (spurious) shift at test time. We investigate the types of shifts under which classífiers that rely on these spurious correlations generalize worse out-of-distribution than classifiers that only use stable (domain- )eenail) correlations. We call these types of shifts well-specified. We propose that when the best in-distribution classifier is also the best out-of-distribution classifier benchmark, this benchmark does not represent the types of scttings domain generalization is concerned with. Thus, our contributions are follows:
invariant classifiers) and another that leverages all available
1.1 Our Contributions
We show that, with high probability, misaligned spurious correlation pre and post-distribution shift necessary and sufficient for well-specified domain generalization benchmarks, Theorem
12
We demonstrate that these well-specified benchmarks and those with accuracy on the line are provably at odds, Theorem 3 Thus, accuracy on the line is a test and sign of misspecified benchmarks.
With over 40 total ID/OODF data splits across trate-of-the-art benchmarks, we show that many state- of-the-art benchmarks exhibit accuracy on the line and may be misspecified for domain generalization. We also identify well-specifeed splits that the field should prioritize. Code We also provide a visualization tool to study future datasets: Link
We outline design principles that give well-spocified ID/OOD splits for future curation of domain generalization benchmarks.
Additionally, our findings have implications for related lasks such as benchmarking algorithms for algorithmic fairness, causal representation learning, etc., in the context of both predictive and generative models.
2 Theoretical Analysis: (Meisspeiiiiiction of Domain Generalization Benchmarks
2.1 Preliminaries
Following previous work Wang et al. 2019 Rosenfeld et al. 2020: Ahuja et al. 2021 Salaudeen & Koyejo 2024), we define domain-general (dg) features Zag C R² where the optimal classifier that only uses these domain-general features is desired. Converscly, spurious features (spu or domain-specific) Z S R contain additional domain-specific information that improves the prediction task in-distribution, but their use can
"In many domain generalization benchmarks, of domains is available. This set typically split into two disjoint subsets, where one for training (in-distributionnIDD) /ID), and the other for testing Nouttof-distriuutioo/0000 https://github. cacilauu.ocmm////// www.aaaaaaaaaaeeeee sewpeeiiieeiiieeeeieeeeeeee. D0_benchmarkss
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degrade performance out-of-distribution. The observed features X' S I are concatenation of Zag and Zapu: where d = k + I. We also define y = [+1]- P's represent probability distributions over X, Y. &, where [&] > 1, denotes the set of distributions of interest. P. E & implies that marginals without Zapu are preserved across P.'s.
Then, we consider classifiers /
y of the form f(X) = XTw = Zgg Wag + 7sp Wspa where Wdg E R and Wspu E RR. We note empirical findings support our parameterization of Zag and Zepu from non-linear transformatton when a final linear mapping is applied, e.g., kernel regressors or common deep neural networks. Rosenfeld et al. 2022a] demonstrates that improving out-of-distribution performance can be done with now linear classifiers on learned (non-linear) representations. Additional work shows that last-layer retraining improves robustness to spurious correlations (Kirichenko et al., 2022; LaBonte al., 2023). These findings suggest that the non-spurious and spurious representations learned by deep models are often linearly separable, suggesting that our assumption is reasonable. Importantly, allow for nonlinear shift in spurious correlation.
Additionally, within F, Fág C F comprises functions fág that only use domain-general features fxg = (an((Za;;)) 0]). Also, denote fx E FV == =xx where fx uses both domain-general and domain-specifie features. The function f(:,-) + R denotes a loss function, and R') = Er. [6(Y f(X)] defines the expected loss for function f E F. Additionally, we define the accuracy of f F on distribution / as
accp(f) = ((, Y 0)] = Pr ((X))Y>0)
(1)
where 1(-) is the indicator function.
Next, we enumerate some assumptions and definitions for our theoretical analysis.
The relationship between spurious features and the label we want to predict is allowed to change across domains and negatively impact out-of-listribution performance, while the relationship between domain-general features and the label is stable across domains.
We first formally define spurious features and domain-general features (Definitions D 2
For example, consider the canonical example of classifying cows and camels. Here, domain general features are animal features, while domain-specific (spurious) features are backgrounds. Predictions derived for features of the cow are generally expected to be stable, while predicting cow vs. camel based on pasture vs. desort may be accurate for some data distributions and catastrophically inaccurate in others. Another example is domain-general physiological features rather than spurious hospital-specific markings in clinical chest X-rays diagnostic tasks (Zech et al. 2018).
Definition 1 (Domain-General Features Zag). For all P., Pj E E,
Y Zag] = Er, [V Zag]-
(2)
Definition 2 (Spurious Features Zupu). There erists Ps, Pj such that,
Ep, [Y| Zepul F Er, Y Zpp and Er, Y Zay: sppu # Ey, [Y Zdg Zspu)=
(3)
We assume both types of features are informative about labels (Assumption and are not redundant (Assumption ) Clearly, if Assumption I does not hold, then the learning problem is trivial; features are uncorrelated with labels. When Assumption does not hold, spurious features are redundant and have no unique information about the labels. Ahuja et al. (2021) study this sctting (Fully Informative Irvariant Features (PIIF); we use Noomain-general instead of 'invariant') and give conditions under which classifiers using spurious correlations focus on the partially informative domain-general fcatures setting, i.c., under Assumption 2 For example, we assume that, in the training domain, there are examples that a classifier is more likely to get correct if it uses spurious features.
achieve equal OOD accuracy the optimal invariant classifier. Hence, we
For instance, there may be occluded animals their natural habitat included in the training set, so using background improves the prediction accuracy on the training set (Beery et al., 2018).. Similarly, the physiological features of disease, like COVID-19, may not be fully represented in some chest X-rays, but
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hospital-specifie markings correlated with the diagnosis of COVID-19 serve predictive shorteut for the label in the training set. We assume both types of features are correlated with and contain unique information about the label.
Assumption 1 (Informative Deneial and Domain-Specifir Features). For all observed training distributions P,
Ep|Y! Zagl * Ep(y] and Ep Zspal # Ep[Y].
(4)
Assumption 2 (Non-Redundant Features).
Zupu X Y Zdg and Zag A. Zspa-
(5)
Also referved to as partially informative Momain-generd features.
Since we consider any feature whose inclusion decreases worst-case performance on the set of distributions of interest & as spurious, the definition of spurious is strongly tied to the & 'worst-case' is with respect to, such as the set of hospitals one expects a classifier to have to operate in. What is considered spurious for + E may differ, even if g' C & or C E'. Notably, this observation implies that domain generalization cannot. practically be divorced from domain expertise in defining E. Clearly, too narrow a & decreases expected robustness (potentially catastrophically). and too broad & may excessively and unnecessarily decrease overall utility (Shen et al. 2024
We define two types of classifiers: (i) the optimal domain-general classifier, which depends on & (Definition and (ii) the optimal domain-specific classifier for a given P E & (Definition 4
Definition 3 (Optimal Domain General Classifier fág). Given set of distributions of interest = Y)) i= i=....,
fág argmax min accp, (f). feFs PEE
(6)
By construction, Sāg Fag does not use spurious features. Definition 4 (Optimal Domain Specific Classifier f&). Given a distribution and
R&
argmax accp() (J).
(7)
JEF
By construction, f& E Fx uses spurious features (Lemma
Our first result, Lemma [ shows that, given informative and non-redundant features (Assumptions 1 2 for any distribution P E E, the optimal E-domain-general and P-domain-specific classifier are different, and the optimal domain-specific classifier achieves a lower (strongly convex) loss in-distribution than the optimal domain-general classifier. This result contextualizes the rest of our results in that optimal domain-specific classifiers use all non-redundant features that improve in-distribution performance.
Lemma 1 (Domain- and Domain-Specific In-Distribution Error Gap). Assume non-redundant//oo trivial features, partially informative domain-generd, features (Assumptions and strongly conver loss L.
min R*) < min R*(), FEF
(8)
fEF
where F X where f(x) = wT dd + u Apas F.F. For f Fag f(x) w w Zdg The proof of Lemma is provided in Appendix A.1
We typically train the sort of classifiers study in this work with cross-entropy loss, which is generally not strongly convex. However, with common regularizers, like weight decay, the objective can become strongly convex Shaley-Shwarts & Ben-Davin 2014). Given that the in-distribution risk minimizer and the domain- general classifier differ, we now show that, given training and test distributions PiD # PooD E E,, respectively, the domain--eneral classifier 28 may also not achieve a higher OOD accuracy than the in-distribution risk minimizer PID Poon-

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Next, we define well-specified benchmarks (Definition 5 Consider the running example of predicting cows and camels. Suppose are given set of training distributions and an OOD distribution. Then we are given two classifiers: a nomainngennerl classifier that only uses animal features for prediction and a domain-specific classifier that also uses background information- both classificrs are trained on the training distributions. We consider this a well-specified benchmark if and only if the domain general classifier achieves a higher accuracy on the OOD dataset. Effectively, well-spocified benchmarks penalize classifiers for using spurious correlations; hence, they can identify algorithms that give (omain-general classifiers.
Definition 5 (Well-Specified Domain Generalization Benchmark). Two ID/OOD splits, Pwy PooD E E, are 'well-specified' if and only if
accPood accpoos(fx)) (fRm0) < accPood (fx)
(9)
where fåg. PPo are from Definitions and raspectivelynnot that this definition is with respect to accuracy.
Analysis Setting. For the rest of this work, we consider sub-Gaussial Zypu with mean papu, covariance Espus and sub- nab-Gaussian parameter K- i--imoortanlly, our results also apply more generally to other classes of random variables (Remark Zy. We define an Lo-Lisschitz function which parametrizes the distribution shift w.r.t. Zapu: Φ: Rixl + R!xT such that Zap = 6 ? and E| B(Z MB Mapsp where M E R and Dg is the covariance of Zapu..
Our overall goal is to identify shifts where achieving higher transfer accuracy is informative about the reliance of classifiers on spurious correlations. Tdentifying the class of such shifts is necessary as they describe TD/ ID/ODD splits for domain- -ennainngeeeralzzation benchmarks where achieving the higher OOD accuracy meaningfully maps to learning Momain-general classifiers, i.e., the shift is well-specified for the task. Without such well-specisied shifts, we would incorrectly assume that domain-general classifier is ineffective because it achieves a lower accuracy out-of-distribution than different empirical risk minimizer. Thus, our next result shows that when shifts (Ø) result in a sulficient misalignment bet ween spurious correlations before and after the shift, the domain- domain-general classifier achieves a higher
accuracy than a domain-specific classifier.
Our results consider uub-Gaussian spurious features. However, our resulls hold for other classes of random variables, e.g., sub- eebeeeponnntall or other random variables in Orlicz spaces (Krasnoselskij 1960). In these cases, our proofs remain largely unchanged, with only the constant factors adjusted to account for the diffeernnt concentrulion properlies of the spurious fealures.
Remark 1.
2.2 Main Results
Overview. We formalize when out-of-distribution (OOD) benchmark is well-specified for evaluating robustness to spurious correlations. Theorems 12 give sufficient, and under symmetry necessary, conditions for well-specification:: the spurious feature label correlation must be sufficiently misaligned between in- and out-of-distribution splits, a property we call spurious correlation reversal. Theorem 3 proves that a strong positive correlation between in- and out-of-distribution accuracies (*accuracy on the line") is essentially incompatible with spuriously correlated correlation reversal. Specifically, the probability of observing a well-specified benchmark with accuracy on the line is zero. Together, these results provide a lest of the absence of accuracy the line to identify benchmarks that meaningfully penalize reliance on spurious correlations.
Our first results give sufficient conditions for well-specisied domain Somaain-generalizttion splits. Theorem I demon- strates that a domain generalization split is well-specified if (i) the OOD spurious correlation is misaligned with the ID spurious correlation and (ii) the variance of spurious features is sufficiently controlled not to undo the effect of misalignment. For symmetrically distributed features, such as Gaussians, Theorem gives similar conditions that are both necessary and sufficient.
Theorem 1 (Sufficient Conditions for Well-Specified Domain Generalization Splits). Assume Spp is sub- Gaussian with meon Papa, covariance Eppu: and parameter k. Define an -Lipschhtt nonlinear transformation
RR + +
and let Spps di Z
Assume further that
E 7 spo) ((pp) M Mapas
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for matriz M Rixi, In-distribution, Z denote Wapu the contribution of zpp to the optimal PıD classifier f Then, for any E (0, 1), if
PID and
spu PooD- Additionally,
( Papu) + v2 La K. llsspull /oopull///og(1/0)) <0,
fwith the understanding that under the Lipschitz assumption, the nub-Gaussian property carries over with parameter Lg K), then with probability at least 8 over Zapu we have
accPood accpooo(fx (fx < accpood (fäg): (fag)
where fåg and f0 are the optimal domain-generde and domain cemmain--pecific. predictions (Definitions conditions give well-specified shift, Defimition 3
These
Proof provided in Appendia A.2
We assume the distribution of spurious features is symmetric; & similar form of Theorem results are both necessary and sufficient, Theorem 2
Theorem 2 (Necesary and Sufficient Conditions for Well-Specified Domain Generalization Splits under Distributional Symmetry). Assume spu is random variable with mean Hspus covariance Exprs and w 2 is symmetric about its mean. Define an Lo Lipschitz nonlinear transformation
Φ Rd + RR,
and let Zpp 0 Zppu).
Assume further that
Z spu
El (pp)))) M Mapu:
Cov(Z (Z Eg. and Zog Zopu Y spu
for some matriz M E Rixt, and that the scalor and w spm Zoon is symmetric about its mean. Then,
d + w M Mapu
w Wdg
accP (P)) < accpooo (1gg
(10)
W dg Wag + w W Eng Wapn
w) Dagwas
Note that the RHS of Equation 10 implies that either:
Spurions Correlation Reversal: "m M Mspr < O (related to sufficiency condition without the symmetry assumplion in Theorem or Small Signal-to-Noise Ratio (SNR): sufficiently large variance for w M Masa > 0.
These conditions are necessary and sufficient for a well-specified shift, Definition Proof provided in Appendia A.3
As an example, consider the classic examples of classifiers using background, pasture, or desert predict cows and camels, respectively. Our results show that. the correlation between background and label has to lead to sufficient disagreement (reversed) between training and test distributions such that their use harms OOD (test) predictions. Without this feature, classifiers without spurious correlations are not expected to transfer better than classifiers with spurious correlations.
Importantly, we are concerned with benchmarking classifiers designed not to rely on spurious correlations for prodiction, even if they are useful and potentially reliable classifiers most of the time. Figure 2 verifies these conditions empirically with simulation experiments. Additionally, semi-syuthetic examples with variants of the MNISST dataset is provided in Appendix B.1 (Arjovsky et al. 2019).
Remark 2 (Multisource (ammainnGenerrllzattoo.... We often have access to set of distributions for evaluation, ID and OOD splits (Gulrajani & Lopez-Paz, 2020 Koh
and the norm is to evaluate el al., 2021). In the contert of Desinition 5 P.o represents the mixture of ID distributions, and the lest split is Poop (which can also a misture). Importantly, a finite mixture of sub- Gaussians is also sub- Gaussian (Wainwright, 2019), so our results directly apply without loss of generality. Notably, we focus on distinct ID/OOD splits. A set of domains may give many splits, yet only a subset of the splits, if any, may be weel-specified.
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transfer accuracy
0.8
trandfer accuracy
0.8
transfer accuracy
0.7150
0.6
00
00060
mo
O000)
OD
00080
cood
0.7125
0.7100
0.4
0.4
0.7075
0.2
0.7050
-40
-20
20
40
-6
-4
-2
-0.010
-0.005
0,000 o. 0,005 0.010
Mags
Mplao
Way Mpap
(a) All random variables are Gaus- sian. When Theorem 12 conditions are satisfied OOD (x-axis: Wapn M pepu
(b) Zapu is defined be a mix-
(c) Rosenfeld al. (2022b) studies do- main interpolation, where the larget domain a mixture of the training domain; variant of ERM is provably worst-shift. optimal. Spurious features (Zipa) are defined to be mix- ture of 4 Gaussian distributions such that the test distributions are a different mixture of the 4 Gaussians. Overall, there is minimal difference in OOD accuracy between domain-general and domain-specific classifiers in this setting (domain in- terpolation).
ture of 4 Gaussian distributions Asub-Gaussian such that the test distributions are not mixture of the same 4 Gaussians. The same conclusions in Figure 2a hold for sub- Gaussian random variables when the lest distribution is an extrapolation.
O; 0), the domain general classifiers outperform the domain spe- eific classifiers OOD, This result ver- ifies that there needs to be sufficient misalignment between in- and out-of- distribution spurious correlations for the domaan-general features to oulper- form the domain- domain-specific classifiers in OOD accuracy.
Figure 2: In our simulation, Nomain-general classifiers (fag) are trained on only domain general features, while domain-specific classifiers are trained on both domain general and domain-specific features from the same distribution. Features are sub- Gabssiansi We evaluate these classifiers on 50 test distributions generated by randomly sampling the distribution shift. (M) such that all marginals without spurious features are the same ID and OOD in the test distribution, but for spurious features: Zypar M Ztpu and, thus, the spurious correlation shifts from Mspu M Hspu. Details on the experiments can be found in Appendix B
So far, have shown that the learned domain-general and domain-speeific classifiers given training distribution differ, and the domain-general classifier achieves higher OOD accuracy (well-specified) when spurious correlations ID and OOD are sufficiently misaligned. For symmetrically distributed spurious features, such misalignment is necessary and sufficient for the evaluation goal. Next, suppose we observe such misaligument and the split is well-specified. Then, we evaluate a set of diverse classifiers on held-out ID and OOD test examples. We should observe weak correlation or a strong negative correlation between the classifiers' ID and OOD accuracy, i.e., no positive accuracy on the line. When we observe accuracy on the line, with high probability, the ID/OOD split is misspecified, Theorem
2.2.1 Tradeoffs between Well-Specification and Accuracy on the Line
First, we formally define accuracy on the line, the correlation strength between in- and distrribiitiiiinn accuracy, Definition
Define a E IR, 2 O, and Φ-1 as the inverse Gaussian cumulative density function. The correlation property is defined as
Definition 6 (Accuracy on the Line; Miller et al.
2021
(acer,a
+ (accrood (f)) S
(11)
where f's distinct classifiers.
If there exists an such that = 0, then there is a perfect correlation between ID and OOD accuracy. As grows, the strength of the correlation decreases. a > 0, then the correlation positive, and if a < 0, the correlation negative. We will call the setting where a > positive accuracy on the line and a < accuracy on the inverse line. Theorem 3 shows that the smaller the e, the smaller the probability that the ID/OOD split is well-specified. The probability of well-specified ID/OOD split when e 0 is also O.
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90
y-x
90
OOD aCc
70
= al,
Slope: -2.25
60
60
2 0.83
50
50
M al, > 0:
50
40
Slope: 2.61
30
30
2: 0.97
20
Slope: -0.05
20
M al.a
M
2 0.00
y-x
y-x
10
20
30
40
50
60
70
90 10 20
50 60 70
90
10
20 30 40 50 60 70
90
acc
ID acc
ID acc
Figure 3: ID OOD accuracy on probit scale. When M is sampled arbitrarily, such that the SNR condition is salisfied (Theorem 2). the strong correlation between ID and OOD accuracy disappears. (left figure). For MID = and MOOD = al, there is only a linear scaling in the spurious correlation and spurious feature variance. In this setting, we observe a correlation between ID and OOD accuracy. However, when spurious correlation reversal not satisfied, a > 0 (middle figure), we have positive accuracy on the line. When a < (right figure), we have the spurious correlation reversal condition and have accuracy on the inverse line, where there is strong but negative correlation between in- and out-of-distribution accuracy. More experimental details can be found in Appendix B
Theorem 3 (Benchmarks with Accuracy on the Line are Misspecified Almost Everywhere). Define
u (M Mapa) + (2 1:)2 [ssppll log(1/6) < 0 (well-specified; Theorem
W.
M MERIX and
Φ Φ- D-ICacCooo (fx)) a = Sc (accuracy on the line; Definition
6
(12)
Then:
(i) Wo has Lebesgue measure zero in gixt (t.e., probability zero for random draw).
(ii For any 0 S S cj, we have We, S Wey.
The proof of Theorem and supporting Lemmas are provided Appendir A.6
Theorem 3 demonstrates that the two conditions in Equation 12 are at odds. In particular, c- 0 (i.e., perfect accuracy on the line), almost every shift is misspecified, and the Lebesgue measure (probability) of the set of well-specified shifts grows monotonically with e, ie., inversely with accuracy on the line. This means that when we observe accuracy on the line, with a high probability, the ID/OOD split is misspecified. In Appendix A.7 we construct an intuitive example such (zero-measure) shifts, where both the conditions for well-spocified splits and accuracy on the line hold.
To summarize, Theorem 2 give necessary and sufficient conditions for well-specified domain generalization 3 demonstrates that there is zero probability that such conditions and perfect
ID/OOD splits, and Theorem accuracy on the line for the ID/OOD split simultancously hold-Figure 3 Morcover, accuracy on the line is at odds with well-specified shifts. Our results suggest that datasets with accuracy on the line may be misspecified for benchmarking domain generalization. line, or weak correlation between ID and OOD accuracy, are better suitod benchmark domain generalization. The accuracy the line property now gives a test for well-specified benchmarks. Building on this insight, apply this test to state-of-the-an domain generalization benchmarks (Section
contrast, benchmarks with accuracy on the inverse

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3 Domain Generalization Datasets and Related Work
We first consider two influential domain generalization benchmark suites: DomainBed (Gulrajani & Lopez Paz 2020) and WILDS (Koh et al., 2021p. DomainBed is collection of object recognition domain generalization benchmarks. For example, PACS (Khosla et al. 2012 Li et al., 2017) includes images of seven classes across four domains: Photos, Art Paintings, Cartoons, and Sketches. Another benchmark is ColoredMINIST (Arjovsky et al. 2019), a semi-synthetic binary classification variation of MNIST (Deng 2012), which introduces color as spurious correlation and defines domains by specific color-label associations. Guirajani & Lopez-Par Paz (2020) found that empirical risk minimization achieved the best transfer performance compared to state-of-the-an domain generalization algorithms across PACS, ColoredMNIST, and other DomainBed benchmarks. They also observed differences in their findings based on their choice of model selection.
WILDS was designed to better represent real-world shifts across vision and language. For example, Camelyon17 (Zech et al.] 2018 AlBadawy et al.] 2018) includes images of tissue that may contain tumor tissue (classes) from different. hospitals with varying conditions (domains). CivilComments Borkan al. (2199) includes comments to online article that may be toxic (class) for different subpopulations defined by demographic identities (domains). These benchmarks illustrate the suite's focus practical and natural real- world applications. However, even across WILDS benchmarks, no state-of-theeart methods have demonstrated consistent superiority over ERM (Koh et al. 2021
For Subpopulation shift, which we consider a special case of the broader domain generalization task, benchmarks are designed specifically to evaluate distribution shift robustness in scenarios where spurious correlations lead to worse performance on underrepresentedd subgroups out-of-distribution. For example, the Waterbirds benchmark (Sagawa et al.) 2019) introduces spurious correlation between bird species and backgrounds, such waterbirds predominantly appearing in water environments. Classifiers that rely on backgrounds rather than bird features when predicting bird type (classes) generalize poorly to new domains where backgrounds are urban areas-birda in urban backgrounds are undersampled in the training data.
Benchmarks addressing other types of distribution shifts where domain generalization desired have also been proposed, e.g., temporal shifts (Yao al. 2022 Joshi et al. 2023 Zhang et al. 2023).
Other conceptualizations of distribution shift datasets have been extended to benchmark domain generalization. However, not all are designed to evaluate robustness to spurious correlations, nor do they represent multi- source domain generalization settings. For instance, some benchmarks are designed to assess robustness against typical data quality variations encountered in real-world settings, like variations of ImageNet (Deng let al., 2009 Hendrycks & Dietterich, 2019 Hendrycks et al., 2021). Such benchmarks are out of scope for our study as they are not standard benchmarks for the domain generalization tasks we study. Nevertheless, we refer to previous work studying accuracy on the line for these datasets (Taori et al. 2020
Developing and curating benchmarks is an arduous process and is of immense service to the research community. DomainBed, WILDS, and other benchmarks are uscful when applied in the appropriate scope. Our work demonstrates that robustness to spurious correlations and related tasks are inappropriate applications of many of these benchmarks. In the next. section, we outline design principles to develop new benchmarks for spurious correlations. Furthermore, we identify benchmarks that our analysis suggests are well-specified.
3.1 Accuracy on the Line
The accuracy on the line phenomena has been observed empirically in previous work for many benchmarks in these suites (Recht et al. 2019
Miller et al. 2021 2020 Taori et al.) 2020 Baek et al. 2022 Saxena et al.
2024). However, Liu et al. (0023 identify worldworld tabular datascts with weak or negative linear correlation,
and Teney el al. (2023) identify non-tabular real-world datasets where ID and OOD performance exhibit. other patterns between in- and ditto--distribution accuracy beyond strongly positive and linear. Notably, they also found that correlations can be inverted in some datascts. [Sanyal et al. (2024) derive noise conditions to achieve shifts with accuracy on the inverse line. Our work uniquely specifies the implications of accuracy the line (or lack thereof) the utility of datasets as domain generalization benchmarks.
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Closest to our work is Bell al. (2024). which studies spurious correlation benchmarks and shows that different datasets often disagree on which methods perform best, hindering reliable conclusions about spurious correlation mitigation. They propose three desiderata: (i) ERM Failure, (ii) Discriminative Power, and (iii) Convergent Validity arguing that good benchmarks must produce group-wise errors under standard empirical risk minimization, distinguish among methods, and agree with other benchmarks that test the same phenomenon. However, they focus primarily on subpopulation shifts and systematically assessing the agreement between existing benchmarks (e.g,, Waterbirds). Although both studies highlight the importance of benchmark design, our primary goal is to formally characterize when distribution shifts fail to penalize reliance on spurious correlations, resulting in an ill- ill-posed domain generalization setup, rather than to compare a wide range of algorithms across multiple datasets to demonstrate agreement or lack thereof.
Other works have studied conditions where the empirical risk minimizer for observed distributions is sufficient for domain generalization. Rosenfeld et al. (202bb) show this to be the case under domain interpolation, i.e., OOD distributions are convex combinations of observed distributions (in an online sctting), see example in Figure 2c Additionally, when domain-general features are fully informative, i.e., spurious correlations are redundant, and under some conditions, Ahuja et al. [2021) also show this to be the case. Our results corroborate their findings.
Worst Case Consideration. This notion of worst-case stress testing to establish robustness under distribution shift is not new. When evaluating the out-of-distribution generalization of deep learning methods developed on the ImageNet task (Deng et al., 2009 (ImageNNet Large Scale Visual Recognition Challenge (ILSVRC) [Russakovsky et al. 2015)) ). Kornblith al. (2019) assess generalization by applying said methods contemporary datasets with presumably different distributions than ImageNet, such as CIFAR-10 (Krizhevsky) 2009), however, the datasots they investigated could be considered quite similar in distribution to ImageNet. Allernatively, Salaudeen & Hardt (2024) adversarially constructed a dataset, ImageNot, designed to shift spurious correlations present in the original ImageNNtt construction strongly. Notably, the findings of Salaudeen & Hardt (2024) align with those of Kornblith et al. (2019) despite the deliberately adversarial construction of ImageNot. We argue that constructing datasets with such adversarial spurions correlation shifts is essential for rigorously probing a classifier's use of spurious correlations.
More generally, other works have proposed alternative approaches to developing domain generalization benchmarks. Satisfying our conditions introduces an additional necessary dimension for creating more meaningful evaluations. For example, [Zhang et al. (023)) argues that many existing benchmarks are limited by having too few domains and overly simplistic settings, which restrict their ability to simulate the significant distribution shifts observed in real-world scenarios. Similarly, Lynch et al. (2023) contend that benchmarks inadequately capture the complex, many-to-me ta-y spurions correlations that can arise in practical applications.
Like previous work, we next investigate the accuracy of the line propertics of statc-of-tho-art domain generalization benchmarks to take an inventory of well-specified datasets. Table I provides a sunmary.
4 Empirical Results
We evaluate the correlation bet in-domain (ID) and out-of-domain (OOD) accuracy for benchmarks in the popular DomainBed (Gulrajani & Lopez-Paz 2020) and WILDS (Koh et al. 2021] benchmark suites, as well as subpopulation shift benchmarks, c.g., WaterBirds Sagawa ct al. 2019).
Datasets.
Specifically, our results include the following datasets:
Camelyon Bandi et al. 2018 Koh et al,
2021), CivilComments (Borkan et al. 2019 Koh et al., (201)) ColoredMNIST (Arjovsky et al. 2019
Guirajani & Lopez-Paz 2020), Covid-CXR AllzateGrrislles et al. 2022; Cohen al. 2020b Tabik et al,
2020 Tahir et al. 2021 Suwalska et al. 2023 FMoW (Christie et al. 2018 Koh et al. 2021
PACS LL
*The extent which test or mcasure correlates with other tests or measures that designod to assess the same similar
constructs, indicating that the different mcasures capturing the same underlying concept.' To satisfy Convergent Validity, benchmark should agree with similar benchmarks and disagree with those that are dissimilar (Bell al. 2024 Salaudeen et al.
2025
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Table v: well-specified (R < 0.3); x: misspecified (R> 0.3). We train on a set of ID) distributions and test on a left-out OOD distribution. We present the Pearson of ID and OOD probit-transfommet accuracies and the slope and intercept of OOD accuracy regressed on ID accuracy. (*) OOD for waterbirds refers to the group where y 0 and = 0. The ID dataset
mixture of groups at train time. Additional datasets and analysis provided in Appendix C which also includes complete tables with all splits for each dataset. With conservative threshold of R < 0.3, only subset of datasets satisfies our derived conditions. We note that 0.3 is commonly regarded as the threshold for "weak" correlation, which is the sole justification for our choice.
Dataset
OOD
R < 0.3
slope
offset
R
p-value
std error
ColoredMNIST
Env 2 acc
V
-1.56
0.47
-0.74
0.00
0.01
CXR
Env acc
V
-0.60
0.56
-0.48
0.00
0.03
SpawriousO20 hard
Env acc
0.32
-0.21
0.50
0.00
0.05
SpawriousM22 hard
Env acc
0.76
-0.26
0.94
0.00
0.01
SpawriousO20 easy
Env 0 acc
0.48
-0.29
0.74
0.00
0.04
SpawriousM2M easy
Env 0 acc
X
0.34
0.26
0.60
0.00
0.00
PACS
Eav 1 acc
X
0.68
-0.68
0.84
0.00
0.01
Terralncognita
Env 1 acc
X 0.83
-1.41
0.74
0.00
0.02
Camelyon
Env 2 acc
X 0.62
0.49
0.78
0.00
0.01
Subpopulation Shift Datasets
CivilComments
Env acc
-0.49
0.16
-0.47
0.00
0.03
WaterBirds
Env (*) ace Env acc
-0.13
1.58
-0.13
0.00
0.03
FMoW
0.76
-0.61
0.87
0.00
0.01
et al.
2017 Guirajani & Lopez-Patt 2020), Spawrious
[Lnnch et al. 2023
TerraIncognita (Beery et al.
2018
Gulrajani & Lopez-Paat Paazz-Paz 2020 and Waterbirds (Sagawa et al. 2019
Model Architectures. For vision datasets, we leverage rreted deep learning architectures such
ResNet-18/ (He et al. ct al. 2020), and ConvNeXt-Tiny [Liu et al. 2022 For language datasets, we utilize pretrained embeddings from BERT (Devlin et al.
2016), MenseNeet121 (Huang al., 2017), Vision Transformers (Dosovitskiy
2019 and DistilBERT (Sanh et al. 2020), and apply lower-capacks machine learning classifiers, such as logistic regression, for downstream classification tasks.
Experimental Setup. The benchmarks we consider include a set of domains, where domains are distinct data distributions. As standard in the literature (Gulrajani & Lopez-Paz 2020 Koh et al. 2021), the in-distribution (ID) data defined as a mixture of subset of the data domains, and the out-of-distributiin (OOD) data is not included in the training domains, that is, we perform leave-one-domain-ouu ID/ 1DOODD splits for experiments, where train on all but domain and use the left-out domain OOD. We generate classifiers for our experiments by varying classifier and training hyperparaneters for each architecture, including the number of training epochs (Appendix C Table 2 Our experiments include training these classifiers end-to-end with varying hyperparameters and data augmentations, as well as finetuning pretrained classifiers and transfer learning.
Table I highlights the prevalence of widely-used domain generalization benchmarks with accuracy on the line, signature of potential misspecification, while Figure qualitatively illustrates benchmarks with weak strongly negative correlation between in and out-of-distribution accuracy. We provide detailed account of our experiments, along with benchmark-spechs discussions, in Appendix C
Sclecting Number of classifiers. To ensure a robust estimate of ID-OOD correlation, we follow and exlend the approach of prior work by training a diverse set of classifiers that vary in architecture, random seed, data order, and hyperparameters. This diversity mitigates the risk of single-classifter artifacts influencing our findings. We adopt a simple heuristic: continue adding classificrs until the ID-OOD Pearson correlation changes by less than 1% with the addition of new classifiers. As shown in Table 3 this threshold is consistently reached well before exhausting our pool of trained classifiers. In practice, we train 2 10x classifiers than
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R=0.98
R=0.78
Test Env ACC
90
Test Env 2 Acc
80
80
70
60
70
50
40
00
30
40
20
10
20
70
99
20
40
50
60
70
80
90
98
Average Train Envs Acc
Average Train Envs Acc
(a) PACS (Env 0)
(b) Camelyon (Env
R Rm-00.m -0.48
99
70
90
Test Env 3 Acc
70
60
Test Env
60
Rm40.47
40
50
30
20
10
40
30
40
50
60
70
73
20 405060 80 Average Train Acc
99
Average Train Envs Acc
(c) Covid-CXR (Env 3)
(d) CivilComments (Env 1).
benchmarks with strong positive, weak, or strong negative correlation between in-distribution and out-of-distribution accuracy. Our results suggest that algorithms that consistently provide models with the best transfer accuracies for these splits are at least partially successful in removing spurious correlations. For Camelyon (b), within some accuracy range, have accuracy the inverse line, indicating the importance of qualitative assessment these trends.
Figure 4: We show some ID/OOD splits of popular
previous studies, often exceeding thousands per dataset and ensuring our correlation estimates are stable and conservative (Miller et al.) 2021). For instance, train over 10,000 classifiers for single CivilComments split, and our criterion is satisfied with less than 7,000 classificrs. We train over 5,000 classifiers for a PACS environment, and our criterion is satisfied with 910. We trained as many classifiers as computationally feasible to ensure the robustness our findings.
4.1 Findings
We find that semisynthetic datasets more reliably satisfy our derived conditions. Particularly, semisynthetic here means real-world datasets that either (i) have real-world but artificial spurious correlations introduced (ColoredMNIST/ccole
Spawrions/baccgroondd and Waterbirds/backkroonnt background)) or (ii) have some selection process that introduces spurious correlations (CivilComments). Other datasets without careful consideration with respect to spurious correlations generally have accuracy on the line. The Covid CXR datascts from different regions sometimes exhibit. a relationship that Teney et al. (2024) refers to as 'No Transfer,' where the OOD accuracy is near constant despite variance in the ID accuracy. Our results further echo Teney et al. (2024): notes on potentially mislcading advice from past studics: particularly the recommendation of focusing on ID performance to improve OOD robustness (Wenzel al. (202)), which our work suggests is only reasonable strategy in settings where distribution shifts are relatively simple and weak.
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<5 / Red
>= / Green
W bird / W bg
bird / bg
=5 Red
OOD
L bird W bg
OOD
< Green
ID
W bird / L bg
ID
(a) ColoredMNIST.
Embeddings of a model trained on Env 1 and The labels are < 5, 2 and the spurious feature is color (red, green). The UMAP of the embed- dings clusters based on color (spurious) rather than digit features (donain--eneral). This leads to incorrect predic- tions OOD where the color-label correlation reverses, as observed in our experiments (Section C.2
(b) Waterbirds. Embeddings trained examples with spurious correlation between bird type and background. The bird type is the label, and the background is the spurious feature. The UMAP of the embeddings clus- ters based background lype (spurious) than bird features (domain-general); This leads to incor- rect predictions OOD where background-label correlation changes, as observed in our experiments (Section C.9)
strongly
Class 0
Class 1
Class
Class 2
Class
Class 3
Class
Class 4
Class 3
Class 5
Env 0
Class 6
Env
Env
Env 2
Env 1
Env
Env
(c) Spawrious 020 Hard. Embeddings a model trained Eny 1 and 2. The label is animal type ,and the spu- rious feature is background. Generally, for some classes (e.g., Class 0), some subset of OOD (Env 0) examples cluster very closely to the same classes in in-distribution data. However, for the same class, many OOD examples incorrectly cluster more closely to class 2; there is also smaller variance in these examples. We ohserve this most classes, demonstrating that the model is relying spurious correlations. This matches our observations of OOD performance degradation due to spurious correla- tions (Section C33.2
(d) PACS. Embeddings of a model trained on Env 0-2. The spurious feature is unknown, but it is purported to exist because each environment has a different image style. We find that there a relatively strong alignment between the in-distribution and out-of-distribntio embeddings for each class. This matches our finding that there is not strong spurious correlation that would cause models to embed examples based on features that have an unstable relationship between labels (Section C.1. like we see in other examples we analyzed with UMAP.
Figure 5: We use a UMAP to visualize the embedding clusters for examples. We the UMAP to held-out. examples of the in-distribution data and also apply to the out-of-distribution data.
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Our results and findings should not be surprising given that the spurious correlations we aim to mitigate for real-world decision-making tend not to be localized. For instance, correlations between gender and occupation likely persist across 'naturally" collected datasets (leresa-Morales et al. 2022]. Hence, they may not harm performance across different data sources. We find that the subpopulation shift benchmarks we assess often have the desired properties derived in this work (Yang et al., 2023). Notably, these benchmarks are (i) constructed so that the spurious correlation from training is no longer accurate at testing and (ii) evaluated based on worst-group (worst-case) performance. Our work shows that in less explicitly defined domain generalization contexts, such types of shifts are also necessary. We provide detailed results and discussion in Appendix
Embeddings Analysis. We qualitatively verify that classifiers learn spurions correlations without targeted algorithms. We use UMAP (Uniform Manifold Approximation and Projection). a fast, scalable dimensionality reduction technique of feature representations that preserves both local and global structure of data (MeInnes iet al. 2018). Figure 3 demoustrates the concordance between our findings and the embeddings learned by classifiers. For instance, for ColoredMNIST, Waterbirds, and Spawrious, embeddings of classes are consistently clustered together in the training distribution; however, out of distribution, the same classes' embeddings are aligned with the wrong class from the in-distribution data, and would likely be labeled incorrectly, The influence of these spurious correlations leads to clustering examples with the wrong class out of distribution when the spurious correlations change. We find that these datasets also have a lower (negative) correlation between in- and out-of-distribution accuracy.
For PACS, however (Figure 5d ; the alignment. wet in- and out-of-distribution embeddings is consistent, meaning that the same classes in- and out-of-distrbution are closely embedded. Our analysis also finds that PACS has strong correlation between in- and out-of-distribution accuracy. Overall, our qualitative analysis of the embeddings learned by in-distribution optimal classifiers supports our theoretical and empirical findings.
4.2 Discussion
Domain generalization considers worst-case shifts (Arjovsky et al.] 2019: Rosenfeld et al. 2022b).
While
worst-case shifts may have a trade-off with average utility (Salaudeen & Koyejo 2024; Miller 2024 it is
crucial in population-sensitive and high-stakes applications like healthcare (Angwin et al. 2016 Chen et al. 2019 Oakden-Rayner et al. 2020). However, domain generalization benchmarks have not been rigorously assessed for their utility in such scenarios. We address this gap by deriving conditions for well-specified benchmarks, helping practitioncrs align benchmark choices with their goals. We showed that well-specified benchmarks, i.e,, those that are aligued with worst-case shifts, will not have accuracy on the line. Hence, benchmark users and curators with a goal of addressing such shifts should seek out benchmarks with weak or negative ID and OOD accuracy correlation.
We emphasize that dataset availability reflects selection bias. The frequency and importance of real- world scenarios where spurious correlations degrade OOD performance, particularly in high-stakes domains like healthcare, policing, and finance (see previously cited examples), cannot be dismissed simply because they are underrepresented in public benchmarks. In these domains, spurious correlations often stem from historical biases and can lead to harmful outcomes if left unmitigated. However, privacy constraints frequently limit the relcase of relevant datasets (Cohen & Mello 2018).
Many benchmark curators define their intended scope carefully. For example, WILDS (Koh et al. 2021 focuses common real-world shifts rather than worst-case shifts. Our findings often validate these intended scopes, yet benchmark users may not always adhere to them. To clarify benchmark suitability, we categorize benchmarks into (i) worst-case and (ii) natural shifts (Koh et al., 2021 Taori et al. 2020). We emphasize that worst-case benchmarks are particularly valuable for auditing uninterpretable classifiers, such as detecting demographic biases in classifier decisions (Ferrer et al., (2021), whereas natural shifts may suffice when prioritizing average OOD performance. Our conditions enable assessing whether spurious correlations (c.g., race in Chest X-Rays (Gichoya et al. 2022)) impact predictions.
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Importantly, many benchmarks that do not satisfy our conditions reffect other types of shifts observed in the real world, where spurious correlations, if they exist, are preserved, but some other source of shift occurs. In such contexts, non worss-otte benchmarks have significant utility. However, these benchmarks do not evaluate strong shifts in spurions correlations, such as when they are removed or inverted via interventions like pandemic or policy.
Building on our insight, for robust algorithm development, both worst-case and natural shifts should be considered to ensure broad applicability. Additionally, new spurious correlation benchmarks should undergo accuracy the line evaluations to support their reliabilit--idedti there is no positive accuracy the line.
In general, it is crucial to benchmark classifiers on both natural and worst-case benchmarks to comprehensively characterize the nobusiness spectrum. This requires identifying and calegorizing benchmarks as natural or worst-case, a framework for which this work carefully develops conditions.
Finally, domain expertise is critical in defining spurious correlations: a narrow potential distribution set may limit robustness, while an overly broad definition of potential distributions can unnecessarily reduce utility (Shen al. 2024). For instance, Chiou et al. (2024) report that training classifier on multiple recording sessions degrades OOD (new session) brain-computer interface (BCT) classification performance compared to training on single-session data.
4.3 Applying Our Findings
Constructing Benchmarks Without Positive Accuracy on the Line. Indeed, some of the datasets we study empirically that satisfy our desired conditions are primarily semisynthetic (e-g., ColoredMNISTT Spawrious, Waterbirds). In contrast, datasets such as Covid-CXR and WILDSCamelyon have in-distribution (ID) and disti--distiibutionn (OOD) splits that do not exhibit positive accuracy on the line. This suggests that careful and intentional data collection and curation are likely necessary to obtain naturally occurring datasets without positive accuracy the line.
One approach to constructing such datasets is to identify settings where a natural experiment or intervention has occurred. For example, the Covid-CXR dataset leverages the natural intervention of the pandemic and regional variations. Such interventions can remove confounds that induce spurious correlations or alter the directions of spurious correlations. Without such interventions, there may be little reason to ex- pect that spurious correlations would fail generalize in the real-world datasets that are most readily available.
Another approach is to split datasets along axes where there are strong spurious correlation effects, such as demographics in human decision-making An example of this is CivilComments, where data is split across demographic attributes that affect human-generated labels of toxicity. not toxic.
Before finalizing a benchmark, we should measure the correlation between ID and OOD perfor- mance ensure that the dataset does not exhibit strong positive accuracy on the line. This validation can help confirm that the benchmark is well-specified and truly tests robustness to spurious correlations.
Qualitatively Assessing Accuracy on the Line. For the Camelyon dataset (Section C.6 we observe a shift in correlation patterns based on classifier accuracy. Specifically, classifiers with greater than 90% accuracy exhibit a negative correlation between ID and OOD performance, whereas classifiers below this threshold show a positive correlation. Since accuracy on the line (Definition 3) is a global property, this datasct does not meet the criteria for accuracy on the line since there is strong deviation from the positive correlation in some regimes. This underscores key limitation: evaluating overall correlation may not sufficiently identify well-specified benchmarks. While this approach is robust against false negatives of misspecification, it may introduce false positives, necessitating qualitative inspection as complementary assessment lool.
On the Slope vs. the Correlation Coefficient. In this work, we focus on the correlation between ID and OOD accuracy rather than the slope between the two quantities. Recall that for a set of pairs x, y, which have Pearson R correlation of r, the slope when y is regressed is r * (ay/da)- depends
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variances dy and oz; which may may not be relevant to the spurious correlation problem. The slope depends strongly on the definition of F (Appendix [ This observation is also related to limitations in attributing the accuracy drop across distribution only to distributional systematic bias (Salaudeen & Hardt 2024).
While s is informative, its relationship with our derived conditions is not as obvious.
Reconciling Worst-Case and Average-Case Generalization. A strict worst-case approach to general- ization can conflict with developing locally (sites) beneficial classifiers, particularly in healthcare (Futoma et al. 2020 Miller 2024). However, failures even within the same site suggest this tension is unavoid- able Oaakden-Rayna et al. 2020), as spurious correlations remain brittle due to other non-local factors like temporal drilts (Ji et al. 2023) or interventions (Birkmeyer et al. 2020). Reliable worst-case robust.ness benchmarks remain essential, even with narrower scope. Still, when worst-case focus severely impacts average utility, additional evaluation alternative benchmarks is warranted. A practical approach is to narrow the classifier's deployment scope to smaller set of distributions, which can improve worst-case performance (e.g,, when the shifts in the smaller set are weaker) without sacrificing as much average utility. However, practically, maintaining reliable predictions may require scope-dependent classifier monitoring and updates.
4.4 Downstream Implications
Implications on Key Domain Generalization and Evaluation Practices. ID/OOD splits within the same dataset can exhibit varying Pearson R correlations, meaning some splits provide more reliable benchmarks than others. However, averaging over all ID/OOD splits (Gulrajani & Lopez-Paz 2020) dilutes this reliability, particularly when only one split is well-specified. The issue worsens when averaging across multiple datasets to compare domain generalization methods (Gulrajani & Lopez-Paa 2020 In subpopulation shifts, worst-group accuracy is the standard evaluation metric (Koh et al,, 2021 adopting a similar norm more generally for domain generalization improves the robustness of evaluation.
A related issue ariscs in classifier sclection via cross-validation, where selecting classifiers based on held-out accuracy, whether ID or a held-out domain, can lead to overfitting to spurious correlations specific to that set. Alternative selection criteria are implied conditional independencies (Salaudeen & Koyejo 2024), cross-risk minimization (Pezeshki et al.) 2023), and confidence-based ensemble aggregation (Chen ct al. 2023 However, classifier selection under distribution shifts remains a challenge.
Furthermore, accuracy the line can act as a pre-check to decide whether algorithmic invariance penalties are worth tuning. If there are spurious correlations between set of observed enviromments, for any single test environment, it may be the case that classifiers that ignore these spurious correlations perform worse than a classifier that utilizes them. Such settings will exhibit accuracy on the line. However, in other environments, the same spurious correlations can lead to failure. We expect such settings not to exhibit accuracy on the line.
Implications on Benchmarking Causal Representation Learning aims to uncover underlying causal structures. One approach to this is independent. causal mechanisms (Pearl 2009 Schölkopf al. 2021] which has motivated many domain generalization algorithms (Arjovsky et al. 2019 Salaudeen & Koyejo 2024 Peters et al.) 2016 Since both tasks require distinguishing stable from spurious correlations, eval evaluating domain generalization also apply to benchmarking causal representation learning. Specifically, when assessing classifiers, including disentangled causal classifiers, based on OOD accuracy, our framework helps determine when domain generalization reliably reflects success in learning causal representations, as discussed in Salandeen et al. (2024).
results
Implications on Benchmarking Algorithmic Fairness which aims to mitigate biases that cause disparate performance across demographic groups; some definitions of fairness are closely linked to domain generalization (Creager et al. 2021). Group sufficiency particularly aligns with the principle of invari- ance [Chouldechova] [017]] Liu et al.) 2019). We emphasize a straightforward but key insight: when using OOD accuracy to benchmark whether classifiers avoid relying on group information, the benchmark must.
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ensure that group information hinders out-of-distribution performance. In this case, a strong positive correla- tion between training and worst-group test accuracy suggests that group information generalizes. In contrast, a weak or negative correlation between ID and OOD accuracy is preferable.
Modern foundation classifiers are susceptible to spurious correlations (Alabdulmohsin et al. 2024 Zhu et al. 2023 Gerych al. 2024 Hamidieh et al. 2024). A Analyzing the CivilComments dataset, we find that spurious correlation shifts in language datasets exhibit similar patterns to vision datasets within our framework, showing strong positive, negative, and weak correlations between ID and OOD accuracy. Furthermore, [Saxena et al. (2024) report strong positive correlations in large language classifiers for predictive tasks (Q/A) under distribution shift. Our results highlight that the benchmark conditions we establish are also crucial for evaluating spurious correlations in foundation classifiers.
State-of-the- Art Algorithms. Many algorithms have been proposed for domain generalization Gulrajani & Lopez-Paz (2020); Koh et al. (2021): Yang al. (2023) On datasets we identify as misspecified, such as PACS, Terra Incognita, W'L.sCCamclyon, and WILDFMoW, empirical risk minimization (ERM) performs
comparably to stale-of-the-art domain generalization algorithms (Gulrajani & Lopez-Paz, 2020 Koh et. al.)
2021), particularly when controlling for factors like the use of unconstrained, large-scale pretrained classifiers
(e.g., CLIP (Radford ct al.) 2021). trained on internet-scale data rather than ImageNet).. Specific results
for WILDS datasets are available al https:/ http///.ssaaooor.ee..... wwl.dd/aaallrr..ddd///llerrrrr/ maintained by the WILDS leam.
In contrast, on datasets we identify as well-specified, several algorithms have consistently outper- formed ERM, even under independent validation. For example, YYng et al. 2023) demonstrates improvements over FRM on Waterbirds and CivilComments.
Limitations and Future Work
Yang et al. 2023) demonstrates that while accuracy on the line may hold, other metrics or a combination of metrics may not simultancously have the same strong positive linear trend. Investigating metrics other than accuracy is left for future work. Additionally, our theoretical analysis focuses on binary classification, though our empirical results include both binary and multi-class classification. We leave extending our theoretical analysis to multiclass (and multilabel) classification for future work. Furthermore, our results are only necessary and sufficient for symmetric distributions where spurious and domain general features are condit ionally independent given the label. We leave studying settings with less restrictive assumptions to future work.
Another direction for future work is to develop a more robust automated method for assessing accuracy on the line, beyond simply computing correlation across all training accuracy levels. Currently, qualitative assessment remains necessary to account for cases where classifiers with higher training accuracy exhibit negative accuracy on the line. Empirically, we found that standard change-point detection methods (Killick et al. 2012) are highly sensitive to noise in accuracy measurements. However, incorporating more robust heuristics could improve their reliability, making them a viable approach for automation.
Additionally, while we have evaluated a wide variety of classifiers in this work, continuing to collect data points of ID/OOD accuracies for benchmarks improves the accuracy of the true relationship. Finally, we leave curating additional benchmarks without accuracy on the line-from diverse, real-world scenarios with high-dinensional spurious features for future work. This work, along with others (Recht al. 2019 Taori Miller el al. 2021), has characterized this property for a variely of popular benchmarks.
et al.
2020
Broader Impact
Our work aims to help the community build evaluation tools that assess robustness to spurious correlations. By formalizing when a benchmark is well-specified for this task, we give researchers a simple test for deciding whether gains in-distribution can, or cannot, predict gains out-of-distribution. Well-specffied benchmarks will push methodological advancements toward developing classifiers that remain reliable and robust once deployed in critical real-world applications, such as in medical imaging and ecology.
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Finally, well-specified benchmarks are only step toward safe deployment. Practitioners must still test for domain-specific failure modes and monitor classifiers in production. We believe the overall impact of this work is positive: exposing benchmark misspeciiication moves the field toward classifiers that succeed for the right reasons and reduces unexpected failures in real applications.
5 Conclusion
Robust.ness to spurious correlations under worst-case distribution shifts is a critical challenge in machine learning, essential for ensuring the reliability and fairness of classifiers (Piohl et al. 2025p. In this work, we identify significant limitations in current benchmarks designed to address this problem. Specifically, many -fate-of-the-art benchmarks, which evaluate out-of-distributton (OOD) accuracy by training classifiers on in-distribution (TD) split and testing
OOD split, fail to guarantee that classifiers free of spurious correlations will transfer better. We define benchmark well-specified if such guarantee exists.
Previous work observed that many benchmarks exhibit the phenomenon of accuracy on the line, where improved ID performance directly correlates with improved out-of-distribution performance. Our theoretical findings suggest that this behavior indicates that such benchmarks are misspecified for evaluating domain generalization and emphasize the importance of prioritizing benchmarks that do not exhibit accuracy on the line when addressing worst-case distribution shifts. We aim to provide a clearer path toward developing classifiers robust to spurious correlations by addressing the evaluation ambiguity. Reliable benchmarks for robustness to spurious correlations contribute to the broader elforts to develop meaningful and valid evaluations of Al systems (Weidinger et al. 2025 Salaudeen et al.
2025
Wallach et al. 2025
Acknowledgements
OS was partly supported by the UIUC Beckman Institute Graduate Rescarch Fellowship, NSF-NRT 1735252, GEM Associate Fellowship, and the Alfred P. Sloan MPhD Program. SK acknowledges support from NSF 2046795 and 2205329, the MacArthur Foundation, Stanford HAI, and Google Inc. We thank A. Anas Chentouf, Tyler LaBonte, Vivian Nastl, Haoran Zhang, and Yibo Zhang for their comments on an earlier draft.
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Appendix Table of Contents
Proofs
28
A.1
Proof of Lemma:
DSmain-Specilie Classifiers have Lower In- Iomann Error under Partially
Informative Domain-General Features
28
A.2
Proof ol Theorem
Sulficient. Conditions for Well-Spen Specilied Domain Genera cenneraliztion Benchmark
Splits
29
3
Proof of Theorem 2
Necessary and Sufficient Conditions for Well-Specified Domain General-
ization Benchmark Splits under Distribution Symmetry
31
A.4
Lemma
4
Accuracy on the Line
32
A.5
Lemma
Tradeolf Between Accuracy on The Line and Well- Nell-Specification
A.6
Proof of Theorem
Benchmarks with Accuracy on the Line are Misspecilied Almost Everywhere.
36
A.7
Example of Shifts with Accuracy on the Line that are Well-Specified
38
B Simulation Experiment Sctup
39
B.1
ColoredMNIST Case Study
40
C Additional Results and Discussion
41
C.1 Model Training
41
CNIST
43
C33
Spawrious
46
PACS
53
C.5
Terra Incognita
56
C.6
Camelyon
Covid-CXR
61
C.8
FMoW
66
.9
Waterbirds
70
0 CivilComments
74
A
Proofs
A.1 Proof of Lemma 1
Domain--mppap Domaii-Specific Classifiers have Lower In-Domain Error under Partially Informative Domain-General Features
Assume Non-trivial and non-redundant features (Assumption and strongly convex é.
mii
[(f((x), < min [xx.))-(((((xx,)))) feFos
(13)
where F: X + R where f(x) = w Zdg
2dg + w &apu f f For f E Fag. f(x) (w) WT <dg.
Proof. From Assumption 22 non-trivial and non-redundant features, classifier that uses both domain-
general and spurious features more expressive than one that does not. Let be the Bayes optional By the
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Bayes optimality of w any achieving the same risk agrees with w almost everywhere, i.c.,
/ ({ w x x x 0,
where j denotes the Lebesgue measure on X.
Let = Zdg 2 Espu and w = wdg ① Wapu such that
w T x W.r Adg + W "'ppu
If we only consider values x where Wag ddg # O and Wspu App + O, then without loss of generality we have that
T
w x = wag adx + W Wepu Aspu + W Zdg.
Given Assumption
p (fx1 Wdgzdg 00) 0,
the risk of wag 2dg is strictly greater than that of TTTa x. Equation 13 follows from the strong convexity of the loss.
A.2 Proof of Theorem --ffffcceen Conditions for Well-Specified Domain Generalization Benchmark Splits
Assume Apm is sub-Gaussian with mean Mepu, covariance Espu, and parameter w. Define a nonlinear transformation
Φ &:R'
that is To -ippshiii and let
ZOOD o( Zppu)
Assume further that
E spu
= E. (0) Z = M pspu:
for some matrix M Rix! In-distribution, Zapu PTD and out-of- distributiin Zoon POOD- Additionally, denote Wepu the contribution of Zap to the optimal P.n predictor fRD. Then, for any 5 E (0, 1), if
/ w (M Pspp) + V2Lg 1i || 12 log/1////////// (1/5) <
(with the understanding that under the Lipschitz assumption the sub- Gaussan property carries over with parameter Ls n),, then with probability at least 1 - over Zapa we have
acc (fRD)) < accPooD (fag)
where fīg and f the optimal domain domain-general and domain predictions (Definitions
Proof. Define
Z o(Z.p)))
From Equation and the law of total probability, the out-of-distributtin (OOD) accuracy of pPID is equivalently
acCoon (PD) = Pr (w Zig + w w.pp Zepu 0)
and
T Zún accoop (f5Ø) Pr
Wig Zdg > >))
It suffices to show that
Pr Wdg Zdg > - w spu
200D
< Prr (w.TTT
[wdg Zdg > 0)
(14)
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with high probability.
Since Z is sub-Gaussian with parameter K, by the Lipschitz property of ø the random variable w Wsp D is sub-Gaussian with mean
E
W Wapp spu
Wapu
Zopar Wapu (M Mspu))
T
and aub-Gaussian parameter at most Lo (i.c., with variance proxy bounded by (Lg k:? || Thus, for any > 0,
t²
Pr
W spu
spp > I (M Mapu) + +t) S expl 2(Li x:)2 ||
Choose
= (2l2)i)) hww.s/plll/////o llo///(1/))))))
Then,
Pr W spu Zopor w (M Mapu) + +1) S 6.
Therefore, with probability at least 1- 6,
ZOOD W spu
W Wapa (M Mapu) 2L Waapl//appllll Wapu / logg/////
Assume that
T W spu (M Mapu) + VLL k6l Wapu w
log(1/5) < 0.
Then, with probability at least 1 - 6, we have
W spu Z
In this case,
Wdp ZaR
-w
Spu ssp j
C
War Zaz > =0.
which implies
Pr
Wdg 7dg > - spp 75p0n
< Pr
Wg Zdg > 0).
Equivalently,
acCooD ( = Pr w dg
Zdg + w spu 70o0 > 0)
Pr Wdg Zdg > - Wapu Zpu
T Zulg > 0)
Pr dg
acCood (fāg).
Therefore, with probability at least 1 - 8,
acCooD (f(io) < (fC
Lemma (Monotonicity of 0-1 Accuracy under Symmetry). Let Wo be continuous, nondegenerate random variable satisfying
Wo -Wo
i.e. centrally symmetric about zero, and assume its CDF Fw, is strictly increasing on with Fw, (0) 2' Fir p= == Pr(Y = 1) E (0, 1), and define the conditional scores
U (Y 0) =WWo,
U U =1) mn +WW.
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with 0 and M1 R. Let
M1 0 M1
Then the 0-1 accuracy of the Shreshold-at-zers classifier y 11U > 0} can be written
acc p Fwo p) Pr(Wo - 0) - Fw: (r) + (1 pp 2
and this accuraey is strictly increasing function of the signal -omia-tt-nooise ratio r.
Proof. We first define the accuracy decomposition. Writing = = > 0, we have
acc Pr) Pr = L,Y = 1) + Pr( = 0, = 0)
= Pr(U > Y p) Pr(U 0 = -).
Then we use change of variables using symmetry. Since U Y 1 μ4 + aWo.
Pr(U > 0|Y 1) Pr(Wo - = 1 - Fw, = (r),
and since U with Pr(Wo - 0) (0) = 1
Pr(U < Y = =0) =
Substituting into the decomposition gives
acc = p W. (r) + -
Because Fw, is strictly increasing and p > 0, G strictly increasing in r. Hence accuracy increases in the signal-to-noise ratio u1/a.
A.3 Proof of Theorem 2 Necessary and Sufficient Conditions for Well-Specified Domain Generalization Benchmark Splits under Distribution Symmetry
Assume spa is random variable with mean Mapu, covariance Expu: and Wag Zoon is symmetric about its mean. Define nonlinear transformation
Φ R - RR
that is Lo Lipschith and let
ob (Sppu)
Assume further that
E P[Zgpan = BB (Zpp) M Uspus
Var( (ar(Zgo)) = Egp and Zág
Zupa Y
for some matrix M E Rixl, and that the scalar and w T Wapu Zoop is symmetric about its mean. Then,
w Wag + w W M Mapu
W WagMdR
acCPood
(1 PiD)
acc (faz)
(15)
V WW Edg Wag + Cp
Wag Edg
Note that the RHS of Equation
15 implies that either:
Spurious Correlation Reversal: w Muspu 0 (related to sufficiency condition without the symmetry assumption in Theorem D. or
sufficiently large variance when II M > 0, i.c., small SNR.
Proof. Directly applying Lemma
2
W Wigghde + w M Mapu
w I Wag
accrood (f) < acc (fag) (ff
dg
I Edg Wag + w Eg Wapu
Wag Edg Wdg
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A.4 Lemma Accuracy the Line
Lemma 3. Assume Z is sub- Sobb-aussian with mean Mapus covariance Espus and parameter Define nonlinear mapping
Φ R - R
that is Lippiscchtt, and let
spu Φ (pp)
Assume further that and that
E| spec
Elo(Z
M Papu: and Eg 2 ( opee) T
M Hapa Mspul || S 61,
(16)
T Wapn
2g Wapu
w uspu Mapu S E2.
(17)
Moreover, assume there ewists a constant B > O such that for sufficiently small t (a Tsybakov-type condition),
(r((xx(X) St) Bt.
Then, for aray 5 0, with probability at least 1 5 over app the following holds for any classifier fx F:
(ccop(fx) accp, (fx) s Be,
where
= (uspal/E1 + +C' Cooog((16) + VEg,
and for some small constant
C = CK max {
Wsppells [| || [[ssspl
Proof. Define
= ((X)=(()-[(X3)
where P and X' Ps. respectively, (i.e. X' is obtained by replacing Zopu with Zgpar)
Since
f(X) Zdg Zdg
w T Zpp and f(X) W Zdg
W spu
Z
spu
we have
A(X) w Wepu Zpp
W Wapu Zopa
W spu
Zap
Zepu
We now decompose A(X) into a deterministic part g(X) and a stochastic part. l(X).
A(X)
w spu
B
LID
w Wap
E2
2OOD]
+
(18)
x(X)
Jp
ZID
E/w
T
sppll)
(wiju
Zepor
2OOD E(w
(19)
spu
spu
spu
A(X)
Since
EE| [ppl Uspu
and E 1 M Papu:
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have
|g(X)|
Wapulspu
T W Wapu (M (spu))
(20)
Wspn (Mupu M Mspu)]
T
(21)
Wspul I M Uspall
(22)
Wspul é1.
(23)
Next, consider the stochastie term 1(X). Since 7pp is sub- nau-Saas with parameter Ki both Wapu 7spa and W Wspu Spu are sub-Gaussian with parameters x|| Wapu and Lan [sspul respectively.
Therefore, for any t > O,
t² 2(15| Noel|Wapull Pr [w. Zasu Zap B[[wpu Zapull > 2exp(
)2
and
12
Pr [w spu ZOOD E W
ZOOD]
>1) S 2 exxp
spu
2( Wspull
Applying the union bound, with probability at least 1 - we have
W spu Zopu
E(w spu Zspul| + w Spu
E W spu 2 I CVloog(1/6)
where with small constant factor 0, Additionally, by assumption,
C' = cK max [| ||Wspull Lo ||
W Wapp (M Mapu) w
Wapu/Mpp
S |/ ci.
and
T Wapu (Eppaap)) Wapu (Espu Wapu) Se2.
Combining these bounds, with probability least 8 we obtain
A(X) < ||Wapulls + (oolool(/8) + V62
Finally, the classifier's accuracy difference is determined by the probability that f(X) and f(X') disagree in sign. Given Tsybakov-type condition, Pr(Ifx $ Bt, this probability is controlled by Bc. It follow that,
accr(f) accp, (n) s Bc.
This completes the proof.
Lemma 4. Assume Zop is sub- Gauusasssin with mean Mapus covariance Espus and parameter R. Define nonlinear mapping
Φ R - R
which is Lg Lipschitz, and let
7 spu = $ 7.
Assume further that and that
E 1000
M Mapus and Eg 7 2000 spo
T
| Mapa Mapull || < E1,
(24)
Wpp Eg Mapu T Eopu Wapu $ 622
(25)
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where the second inequality is understood to control the difference in the covariance (or concentration) of the spurious features after transformation. Moreover, assume there evists constant B >> 0 such that for sufficiently small t (a Tsybakov-typp type condition),
Pr (Xxx())) Bt.
and there erists > 0 such that
accp [a, al and a acop, (fx) E [a, - a].
Then for any 8 (0,1), with probability al least 1-d, 8,
xh (acep((x)) Φ (accpp Se,
(26)
for any classifier fx E F, where
ē= LB [spullls + CVlog(1/5) + VEZ) + t:
with for a small constant factor 0
C = CK max m Wepulls ||
Wepul }.
and
<= <= al vE(0,1-00
4 (x))
where T is the Gaussion cumulative distribution function, and L is its Lipschitz constant on [a, 1 - a), i.e, for p.q E [a, 1 1 a),
(p)(p) +b (a) s Llp g/-
Proof. By Lemma with probability at least 1 - have
(fc)((x)) a acccp, (fx) < B [sspullc1 + C Coloo((1/) + voz).
Since a/cp(fx) and a aacop, (Jx) lie in (ar, a). the function $-1 is Lipschitz on this interval with constant. L. Therefore, if p afc)(fx) and q = accp, (fx), then
④ (p) -- (g) s Llp - g)-
-1
Taking into account the scaling factor a yields
Φ (p) a - < Llp q + |1 al Φ (g)
Since 1p al is bounded by the result of Lemma
we obtain
Φ (accp(fx)) - a (accp, (fx)) < LB [Issplles
(27)
+ CV/2lo((2/)) + V(2) +11 1-al max
0-1 (z)
(28)
Defining the right-hand side as ē completes the proof.
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A.5 Lemma 5 Tradeoff Between Accuracy The Line and Well- Well-Sppeification
Lemma 5. Assume Zo is sub- Gaussiin wilh meam Mapus covariance Espu: and parumeter Define nonlinear mapping
Φ:Ru R - R
which is Lg Lipschitz, and let
7. 2000 $5 (pt
Assume further that and that
7 /pp
El(a( spp
M papus and Es = Zap (2 spu
|M Uspa Mspull S 61,
(29)
T Wapn Eeg Wapp W Wapn Expra Wapu S E2.
(30)
Fir Wapu IR that w spn 0. Suppose that M satisfies the spurious correlation reversal condition
T Sapu (M Mapu) + V 2(( Wapul logg1//)) log(110) S -7 0,
for some margin 7 0. Moreover, assume there erists a conslant B > such that for sufficiently small (a Tsybakov-type condition),
Pr(1x((X)| s Bt,
and that there exists some a > such that
accp(fx), a accp, (fx) E (ax, at.
Then, with probability at least 1 5,
② 1 (accp(((fx)) - a x (accp, (fx)) 2 C' Kapss (lo((1/)) | Mapu Mopull -5,
(31)
where
Ç <=
max x [_,1-aa
Φ (x)
and C is positive constant (depending and the local slope of T Φ- Lipschitzness of o, and concentration of Zope) ). Moreover,
| Uspu Mssull 2 [Wsppl ((7 ( Wapulspp))
that the right-hand side of 31 is strictly positive whenever y + w WapuHopp >> 0.
Proof. Since the spurious correlation reversal condition yields
Wapu (M (app) S -7 <,
and since w Wapu
0, we have
w spu (M Pepu Mspu) = W Jpn (M (spp) T Wapu/ssp S - WsmnuMpuu- T
By the Cauchy Schwarz inequality,
|Wssplll II |IM Pspu Mepull 2 W (M Uspu Mapu)) 2 7 + WapuPspur
that
7 w Wapullppu
M pspu Pspul
||
>
Wapu
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Next, note that
accp(fx) = --r(ww (Wwwr Zdg > -W "spu Sspu
and accp, (fx) = Pr
Wd Zdg > - Wspp Zapu
Since (M
(M Mapu) is very negative relative the random fluctuations of W spu (by the sub-Gaussian concentration inequality with parameter x) and since WspuHspu > O, one can apply standard concentration arguments to show that with probability least -
[fcp(fx) accp, (fx) 2 Cok ||aaplll (oo((1/))) ||
Pspu -Mspull
for some constant Co Since by assumption accp(fx) and a accep, (fx) lie in [a, a], the inverse Gaussian CDF $-1 is T--.ipschitz this interval. Thus, have
④ -- (accp((())) a a 1-'00p06 (acop, (/x)) 2 L (acop((x) aacep, (/x) -1 aal max
x x-'(x)
a aE|a,1-00
2 L (Con | log(1/8) |M Mapu Mapu ll)
11 al max # 20,11o0 o]
(z)
Defining C CoLK and -1 -1-1a maxxpea.11-a]
② (-)((x))
(x) completes the proof:
- (acccp(((xx)) - a o (accp,, (fx))
2 C W [ssppl
(lo((1/8)
|| Uspu Mspull - S-
Tt Wappll
Since
|| Papu
Mspull
the right-hand side is strictly positive whenever Y + Wapulapu 0. ]
A.6 Proof of Theorem
Benchmarks with Accuracy on the Line are Misspecified Almost Everywhere.
Assume that Zppu is sub-Gaussian with mean Mapus covariance Espu: and parameter k. Define nonlinear mapping
Φ R - RR,
which is La Lipschitz, and let.
Zapo of Sppu)
Assume further that and that
E [
B[cb( Zap = M Mapus
| Mapu Mspull E1
(32)
W spu Eg Wapu
w Wap Eepu Wspu < S2.
(33)
Suppose that M E Igxx satisfies the spurious correlation reversal condition
w Wipu (M (aap) + /2 (Ls x)² || log(1/8) O,
and assume that there exists constant B > 0 such that for sufficiently small (a Tsybakov-type condition),
Pr Pr(1fx(X)| Bt,
and there exists some such that
accp(fx), aacop, (fx) E (a: a),
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where accp, (fx) is the accuracy when the out-of-distribution features are given by o(Zspu). and & denotes the Gaussian cumulative distribution function.
Define
W Wapu (M Mapu) / ()) Wsspul2 1og(1/6)
We = M MERIX
and
(34)
Φ - (accp(((f)) - a - S
Then:
(i) Wo has Lebesgue measure zero in R/x
(ii) For any 0 S e, S Ej, we have We, C W.
In particular, 0 (i.e., perfect accuracy on the line), almost every shift misspecified, and the Lebesgue measure of the set of well specified shifts grows monotonically with e.
Proof. From Lemma have the inequality
i (acep((fx) a (accp, (fx)) c(1 (I logg(11) |M Papu Hapull) -11 al
max =E|a,1-a
+ (x)
where C' is positive constant (depending the concentration bounds), and L is the Lipschitz constant of h-1 on [ar 1 - a]. Suppose
Φ x (acep(fx)) 4 (accpp, (fx)) se,
then for sufficiently small, it must be that
+ 11 max
$ Φ-1
Mspu Mapull S
(35)
C' || |l Hollloloo//// log (1/0)
Thus, as - 0 we must have
| Mapu Mapull 0,
i...
Mapu Mapu
W Wapa (app))) w Wapu 2 0.
The second equality follows from Wapu being the optimal contribution of Z to PPD
Let
S = (M E R Ixl MPspp = Mspp)
Since pepu # 0, S is an affine subspace of Rixl with dimension strictly less than 12 and hence has Lebesgue Since
measure zero (Sard's theorem; Sard 1942)).
Wo S.
it follows that Wo has Lebesgue measure zero.
The monotonicity claim follows immediately from Equations 34 tion
if 0 < G, Cj, then by defini-
We, C Wey
This completes the proof.
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A.7 Example of Shifts with Accuracy the Line that are Well-Specified
Let Zepu E R be Gaussian with mean Mspu E Za * O, and covariance Σxpn- Fix Wspu E R with W spuu # 0 and let > O. Assume the mapping is linear, i.e. Φ(u) = M u, so that
spu
(a = M Z
Then for any e > and E (0, 1), there exists a matrix M E Rax such that:
spu Mapp + / /2 M MT Wapu log(1/6) < O,
(36)
Φ - (accp((xx) - a Φ (accPaN [[x))))x(()) S
(37)
with probability at least 1 - 5, where accp(fx) and accp (fx) are defined as in Lemma For the construction, let.
Wapn
Wspp
be the unit vector in the direction of Wapus and define its reflection matrix
2ec T.
Note that Rv = - and R² I. Choose a scalar a > 0 and define
M = aR.
Then, we compute:
Wapu (Mpspp)) W (aR(lspu) (W R(tapu) W spuu/spus
T
(M Esspu M Wspp a² w W (RE (REngu Wspp a² w spp Expp Wapu: W spu
since Rwapu wsu and orthogonal.
The spurious correlation reversal condition 336 bocomes
W spuuapu
V 2 a² wT Espu log(1/8) < O.
This can be written as In particular, since w WapuMapp > 0, it suffices to choose such that
a Wspullspu + a / WTpu Expu Wspu 1og(1/6) < 0.
/2 T spu Espu Wepa log(1/0)
2
Wapufapp
At the same time, we want the errors induced by M to be small. Define the following error terms:
E = |M Hspull = |aRRtsp Mapall,
and
6g = la² - 1| spu Espu Wspul.
We want to choose close to 1 (so that and are small) while also satisfying the above inequality, Hence, we set
2 w spu Espu Wepu log(1/8)
max +
Wapu/spu
for some small 17 chosen so that 1| is below the desired threshold. By choosing a accordingly, we ensure that:
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1. The spurious correlation reversal condition (36 holds.
2. The induced errors 61 and 62 are small enough so that, by Lemma we have
② (accp - a ((cceP)) (fx)) - B ([Wapullee + C CV/2log(2/0) + V Se.
Thus, M = aR satisfies both conditions 336 and the set of such M has Lebesgue measure zero in
37 with probability at least 1 1-8. 8. Note, however, that
B Simulation Experiment Setup
Simulation Experiments. We evaluate our results so far empirically. We define an initial distribution with Zdg E IR2 as a Gaussian with mean Y Pdg: where Adg = [1:1] and unit variance, and Zopu E R2 as a Gaussian with mean Y- /uspu; where Mapu = [1: 1] and unit variance. The input X R and label y E (0, We define domain by M where Zapu M Zpu and all other random variables' distribution is preserved. We consider settings where the training domain is (i) Gaussian and (ii) Sub-Gaussian (mixture of Gausians). We define set of 50 Gaussian test domains defined by randomly sampled M.
We train two types of classifiers: domain general, which are logistic regression classifier trained and evaluated with only Zdg features but still trained only on the training distribution, and domain specific, which are logistic regression classifiers trained an evaluated with X features but still trained only on the training distribution. Dotails on the experiments can be found in Appendix
Figure 2a demonstrates the setting where the training domain is defined by M = Tp.. i.e., Zspu is multivariate Gaussian. In this setting, observe the expected behavior derived in Theorem [ That is, when the spurious correlation reversal and controlled spurious feature variance hold out-of-distribution, the domain- domain-general general classifiers outperform the domain-specifie classifiers.
Figure 2 demonstrates the setting where the training domain is a mixture of M's, i.e., a mixture of Gaussians making a Sub-Gan Gabssaassiin distribution. Figure 2c demonstrates the setting where M is unconstrained. Here, the test domains can be written as an interpolation of the training domains, i.e., there are positive and negative definite M's mixed to create the training domain. In this sctting, Rosenfeld et al. Rosenfeld et al. (202bb) show that the training domain empirical risk minimizer solves the worst-case domain generalization problem. Indeed, figure 2cls results show that there is not generally difference in OOD performance between the domain-general and domaan--pecific classifiers.
Figure 2c demonstrates the setting where the testing domains are not a convex combination of the training domain test domains can be outside the bounds of the training M's. Here, there is a clear difference in the OOD performance between the domain-general and domain-specific classifiers. Furthermore, the expected conditions derived in Theorem [ are observed. That is, when the spurious correlation reversal and controlled spurious feature variance hold out-of-distributiin, the domain-general classifiers outperform the domain-specific classifiers.
Clearly, in natural datasets, it often impractical to conduct such experiments to determine when a domain-general classifier achieves the best transfer accuracy on a benchmark. Typically, the domain-general features are unknown, and we lack the ability manipulate natural datascts. However, we demonstrato below that the absence of a strong positive correlation between in- and out-of-distributiio accuracy for arbitrary predictors referred to accuracy the line (Miller et al 2021), Definition 6 can identify well-specified benchmarks that reliably evaluate domain generalization via transfer accuracy. We will show that well-specified domain generulization benchmarks echibit either weak in- and out-of- distrribiiiiiiiii accuracy correlation or a strong inverse correlation.
wrefictors-refereedd
Parameters. We use the following parameters across our experiments: م == [1, 1]. Edg diag([1, 1]), Pspu = [1,1] 1]. We expect our results hold independent of these parameters. We chose these parameters for the ease of intuition of the results on the simulated dataset. We sample size of 1000 for each domain.
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Algorithm 1: Generative Mechanism for ColoredMNIST
Input MNNSTT dataset with grayscale images &dg and binary labels y E {0,1} Output: ColoredMNIST dataset with colorized images a and labels y
Define color mapping probability ((Zspoly) based on chosen spurious correlation Sample y y~ P(y) from the original MNIST dataset
Sample grayscale image 2dg corresponding to y
// Introduce spurious correlation
With probability assign color Zspu based on P(zspuly)
With probability 1- p, assign color žspu randomly (breaking correlation)
Apply color transformation T(zag: Aspu) to obtain
return (z, y)
9
838
Figure 6: Colored MNIST image examples
Y
(0rn().5)
PM
7ág NY pdg; Edg)
(38)
Zepu = NY Mpag MEspu MT)
In Figure 2a We pick P₁ train domain and then randomly sample M's to construct Par's. We then train domain-specific logistic regression classifier for fx X + y and domain-general logistic regression classifier for fag: Zdg + y for Pt. We retrain each classifier
B.1 ColoredMNIST Case Study
The ColoredMNIST dataset (Arjovsky et al., 2019) intuitively illustrates the complexity of benchmarking domain generalization. ColoredMNIST modifies the gray-scale MNIST (Deng) 2012) dataset by adding color spurious correlation. The digits labels are binary with +1 when 'digit 2 5' and -1 otherwise. The observed (training) labels, however, contain 25% label noise, i.c., predictor that uses digit information can achieve 75% accuracy at most, in/out-of-distributin
Additionally, the digit images are colored. The color of the digit matches the noisy observed labels with probability Pe, inducing a spurious correlation or shorlcut of strength Pe- Pe defines a distinct distribution.
Since the observed labels are noisy versions of the true digit labels, the color potentially correlates more with the observed labels than the digit itself. For example, consider training domain where p P:(Y +1
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09
acc(f())
test acc etv p=0.75
84-
test p= 10.90
69
....9: ymx
84
y
69
77
50
env
69
66
test
50
acc(fe)
acc(f()
16-
40
50
y
50
69
50
16
50
69
84
train arc
p=0.10)
train acc p=0.10)
train p=0.10)
train
p=0.10)
(a) m = 0.79
(b) m 0.50
(c) m -0.52
(d) m -0.98
Pearson A 0.97
Pearson R = 0.97
Pearson R -0.96
Pearson R -0.98
Figure 7: Correlations between classifier performance In-Distribution vs. Out-of-Distribution ColoredM- NIST variations. is the slope of the line, and R is the Pearson correlation coefficient. The axis-parallel dashed lines denote the maximum vithin-domain accuracy of 75%, and across training and test (target) domains. Classifiers achieving above 75% accuracy use color as a prodictor. Figures 7a and 7b represent shifts where color-based predictors achieve the highest OOD accuracy above 75% accuracy. Without domain knowledge, one might conchude that the best ERM solution is the most genaiil.ennral. However, Figures 7c and 7 show that these classifiers are not domain- -enaii-general; some features that improve ID accuracy hurt OOD performance.
represents invariant performance
color = +1) = 0.9. A color-based predictor would achieve 90% accuracy in-domain, while a domain-general predictor that ignores color would achieve 75% accuracy at maximum. Furthermore, under shift where color = +1) 0.75, color-based classifier trained on p' classifier will still outperform the domain-general classifier in OOD accuracy. However, when pd P2 = color +1) 0.75, the same color-based classifier will transfer worse than the domain-general classifier. This simple example underscores that for domain generalization benchmark to be well-specified, with respect to OOD accuracy, spurious correlations from training to test domains must change enough for the domain-general classifier to achieve the highest possible OOD accuracy.
pl. = PY
Figure 7 demonstrates that domain- classifiers need not transfer the best OOD. To demonstrate this, test of classifiers on a ColoredMNIST training domain where P. 1 color green) 0.1 and across various lest domains with Pto domain domain-general general and coser-based classifier transfer gap in Figure 7 which ID-OOD shifts allow for reliable domain--enerrlizatiin evaluation.
color green) = P/. The observation of the variance in the underscores this work's key question
We leverage a ConvNet architecture for the ColoredMNIST dataset (Table 4 we vary hyperparameters
enumerated in (Gulrajani & Lopez-Paz 2020). We vary the hyperparameters in Table 2 and whether or not we use data augmentation.
C Additional Results and Discussion
C.1 Model Training
Data Augmentation. When data augmentation is applied, the transformation consists of series of preprocessing steps applied to images before they are used for training. First, the image undergoes a random resized crop to a size of 224 x 224 pixels, with a scaling factor ranging from 70% to 100% of the original size. Next, a random horizontal flip is applied to introduce variability in orientation. The transformation also includes color jittering, which adjusts brightness, contrast, saturation, and huc with a factor of 0.3 each, followed by random grayscale conversion, which randomly turns images into grayscale with certain probability.
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Table 2: Models Generation
Hyperparameter
Range
Learning Rate (Ir)
10 5 to 3.5
Weight Decay
10 6 10-2
Batch Size
2 (8) to 25.5 (=x 45)
Data Augmentation
(True, False]
Transfer Learning
{True, False]
Model Architecture
[ResNet18, ResNet50, DenseNet121, ViT-B-16, and ConvNeXt ConvNeXt_Tiny] {0.0, 0.1,0...0.5)
Dropout
Epoch
Experimental Setup. We follow the following general experimental procedure. When experiments deviate from this, is specified in their respective sections.
Each dataset consists of E domains, each corresponding to unique data distribution. Our experiments involve ID/OOD splits using leave-one-domain-ooul approach. Specifically, for each domain indexed E [1. we train on the subset Etrain = (ei....,e-D. eep.,eee... and test on the held-out domain Eiest = {e.}
For each i, we train the following classifiers Peicni; R8sNet18, ResNet50, DenseNet121, and Con- vNeXt Tiny. For each classifier, we consider ImageNet pretrained variants: (i) Finc-tuned end-to-end training pf and (ii) Transfer learning retraining only the last layer on Petrai, classifiers generated with hyperparameters in Table 2 also take classifiers at different checkpoints during training.
Table Dataset environments with minimum samples such that change in ID and OOD accuracies correlation changes less than 1% with new models, and total samples included. We do this for each dataset and ID) ID/OOD split. Our analysis accounts for sampling error and ensures this threshold is met with 95% confidence, determined by bootstrapping with 1000 resamplings.
Dataset
OOD
Minimum Samples
Total Samples
Commentsmntss
Env 0
6,710
10,350
CimilComeents
Env
7,210
10,350
CivilComments
Env
7,610
10,350
CivilComments
Env
10,310
10,350
Civiloomennss
Env
6,210
10,350
CivilComments
Fnv
9,810
10,350
CivilComments
Env
9,810
10,350
CivilCommenss
Env
10,310
10,350
CivilComments
Env
10,310
10,350
CivilComments
Env
4,910
10,350
CivilComments
Eny
5,910
10,350
CivilComments
Env 11
5,510
10,350
CivilComments
Env 12
4,810
10,350
ComilComeents
Env 13
5,610
10,350
Civil CommmCtssttss
Env 14
2,510
10,350
CivilComments
Env 15
5,910
10,350
CNISTTdMNIST
Env
14,110
19,758
CoISTedMNIST
Env
3,110
20,130
CNIST
Env
1,810
20,020
Covid-CXR
Env
3,410
7,140
Covid-CXB
Env
1,410
7,140
Covid-CXR
Env
3,310
7,140
Continued on next page
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Table continued from previous page
Dataset
OOD
Minimum Samples
Total Samples
Covid-CXS
Env
4,110
7,140
Covid-CXR
Env
4,610
7,140
PACS
Env
2,310
4,407
PACS
Env
3,110
4,761
PACS
Env
910
5,043
PACS
Env
1,210
4,806
SpawriousM22 Easy
Env
1,310
6,630
SpawriousM2M Easy
Env
3,010
6,630
SpawriousM2M Easy
Env
1,610
6,732
SpawriousM2M Hard
Env
5,610
5,916
S SpawriousM12 Hard
Env
2,910
5,916
S SpawriousM2M Hard
Env
2,910
5,916
S Spa Easy
Env
4,810
6,630
S OawriousO20 Easy
Env
5,610
6,630
S Spawrious020 Easy
Env
5,610
6,630
S SpawriousO24 Hard
Env
4,810
5,916
SpawriousO20 Hard
Env
4,310
5,916
SpawriousO20 Hard
Env
5,910
5,916
Terra Incognita
Env
910
3,186
Terra Incognita
Env
1,410
3,186
Terra Incognita
Env
1,010
3,135
Terra Incognita
Env
1,710
3,165
WILDS Camelyon
Env
1,610
8,400
WILDS Camclyon
Env
2,010
9,228
WILDS Camelyon
Env
3,110
7,408
WILDS Camelyon
Env
2,810
8,548
WILDS Camelyon
Env 4
4,710
7,480
WILDS FMoW
Env
810
27,340
WILDS FMoW
Env
2,310
27,245
WILDS FMoW
Env
710
28,725
WILDS FMoW
Env
4,110
25,550
WILDS FMoW
Env
4,310
23,930
WILDS FMoW
Env
7,010
26,540
WILDS Waterbirds
Env
2,010
8,823
WILDS Waterbirds
Env
510
9,027
C.2 ColSredMNIST
ColOredMNIST (Arjovsky et al., 2019). variant of the MNIST handwritten digit classification dataset (LeCun 1998). Domain (0.1, 0.2, 0.9) contains a disjoint set of digits colored either red or blue. The label is noisy function of the digit and color, such that color bears a correlation of d with the label and the digit bears correlation of 0.75 with the label. This datasct contains 70,000 examples of dimension (2, 28, 28) and classes.
Experimental Details. We leverage ConyNet architecture for the ColoredMNIST dataset. (Table 4p.
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Table 4: MNIST ConvNet architecture.
#
Layer
Conv2D (in=d, out=64)
ReLU
3
GroupNorm (groups=8)
4
Conv2D (in=64, out==128, stride=2)
5
ReLU
6
GroupNorm (groups=8) Conv2D (in=128,, out=128)
7
8
ReLU
9
GroupNorm (groups-8)
10
Conv2D (in=128,, out=128)
11
ReLU
12
GroupNorm (8 groups)
13
Global average-pooling
Discussion. Despite colored MNIST's apparent simplicity, the spurious correlation between color and the label is quite strong- particularyyy generalization to text environment 2, going from domains with spurious correlation probability of 0.1, 0.2 0.9. in Gulrajani & Lopez-Pav) (2020) evaluation of standard domain generalization methods at the time, they found that no model could mitigate the effect of this spurious correlation. We note that this ID/OOD split has a strong accuracy the inverse line. In test environment 1, observe that the training distributions are such that the spurious correlat cancel out (0.1 vs. 0.9), and the domain-general model is also the best ID empirical risk minimizer.
Knowledge of the spurious correlation mechanism in each domain makes relatively easy to identify the type features a model uses due to the predictability of expected accuracy between models that use color and those that don't. Due the potential ambiguity of benchmarking results when spurious correlation mechanisms are unknown, semisynthetic benchmarks are vital in the evaluation process.
Table 5: ColoredMNIST ID vs. OOD properties.
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
1.90
-0.58
0.82
0.00
0.01
Env 1 acc
0.96
0.01
0.94
0.00
0.00
Env 2 acc
-1.56
0.47
-0.74
0.00
0.01
Table 6: MolOTedMNIST ID vs. OOD properties.
OOD
ID
slope
intercept
Pearson R
p-value
standard crror
Env 0 acc
Env acc
1.23
-0.12
0.99
0.00
0.00
Env 0 acc
Fav 2 acc
-0.73
0.87
-0.36
0.00
0.02
Env 1 acc
Env 0 acc
0.91
0.04
0.98
0.00
0.00
Env 1 acc
Env 2 acc
0.67
0.16
0.72
0.00
0.01
Env 2 acc
Env 0
acc
-1.34
0.53
-0.82
0.00
0.01
Env 2 acc
Env acc
-1.65
0.29
-0.64
0.00
0.02
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70
R=0.82
80
R=0.94
60
Rampy 00
70
40
30
Test Env Acc
20
Test Env
10
50
30
10
20
30
40
50
60
7075
20
30
40
50
60
70
80
Average Train Envs Acc
Average Train Envs Acc
R=-0.74
Test Test Env 2 Acc
85
70
40
30
20
10
1
1
20
30
50
70 80 85
Average Train Envs Acc
Figure 8: ColoredMNIST: Average train Env Accuracy VS. Test Env Accuracy.
89
Env Acc. 0=36.36
80
overall. R=0.42
6P
Env Acc
60
60
Test Env
40
Eny Acc. R=0.98
30
Env 2 Acc. R=0.72
10
Mg
overall. 0=0.85
y=x
10
20
30
40
50
60
70
80
90
10
20
30
40
50
60
70
80
89
Train Envs Accs
Train Envs Accs
Env 0 Acc. R=-0.82
Env 1 Acc. R=
overall. R=-0.44
Test T Enw Acc
90
y=x
80
70
60
30
20
10
10
20
30
40
50
60
70
80
90
Train Envs Accs
Figure 9: ColoredMNIST: Train Env Accuracy vs. Test Env Accuracy.
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C.3 Spawrious
Spawrious (Lynch et al., 2023). The Spawrious image classification benchmark suite consists of six different datasets, including one-Lo-one (O20) spurious correlations, where a single spurious altribute correlates with a binary label, and many-to-mane (M2M) spurious correlations across multiple classes and spurious attributes. Each benchmark task proposed with three difficulty levels: Easy, Medium, and Hard. The dataset contains images of four dog breeds (bulldog, dachssund, labrador, corgi) found in six backgrounds b (beach, desert, dirt, jungle, mountain, sand]. Images generated using text-to-image models and filtered using image-to-text model for quality control. This benchmark suite consists of 152,064 images of dimensions (3, 224, 224).
For the 020 task, the class (dog breed) and background combinations are sampled such that 4% of the images per class contain a spurious background 6" and (100 - 4)% contain a generic background 60e, While the generic background is held constant for each class, each spurious background is observed in only one class (Ptrain (b," (y) if = j and 0 if * j). Two separate training domains are defined by varying the value of μ.. These induced spurious correlations pair for each class (Prest(b: c:) 1).
reverted to yicld test domain with an unseen class-background
For the M2M task, disjoint class and background groups are constructed B1, B2, C1, C2, each with two elements. To introduce the training domains, class- backgroukd combinations (c, b) are selected with c E C, and b E B,. Each training domain consists of a single background per class such that Pirain (bp (x) = e, with domain index E {0, 1}. by E B,, C E C,, In contrast, the test domain is generated by selecting combinations from C. and b Bj with O& E C, {bi,
and sampling backgrounds such that Ptest(b1 (R) Ptest (&) 0.5 for
By.
The difficulty level (Easy, Medium, Hard) differs due to the splits in the available class- background combina- tions. These splits were empirically determined, and the full details of the final data combinations are found in Table 2 Lynch et al. (2023).
Discussion. We observed that test environments 1 and 2 have a strong correlation between ID and OOD accuracy and have a slope of 1, making them misspecified for benchmarking domain generalization. Appropriately, [ynch et al. (2023) propose transferring test environment '0' the spurious correlation task. The correlation for test environment 0 is much weaker than the others, indicating that ID improvement does not as strongly imply OOD improvement. While there is still a positive linear correlation, the interpretation of these benchmarking results is informative because of the knowledge of the spurious correlation mechanism. Lynch et al. (2023) give examples of informative analysis of benchmarking results on this dataset. Notably, the O20 easy setting has a weaker correlation by desigu, and the accuracy on the line strength increases. We see similar behavior for the M2M setting. However, this task is much harder than the 020 task, which is reflected in weaker accuracy on the line.
C.3.1 SpawriousO20 Easy
Table 7: SpawriousO20 easy ID vs. OOD properties.
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
0.48
-0.29
0.74
0.00
0.04
Env 1 acc
1.05
-0.13
0.98
0.00
0.02
Env 2 acc
0.95
-0.11
0.97
0.00
0.02
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Table 8: SpawriousO20 easy ID vs. OOD properties.
OOD
ID
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
Env acc
0.50
-0.33
0.75
0.00
0.04
Env 0
Env 2 ace
0.47
-0.23
0.72
0.00
0.04
Env 1 acc
Env 0 acc
1.09
-0.33
0.93
0.00
0.04
Env 1 acc
Env 2 acc
1.01
0.11
0.98
0.00
0.02
Env 2 acc
Env 0 acc
0.93
-0.12
0.94
0.00
0.03
Env 2 acc
Env acc
0.94
-0.02
0.98
0.00
0.01
99
99
R=0.74
R=0.98
Test Env 1 Acc
90
Test 0
80
70
70
60
60
50
50
40
40
30
30
20
Average
ACC
Average Train Envs
99
R=0.97
2 Acc
90
80
70
60
50
40
30
20
20
30
40
50
90
99
Average Train Envs Acc
Figure 10: SpawriousO20 easy: Average train Eny Accuracy vs. Test Eny Accuracy.
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99
Env
99
Env Acc. R=0.93
Env Acc. R=0.72
Env 2 Acc. R=0,98
overall. R=0.73
overall. R=0.89
Test Env Acc
Test Env Acc
90
y=x
90
y=x
80
70
60
50
50
-0
30
20
20
30
40
50
60
70
80
90
99
30
40
50
60
70
80
90
99
Train Envs Accs
Train Envs Accs
R=0.94
Env 1 Acc. R=0.98
overall. R=0.95
Test Enw 2 Acc
90
y=x
80
70
60
50
40
30
30
40
50
60
70
80
90
99
Train Envs Accs
Figure 11: SpawriousO20 easy: Train Env Accuracy Test Eay Accuracy.
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C.3.2 SpawriousO20 Hard
Table 9: SpawriousO20 hard ID vs. OOD properties.
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
0.32
-0.21
0.50
0.00
0.05
Env 1 acc
0.98
0.06
0.96
0.00
0.02
Env 2 acc
0.94
-0.07
0.96
0.00
0.02
Table 10: SpawriousO20 hard ID vs. OOD properties.
OOD
ID
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
Env 1 acc
0.32
-0.23
0.49
0.00
0.05
Env 0 acc
Env 2 acc
0.32
-0.19
0.50
0.00
0.04
Env 1 acc
Env 0 acc
0.90
0.14
0.89
0.00
0.04
Env 1 acc
Env 2 acc
1.01
0.12
0.97
0.00
0.02
Env 2 acc
Env acc
0.92
-0.05
0.93
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Figure 12: Spawrious hard: Average train Env Accuracy Test Env Accuracy.
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99
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Figure 13: SpawrionsO20 hard: Train Env Accuracy vs. Test Env Accuracy.
C.3.3
SpawriousM2M Easy
Table 11: SpawriousM2M easy ID vs. OOD properties.
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
0.34
0.26
0.60
0.00
0.01
Env 1 acc
0.65
-0.08
0.95
0.00
0.00
Env 2 acc
0.65
0.02
0.93
0.00
0.00
Table 12: SpawriousM2M easy ID vs. OOD properties.
OOD
ID
slope
intercopt
Pearson R
p-value
standard crror
Env 0 acc
Env acc
0.35
0.23
0.61
0.00
0.01
Env 0 acc
Fnv 2 acc
0.32
0.29
0.58
0.00
0.01
Env 1 acc
Env 0 acc
0.64
-0.09
0.95
0.00
0.00
Env 1 acc
Env 2 acc
0.63
-0.05
0.94
0.00
0.00
Env 2 acc
Env 0 acc
0.67
-0.02
0.94
0.00
0.00
Env 2 acc
Env acc
0.61
0.08
0.90
0.00
0.01
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Published in Transactions on Machine Learning Research (08/2025)
99
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Figure 14: SpawriousO20 easy: Average train Env Accuracy Test Eay Accuracy.
99
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Env Acc. R=0.58
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overall. R=0.60
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Figure 15: SpawrionsO20 easy: Train Env Accuracy vs. Test Env Accuracy.
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C.3.4
SpawriousM12 Hard
Table 13: SpawriousM2M hard ID vs. OOD properties.
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
0.16
-0.04
0.29
0.00
0.01
Env 1 acc
0.76
-0.26
0.94
0.00
0.01
Env 2 acc
0.66
-0.10
0.91
0.00
0.01
Table 14: SpawriousM2M hard ID vs. OOD properties.
OOD
ID
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
Env 1 acc
0.17
-0.07
0.33
0.00
0.01
Env 0 acc
Env 2 acc
0.14
-0.02
0.27
0.00
0.01
Env 1 acc
Env 0 acc
0.78
-0.27
0.95
0.00
0.00
Env 1 acc
Env 2 acc
0.73
-0.23
0.92
0.00
0.01
Env 2 acc
Env acc
0.68
-0.11
0.92
0.00
0.01
Env 2 acc
Env acc
0.62
-0.08
0.89
0.00
0.01
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Figure 16: SpawriousMM2 Hard: Average train Eny Accuracy Test Env Accuracy,
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99
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Env Acc. R=0.27
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Train Envs Aces
Figure 17: SpawriousM2M Hard: Train Env Accuracy vs. Test Env Accuracy.
C.4 PACS
PACS (Li al., 2017). dataset comprised of four domains d E {arl, parrtoonssphooos sketches). This dataset contains 9,991 examples of dimension (3, 224, 224) and classes.
Discussion. In general, find that PACS does not st.onggyy represent worst-case shifts for any split. Our results suggest that this benchmark may not accurately benchmark an algorithm's ability to give models free of spurious correlations.
Table 15: PACS ID vs. OOD properties.
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
0.74
-0.31
0.98
0.00
0.00
Env 1 acc
0.68
-0.68
0.84
0.00
0.01
Env 2 acc
1.00
0.32
0.86
0.00
0.01
Env 3 acc
0.76
-0.87
0.86
0.00
0.01
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Table 16: PACS ID vs, OOD properties.
OOD
ID
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
Env acc
0.71
-0.10
0.96
0.00
0.01
Env 0 ace
Env 2 acc
0.64
-0.47
0.91
0.00
0.01
Env 0 acc
Env 3 ace
0.64
0.14
0.90
0.00
0.01
Env 1 acc
Env 0 acc
0.75
-0.59
0.89
0.00
0.01
Env 1 acc
Env 2 acc
0.51
-0.67
0.71
0.00
0.01
Env 1 acc
Env 3 acc
0.71
-0.29
0.98
0.00
0.00
Env 2 acc
Env 0 acc
1.04
0.23
0.87
0.00
0.01
Env 2 acc
Env acc
0.90
0.45
0.82
0.00
0.02
Env 2 acc
Env acc
0.78
0.76
0.75
0.00
0.02
Env 3 acc
Env 0 acc
0.76
-0.79
0.83
0.00
0.01
Env 3 acc
Env 1 acc
0.80
-0.75
0.92
0.00
0.01
Env 3 acc
Env 2 acc
0.50
-0.83
0.64
0.00
0.02
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Average Train Envs Acc
Figure 18: PACS: Average train Env Accuracy vs. Test Env Accuracy.
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99
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Env 2 Acc. R=0.71
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80
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Train Envs Accs
Train Envs Accs
Figure 19: PACS: Train Eny Accuracy vs. Test Eny Accuracy.
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C.5 Terra Incognita
Terra Incognita (Beery et al., 2018). A datasct that contains photographs of wild animals taken by camera traps at locations d E [L100, L38, L43, L46]. This dalasel contains 24,788 examples of dimensions (3, 224, 224) and 10 classes: Bird, Bobcat, Cat, Coyote, Dog, Empty, Fox, Horse, Mouse, Opossum, Rabbit, Raccoon, Rat, Skunk, Squirrel, Weasel.
Discussion. In general, we find that Terra Incognita does not strongly represent worst-case shifts for any split. Ahuja al. (2021) consider Terra Incognita domain-general features to be fully informative, i.c., labels did not need to rely spurious features such as the background to generale labels. Our results suggest that this benchmark may not accurately benchmark an algorithm's 's ability to give models free of spurious correlations. For Env 1, we observe that the slope of the line varies quite bit. Particularly, there is a near-zero slope for models greater than 80% accuracy.
90
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Average Train Envs Acc
Average Train Envs Acc
Figure 20: Terra Incognita: Average train Env Accuracy vs. Test Env Accuracy.
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90
Env Acc. R.0.88
Env Acc. 0.72
3
Env Acc. R=0.91
80
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Env 3 Acc. Rm0.74
60
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overall, R- 0.0.68
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Figure 21: Terra Incognita: Train Env Accuracy vs. Test Env Accuracy.
C.6 Camelyon
Camelyon (Bandi et al., 2018 Koh et al., 2021). A dataset that contains histopathological images of lymph node tissue, collected from hospitals, denoted as Hospital A, Hospital B. This dataset contains 227,680 examples of dimension (3, 96, 96) and 2 classes (tumor, non-tumor).
Discussion. We find that overall, there is strong correlation between ID and OOD accuracy. However, observe that for some ID/OOD splits, regime of training accuracy has negative correlation (environments and 2), suggesting that within a certain accuracy range, these splits may be well-specified for benchmarking spurious correlations for models in the regime with negative correlation. This highlights the importance of qualitative evaluation as opposed to quantitative evaluation.
Table 17: WILDSCamelyon ID vs. OOD propertics.
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
0.78
0.33
0.90
0.00
0.01
Env I acc
0.71
-0.00
0.88
0.00
0.01
Env 2 acc
0.62
0.49
0.78
0,00
0.01
Env 3 acc
0.63
0.49
0.88
0.00
0.01
Env 4 acc
0.63
0.40
0.78
0.00
0.01
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Table 18: WILDSCCmmlyon ID vs. OOD propertics.
OOD
ID
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
Env acc
0.73
0.44
0.88
0.00
0.01
Env 0 acc
Env 2 acc
0.79
0.25
0.90
0.00
0.01
Env 0 acc
Env 3 acc
0.83
0.17
0.90
0.00
0.01
Env 0 ace
Env acc
0.74
0.28
0.91
0.00
0.01
Env 1 acc
Env acc
0.71
-0.00
0.89
0.00
0.01
Env acc
Env 2 acc
0.69
0.02
0.85
0.00
0.01
Env 1 acc
Env 3 acc
0.74
-0.05
0.89
0.00
0.01
Env 1 acc
Env 4 acc
0.69
-0.03
0.89
0.00
0.01
Env 2
acc
Env 0 acc
0.64
0.41
0.81
0.00
0.01
Env 2 acc
Fnv acc
0.59
0.58
0.74
0.00
0.01
Env 2 acc
Env 3 acc
0.67
0.37
0.79
0.00
0.01
Env 2 acc
Env 4 acc
0.61
0.44
0.81
0.00
0.01
Env 3 acc
Env acc
0.66
0.41
0.90
0.00
0.01
Env 3 acc
Env 1 acc
0.57
0.64
0.84
0.00
0.01
Env 3 acc
Env 2 acc
0.63
0.47
0.87
0.00
0.01
Env 3 acc
Env 4 acc
0.61
0.45
0.88
0.00
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Env 4 acc
Env 0 acc
0.63
0.35
0.79
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Env 4 acc
Env 1 acc
0.60
0.49
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Env acc
0.62
0.39
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Env 4 acc
Env 3 acc
0.67
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98
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Figure 22: Camelyon: Average train Env Accuracy vs. Test Env Accuracy.
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Env Acc. R=0.88
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Figure 23: Camelyon: Train Env Accuracy vs. Test Eny Accuracy.
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C.7 Covid-CXR
Covid-CXR AllzatoGrrsslle et al., 2022; Cohon et al., 2020b: Tabik et al., 2020 Tahir et al., 2021; Suwalska et al. 2023). datasel that aggregates live different benchmark Covid-19 datasets, represented the domain E {Cov-Caldas CaaaddasCollaaaa (Columbia), Covid- Serrdd-heet--.. -ayy (Global), Covid-GR, (Spain), Covid-Qu-Eix (Global), and PolCovid (Poland)}. These open-source Covid-19 datasets are collected from around the globe, accessible online. We are provided with chest X-ray image (x) and binary diagnosis (y). The objective is to maintain consistent, performance across different datasets d, which may incorporate data from singular or multiple sources.
Dataset
# Train
# Test
% Pos Train
% Neg Train % Pos Test
% Neg Test
Cov-Caldas
3247
688
0.566
0.434
0.565
0.435
Covid-Chest-X.. X--ay
625
157
0.376
0.624
0.376
0.624
Covid-GR
681
171
0.501
0.499
0.497
0.503
Dovid-Qu-Bll
21715
6788
0.353
0.647
0.353
0.647
PolCovid
4343
450
0.249
0.751
0.333
0.667
Table 19: Composition for Covid-19 Datasets.
Cov-Caldas (Columbia, Alzate- Grisales et al. 2022 This dataset was sourced from single institution, S.E.S. Hospital Universitario de Caldas, located in the State of Caldas, Colombia. Labels were assigned based on positive results from conventional laboratory tests, such as PCR.
Covid-Chest-X-Ray (Global, Cohen et al. 2020b ): This dataset, compiled from web sources, publications, and volunteer contributions, includes data on five types pneumonia and Covid-19, along with metadata such sex, age, and symptoms. The images are sourced from medical websites and are part of an open-source public project, where contributors can submit pull requests add new images. The data is compiled from 138 unique locations.
Covid-GR (Spain, Tabik et al. (2020)): In collaboration with four expert radiologists from Hospital Universitario Clínico San Cecilio in Granada, Spain, the authors developed protocol for selecting and annotating chest X-ray (CXR) images for the dataset. A CXR image is labeled as Covid-19 positive both the RT-PCR test and the radiologist's 's assessment confirm the diagnosis within hours.
Covid- Covid-Qu-Ex Qu-Ex (Global, Tahir et al. (2021)) This dataset aggregates chest X-rays from six subdatasets, including the Covid-19 CXR dataset, RSNA CXR dataset (non-Covid infections and normal CXRs), dataset, PadChest dataset, Montgomery and Shenzhen CXR lung mask datasets, and Ta CXR infection mask dataset. Designed to serve as benchmark, it combines multiple publicly available datasets and repositories, which were originally dispersed and formatted differently. The authors performed quality control to ensure consistency. This datasct, specifically portions of their cited Covid-19 CXR datasel, overlaps with Covid-Chhet-X--Ray (Env 1), thus deviating slightly from our standard experimental procedure of
PolCovid (Poland, Suwalska et al. (2023)): ): Chest X-rays were collected from 15 Polish hospitals using a variety of devices and parameters due to differences in equipment between medical centers. The dataset includes patients with Covid-19, pneumonia, and healthy controls.
Using the DomainBed suite by Gulrajani & Lopez-Paz 2020), we employ the ResNet-50 architecture (with and without AugMix data augmentation), along with ResNet-18. 18, DenseNet- 121, and ConvNeXt-Ting, on the Covid-CXR datasct. Each of the five datasets treated as distinct domain, d. For each model, we apply two pretrained variants using ImageNet: (i) Fine-tuned and (ii) Transfer learning.
Experimental Details.
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Published in Transactions on Machine Learning Research (08/2025)
Discussion. Covid-CXR is real-world medical dataset containing chest X-ray images of Covid-19 patients, where spurious correlations emerge naturally due to the complex and multifaceted nature of medical data. These correlations are not explicitly observed or intentionally introduced but arise from underlying factors such as imaging artifacts, patient demographics, or co-occurring medical conditions that are often unmeasured or unaccounted for in the dataset. As observed, there exist strong, weak, and inverse correlations between ID and OOD performance. While environments 0, 2, and are primarily positively correlated OOD performance, environments and demonstrate the inverse accuracy-on-thee-ine phenomenon. Additionally, several observations show weak correlations, with slopes close to zero. Existing literature suggests that horizontal line (i.e. slope of 0) is indicative of severe distribution shift, where it prevents any meaningful transfer learning between the training data and the OOD data [Tnnyy et al. (2024). This could be attributed to the significantly more severe distribution shifts present in this dalaset. These shifts, as discussed by Cohen jtt al. (2000a), may arise due to errors in labeling, discrepancies between institutions and radiologists, biases in clinical practices, and interobserver variability.
Moreover, Covid-Qu-Ex (Env 3), the largest and most diverse datasel of those explored in this work, demonstrates low OOD transfer accuracy. For OOD performance for ID Env 3, we observe slopes closest to 0, indicating weak or near-zero correlation. The high ID accuracy (up to ~ 99%) suggests that the model may be learning misleading features in OOD domains. However, for environments evaluated on this dataset, a strongly negative relationship is present, suggesting that improved ID) performance may be associated with reduced OOD accuracy. In this dataset, the authors scale by concatenating existing chest X-Ray datasets. But, discussed in Cohen et al. (2020a), Shen al. (2024), and Teney al. (2024), simply increasing the amount of data may not address the core issue of distribution shift,, more data could exacerbate overfitting
to domain-specific artifacts noise, rather than improving gencralization. Shhn et al. (2024) coins this
the Data Addition Dilemmaa, where adding data can both improve and worsen performance.
Table 20: CXR ID OOD properties.
OOD
slope
intereept
Pearson R
p-value
standard error
Env 0 acc
0.54
-0.27
0.55
0.00
0.02
Eny 1 acc
-0.38
0.13
-0.50
0.00
0.02
Env 2 acc
0.44
0.05
0.54
0.00
0.02
Env 3 acc
-0.60
0.56
-0.48
0.00
0.03
Env 4 acc
0.53
-0.04
0.31
0.00
0.04
62

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Published in Transactions on Machine Learning Research (08/2025)
Table 21: CXR ID vs. OOD properties.
OOD
ID
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
Env acc
0.56
-0.23
0.57
0.00
0.02
Env 0 acc
Env 2 ace
0.31
-0.21
0.41
0.00
0.02
Env 0 acc
Env 3 acc
0.16
-0.26
0.84
0.00
0.00
Env 0 acc
Env acc
0.43
-0.39
0.68
0.00
0.01
Env 1 acc
Env ace
-0.43
0.09
-0.61
0.00
0.01
Env acc
Env 2 acc
-0.16
0.06
-0.27
0.00
0.01
Env 1 acc
Env 3 acc
-0.07
0.06
-0.51
0.00
0.00
Env 1 acc
Env 4 acc
-0.21
0.12
-0.46
0.00
0.01
Env 2
acc
Env 0 acc
0.39
0.07
0.54
0.00
0.01
Env 2 acc
Fnv acc
0.36
0.06
0.46
0.00
0.02
Env 2 acc
Env 3 acc
0.09
0.06
0.61
0.00
0.00
Env 2 acc
Env 4 acc 0.28
-0.04
0.51
0.00
0.01
Env 3 acc
Env acc
-0.68
0.54
-0.49
0.00
0.03
Env 3 acc
Env 1 acc
-0.47
0.51
-0.46
0.00
0.02
Env 3 acc
Env 2 acc
-0.38
0.51
-0.37
0.00
0.02
Env 3 acc
Env 4 acc
-0.26
0.54
-0.29
0.00
0.02
Env 4 acc
Env 0
acc
-0.03
0.19
-0.02
0.48
0.04
Env 4 acc
Env 1 acc
0.56
-0.01
0.40
0.00
0.03
Env 4 acc
Env 2 acc
0.47
-0.07
0.37
0.00
0.03
Env 4 acc
Env 3 acc
0.10
0.04
0.36
0.00
0.01
63

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Published in Transactions on Machine Learning Research (08/2025)
75
74
R=0.55
70
R=-0,50
70
Test Env 1 Acc
Test 0
8'
01
50
40
30
Average Train Envs Acc
Average Train Envs Acc
70
R=0.54
R=-0.48
60
70
Test TestEnu Env 3 Acc
Test Env 2
60
510
9
50
30
30
30
40
50
60
70
40
50
60
70
73
Average Train Envs Acc
Average Train Envs Acc
74
R=0.31
70
60
Test Env
50
40
30
20
Average Envs Acc
Figure 24: Covid-CXR: Average train Env Accuracy Test Env Accuracy,
64

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Published in Transactions on Machine Learning Research (08/2025)
Env Acc. R=0.57
99
Env 0 Acc. R=-0.61
98
Env Acc. R=0.41
Env 2 Acc. R=20.27
R=0.84
0=0.51
90
Env Acc. R=0.68
Env Acc
Env 4 Acc. 0=0.46
overall. R=0.39
overall. R=-0,25
Test Env
80
80
y=x
70
60
60
50
40
30
30
40
50
60
70
80
98
60
90
Train Envs Accs
Train Envs Accs
99
Env Acc. R=0.54
Env Acc. R=0.46
Env Acc. R=0.61
Test Env 3 Acc
90
Env Acc. R=0.51
70
Tett Env 2
overall. 0=0.30
80
y
60
Env 0 Acc. R--0.49
Env 1 Acc. R--0.46
50
Env 2 Acc. R=.0.37
50
40
Env Acc. R=-0.29
40
overall, R=-0.25 o.
y-x
90
99
50 Train Aces
80
Train Envs Accs
99
Env 0 Acc. R=10.017
R=0.4
Env 2 Acc. R=0.37
Acc
90
Env 3 Acc. R=0.36
overall. R=0.15
80
y=x
70
60
50
40
30
30
40
50
60
70
80
90
99
Train Envs Accs
Figure 25: Covid-CXR, Train Env Accuracy vs. Test Eny Accuracy.
65

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Published in Transactions on Machine Learning Research (08/2025)
C.8 FMoW
FMoW (Bandi et al.; 2018; Koh et al., 2021). A dataset consisting of 141,696 RGB satellite images from 2022 2017 (resized to 224 x 224 pixels), where the label is one 62 building or land use categories. This dataset simultaneously considers a domain generalization task, where two domains are defined by the year of image acquisition & (be foro200016, after2016), and subpopulation shift task, where the domains are denoted by the geographic region of the image {Africa, Americas, Oceana, Asia, Europe}.
Discussion.
We find that WILDFMoW has accuracy the line for all splits, suggesting that this benchmark may be misspecilied for benchnarking spurious correlations.
Table 22: WILDSFMoW ID
OOD properties.
OOD
slope
intercept
Pearson R
p-valuc
standard error
Env 0 acc
0.75
-0.62
0.98
0.00
0.00
Env 1 acc
0.78
-0.34
0.96
0.00
0.00
Env 2 acc
0.65
-0.48
0.94
0.00
0.00
Env 3 acc
0.83
-0.34
0.99
0.00
0.00
Env 4 acc
0.96
-0.18
0.99
0.00
0.00
Env 5 acc
0.76
-0.61
0.87
0.00
0.01
66

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Published in Transactions on Machine Learning Research (08/2025)
Table 23: WILDSFMoW ID vs. OOD properties.
OOD
ID
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
Env 1 acc
0.78
-0.52
0.98
0.00
0.00
Env 0 acc
Env 2 acc
0.69
-0.72
0.97
0.00
0.00
Env 0 acc
Env 3 ace
0.71
-0.52
0.97
0.00
0.00
Env 0 ace
Env 4 acc
0.48
-0.89
0.96
0.00
0.00
Env 0 acc
Env 5 ace
0.43
-1.11
0.94
0.00
0.00
Env 1
Env 0 acc
0.75
-0.66
0.97
0.00
0.00
Env 1 acc
Env 2 acc
0.78
-0.43
0.96
0.00
0.00
Env 1 acc
Env 3 acc
0.80
-0.20
0.98
0.00
0.00
Env acc
Env acc
0.53
-0.22
0.95
0.00
0.00
Env 1 acc
Env acc
0.49
-0.88
0.94
0.00
0.00
Env 2 acc
Env acc
0.58
-0.50
0.94
0.00
0.00
Env 2 acc
Env 1 acc
0.70
-0.47
0.93
0.00
0.00
Env 2 acc
Env 3 acc
0.64
-0.99
0.92
0.00
0.00
Env 2 acc
Env 4 acc
0.43
-0.80
0.91
0.00
0.00
Env 2 acc
Fnv 5 acc
0.39
-1.01
0.92
0.00
0.00
Env 3 acc
Env 0 acc
0.74
-0.30
0.98
0.00
0.00
Env 3 acc
Env acc
0.91
-0.25
0.99
0.00
0.00
Env 3 acc
Env acc
0.79
-0.49
0.97
0.00
0.00
Env 3 acc
Env 4 acc
0.53
-0.67
0.96
0.00
0.00
Env 3 acc
Env 5 acc
0.49
-0.96
0.92
0.00
0.00
Env 4 acc
Env 0 acc
0.89
-0.13
0.98
0.00
0.00
Env 4 acc
Env acc
1.02
-0.08
0.99
0.00
0.00
Env 4 acc
Env 2 acc
0.92
-0.33
0.98
0.00
0.00
Env 4 acc
Env 3 acc
0.95
-0.08
0.99
0.00
0.00
Env 4 acc
Env acc
0.54
-0.87
0.90
0.00
0.00
Env 5 acc
Env acc
0.74
-0.56
0.89
0.00
0.01
Env 5 acc
Env acc
0.82
-0.54
0.85
0.00
0.01
Env 5 acc
Env 2 acc
0.70
-0.74
0.84
0.00
0.01
Env 5 acc
Env 3 acc
0.74
-0.55
0.85
0.00
0.01
Env 5 acc
Env 4 acc
0.52
-0.90
0.86
0.00
0.00
67

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Published in Transactions on Machine Learning Research (08/2025)
59
50
50
R=0.96
40
Test Env ACC
30
Test Env Acc
30
Poods
1
10
20
30
40
50
59
1
10
20
30
40
50
59
Average Train Envs Acc
Average Train Envs Acc
30
R=0.99
40
40
Test Env ACC
30
Test Env 3 Acc
30
20
10
20
30
40
5055
1
10
20
30
40
50
58
Average Train Envs Acc
Average Train Envs Acc
58
50
R=0.99
40
30
Test Env Acc
20
10
1
1
10
20
30
40
50
58
Average Train Envs Acc
Figure 26: FMoW: Average train Env Accuracy vs.
Test Env Accuracy.
68

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Published in Transactions on Machine Learning Research (08/2025)
90
Env Acc. R=0.98
90
Env 0 Acc. R=0.97
80
Env Acc. R=0.97
80
Env 2 Acc. R=0.96
R=0.98
50
60
,
Env Acc. R=0.96
Test Env 1 Acc
50
60
Env 4 Acc. R=0.95
Env Acc. R=0.94
40
Env 5 Acc. R=0.94
Test Env
30
overall. R=0.80
30
overall, R=0.78
20
y=x
20
y=x
10
10
20 30
40
90
10 Train Envs Accs
60
70
80
Train Envs Accs
90
Eny Acc. R=0.94
Env O Acc. R=0.98
80
Eny Acc. R=0.93
80
Env Acc. R=0.99
70
Env Acc. R=0.92
70
Env Acc. R-0.97
60
Env Acc. R=0.91
Env 4 Acc. R-0.96
40
Env Acc. R=0.92
40
Env 5 Acc. R-0.92
30
overall. R-0.74
overall. R=0.79
Test
20
y-x
y-x
10
20
40
90
Train Envs Accs
Train Accs
90
Env 0 Acc. R=0.98
80
R=0.99
70
Env 2 Acc. R=0.98
ACC
60
Env 3 Acc. R=0.99
50
Env Acc. R=0.9
30
overall. R=0.77
y=x
10
10
20
30
40
50
60
70
80
90
Train Envs Accs
Figure 27: FMoW: Train Eny Accuracy vs. Test Env Accuracy.
69

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Published in Transactions on Machine Learning Research (08/2025)
C.9 Waterbirds
Waterbirds (Sagawa et al., 2019 Koh et al., 2021): The Waterbirds dataset is modification of the CUB dataset (Welinder et al. 2010) constructed to induce a subpopulation shift in the association between bird type b E & {waterbird, d, ]andbird] in the foreground and the image background C {water, land]. Waterbird labels are assigned to scabirds and waterfowl; all other bird types are labeled as landbirds. Image backgrounds are obtained from the Places datasel. (Zhou al., 2016 and subset to include water backgrounds (categories: ocean natural lake) and land backgrounds (categories: bamboo forest broadleaf forest.. The training set consists of 95% of all waterbirds with a water background and the remaining 5% with land background. Similarly, 95% of all landbirds are displayed on a land background with the remaining 5% on a water background. The validation and test. sets include an equal distribution waterbirds and landbirds on each background. This dataset consists 11,788 examples of size (3 x 224 X 224).
Experimental Details. Environment 0 consists of the full training split from the Waterbirds dataset and Environment is concatenation of the validation and test splits from the Waterbirds dataset (Sagawa et al,, 2019). In addition plotting the average ID vs. OOD accuracies for each test environment, we also include average ID group-specific accuracies and pairwise combinations of group-specifie ID vs. group-specific OOD accuracy.
Discussion. In our evaluation of average ID and OOD performance, we find that the Waterbirds dataset does not strongly represent worst-case shifts. However, plotting group-specific OOD accuracies, we find no linear correlation for Enviromment group (y = 0, = 1), representing a degradation in worst-group accuracy (WGA) with improvements Enviromment 0 ID average accuracy. We examine the pairwise group-specific accuracies for each environment and generally find that OOD group accuracy positively correlates with ID) accuracy for groups sharing the same label. Since Waterbirds groups defined by label (bird), attribute (background) pairs, this result is consistent with the studies that suggest. worst-class accuracy (WCA) is good proxy for WGA when group membership is unknown (Yang et al. 2023). Similarly, OOD group accuracy negatively correlates with ID accuracy for groups of the opposite label, demonstrating the tradc-off between majority and minority group/ performance under subpopulation shifts.
Table 24: WIL.DSWaterbn average ID vs. OOD properties.
ID
OOD
slope
intercept
Pearson R
p-value
standard crror
Env 0 acc
Env 1 acc
0.83
0.35
0.92
0.00
0.01
Env 1 acc
Env 0 acc
0.47
0.16
0.69
0.00
0.02
97
R=0.69
Test Env Acc
90
Test Env 1 Acc
90
80
70
70
60
50
40
50
60
70
80
90
97
50
60
70
80
90
95
Average Train Envs Acc
Average Train Envs Acc
Figure 28: Waterbirds average ID vs. average OOD accuracies
70

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Published in Transactions on Machine Learning Research (08/2025)
Table 25: Waterbirds average vs. group-specifie OOD properties.
ID
0OD
slope
intercept
Pearson
p-value
standard error
Env avg acc
Env y=0,a=0 acc
-0.13
1.58
-0.13
0.00
0.03
Env avg ace
Env y=0,a=1 acc
0.00
1.32
0.00
0.95
0.03
Env avg ace
Env 1 y=1,a=0 acc
0.22
1.17
0.84
0.00
0.00
Env avg
Env 1 y=1,a=1
0.32
1.13
0.81
0.00
0.01
Env avg
Env 0 y=0,a=0
0.70
-0.02
0.65
0.00
0.03
Env avg acc
Eny 0 y=0,a=1 aCc
-0.07
1.61
-0.11
0.00
0.02
Env avg acc
Env y=1,a=00 acc
0.27
1.63
0.56
0.00
0.01
Env avg acc
Env 0==1a==1 acc
0.30
1.21
0.60
0.00
0.01
Table 26: Waterbirds pairwise group-specific ID vs. OOD properties.
ID
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0= acc
Env y=0,a=0 acc
0.76
0.58
0.86
0.00
0.01
Env 0 y=0,a=1 acc
Env y=0,a=0 acc
0.72
0.73
0.73
0.00
0.02
Env 0
y=
Eny y=0,a=0
-0.15
2.19
-0.43
0.00
0.01
Env 0
y= acc
Eny y=0,a=0 acc
-0.13
2.17
-0.36
0.00
0.01
Env 0 0-=0,a=0 acc
Env 1=0,a=1 acc
0.45
1.05
0.42
0.00
0.03
Env 0 0y=0,a= acc
Env y=0,a=1
0.65
0.72
0.54
0.00
0.03
Env 0 y==1aa00
Env y=0,a=1
-0.03
1.96
-0.06
0.08
0.01
Env 0 y==1aa= acc
Eny y=0,a=1 acc
-0.05
1.99
-0.10
0.00
0.01
Env y=
acc
Env y= acc
-1.16
3.05
-0.33
0.00
0.10
Env 0 y=0,a=1
Env y=1,a=0
-0.96
2.58
-0.25
0.00
0.12
Env 0==1,a=0 acc
Env y==1a=0 acc
1.24
0.13
0.93
0.00
0.02
Env 0 y=1,a==1 acc
Env y= ace
1.34
0.12
0.92
0.00
0.02
Env 0 y=
Env y=1,a=1
-0.70
2.02
-0.32
0.00
0.07
Env 0 y= acc
Env 1==1a==1 acc
-0.87
2.24
-0.34
0.00
0.08
Env 0 y=1,a=0 acc
Env 1==1a=1 acc
0.84
0.19
0.95
0.00
0.01
Env 0 y==1a==1
Env 1==1a=11
0.92
0.20
0.97
0.00
0.01
Env 1 0 acc
Env 0 y= acc
0.77
0.55
0.96
0.00
0.01
Env 1 y=00a==1 acc
Env 0 y=00===0 acc
0.16
2.18
0.35
0.00
0.01
Env 1 y==1,a=0
Eny y==0a=0
-0.07
2.23
-0.18
0.00
0.01
Env 1y=1,a=1 acc
Env 0 1-==0a=0 acc
0.02
2.26
0.05
0.10
0.01
Env 1 1y=0,a=0 acc
Env 0 10y=0,a=1 acc
0.54
-0.49
0.39
0.00
0.04
Env y=00a=1
Env y==0,a=1
0.73
0.23
0.88
0.00
0.01
Env y== acc
Env y==0aa=1 acc
-0.13
0.61
-0.23
0.00
0.02
Env 1
1y=1,a=1 acc
Eny 0 1-==0a=1 acc
-0.50
1.25
-0.69
0.00
0.02
Env 1
1y=0,a=0 acc
Env yy=1,a=0 acc
-0.33
0.50
-0.18
0.00
0.05
Env 1
1==0,a=1
Env yy=1,a=0
-0.40
-0.01
-0.32
0.00
0.04
Env 1
y==1,a=0 acc
Eny 1y=1,a=0 aec
0.61
0.17
0.84
0.00
0.01
Env 1
1y=1,a=1 acc
Env y==1a=0 acc
0,88
-1.21
0.83
0.00
0.02
Env 1y=0,a=0
Env y==1a=1
0.05
1.08
0.03
0.38
0.05
Env 1 1y=0,a=1 ace
Env 0 y0==1a=1 acc
-0.50
1.50
-0.47
0.00
0.03
Env 1y=1,a=0 acc
Env y0 acc
0.44
1.44
0.60
0.00
0.02
Env
1y=1,a=1
Env y==1===1
0.94
0.13
0.96
0.00
0.01
71

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Published in Transactions on Machine Learning Research (08/2025)
Test Env Group 0.a=0 Acc
Rm0.71
Env 1 Group y=0,a= Acc
Rm-0.14
99
90
80
80
70
Test Env
70
60
50
60
40
50
40
50
80
90
99
80
90
Train Envs Accs
Train Envs Accs
99
---0.06
Test Env 1 Group y=0 Acc
R.1144
0.8=1
0000
90
60
80
40
30
70
Test
20
10
50
20
30
50
60
90
99
90
Train Envs Accs
Train Accs
97
Test O Group y= 1,a00 Acc
R=0.58
Test Env 1 Group y-1,a=0 Acc
0.87
99
10
20
40 50
70
80
90
97
10 20 30
70
90
Train Envs Accs
Train Envs Accs
Test Env 0 Group y= 1,a=1 Acc
99
R=0.68
Test Env 1 Group 1-a.y-111111111 Acc
R=0.82
90
1
10
10 20 4050 70 80
90 99
1
10
20
30
40
50
60
70
80
90
95
Train Envs Aces
Train Envs Accs
Figure 29: Waterbirds average ID vs. group-specific 00D accuracies
72

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Published in Transactions on Machine Learning Research (08/2025)
Acc
99
Test Env Group y=0, ==,0=0 aa=0
90
80
70
70
Env 1 Group y=0,a=0 Acc. R=0,96
Test Env 1 Group
60
Eny Group y=0,a=0 Acc. 0
Env 1 Group y=0,a=1 Acc. R=0.4
Env Group y=0,a=1 Acc. R=0.73
Env 1 Group y=1,=0 Acc. 0.18.18
Env Group y= Acc. R=
20
10
Group y=1,a=1 Acc. R=0.067
Env Group y=1,a=1 Acc. R=
overall. R=0.09
overall, R=-0.03
y=x
10
20 6050 70 80 90
99
10
20 30 405060 70
90
Train Envs Accs
Train Envs Accs
Test Env Group y=0 a==1 Acc
99
Test Env Group 1 Acc
08
90
90
80
Env Group y= Acc. R=0.4
50
Env 0 Group y=0,a00 Acc. R=0.44
Env Group y=0,a=1 Acc. R=0.88
40
Env 0 Group y=0,a=1 Acc. R=0.59
30
20
Group y=1,a=0 0-29
20
Group y=1,a=0 0=00.09
Eny Group y=1,==1 1-11a=1Acc Acc.
R=60.65
Env 0 Group y=1,a=1 Acc. 0=09.
overall. R=00.00
overall. R=0.04 y=x
y=x
10 20 30 405060 70 80 90
99
1
10 20 30 405060 70 80 90
99
Train Envs Accs
Train Envs Accs
Test Env Group y=1,a=0 Acc
Test Env 1 Group y=1.== Acc Acc
Env 0 Group y=0,a=0 Acc. R=-0.31
Env 0 Group y=0,a=1 Acc. R=-0.24
90
Env 0 Group y=1,a=0 Acc. R=0.93
Env Group y=1,a=1 Acc. R=0.91
99
overall. R=0.30
Eny Group y=0,a=0 Acc. R=10.16
Env Group y=0,a=1 0=29.29
y=x
Env Group y=1,a=0 Acc. R=0.83
Erv 1 Sr Acc. 0.8.84
overall. R=0.20
A7
y=x
10
20 30 405060 70 80 90
99
1
10 20 30 405060 70 80 90
Train Envs Accs
Train Envs Accs
Test Env Group y=1. y=1.a= 1,a11 ACC
Test Env 1 Group 1l-aa Acc
Env 0 Group y=0,a=0 Acc. Rm-0.3
99
Env 0 Group y=0,a=1 Acc. R=-0.35
Env Group y=1,a=0 Acc. R=0.95
90
99
Env 0 Group y=1,a=1 Acc. R=0.97
80
60
70
Env Group y=0,a=0 Acc. R=0.029
90
overall. R=0.31
28
Env y=0,a=y Acc. R=-0.48
130
Env 1 Group y=1,a Acc. R=0.6
20
Env Group y=1,a=1 Acc. R=0.96
overall. R=0.16
10
y=x
10 405060 70 80
99
10
30
40 70
90
Train Envs Accs
Train Envs Accs
Figure 30: Waterbirds pairwise group-specific accuracies
73

---

Published in Transactions on Machine Learning Research (08/2025)
C.10 CivilComments
CivilComments (Koh et al., 2021 A datasct comprised of multiple subpopulations, which correspond to different demographic identities. The domain is a multidinennsooall binary vector with 8 dimensions, each corresponding to whether a given comment mentions one of the demographic identitie: male, female, LGBTQ, Christion, Muslim, other religions, Black, and White. For each of the identitics, we form 2 groups corresponding to the loxicity label, generating total of 16 groups (e.g,, male toxic, male non-toxic, elc.). By construction, since each comment may belong to more than one group, this experimental procedure differs slightly from the standard subpopulation shift framework discussed in this work. Regardless, the experimental results with and without group overlaps display similar accuracy on the line and accuracy on the inverse line patterns.
Experimental Details. dataset.
We leverage both BERT and DistilBERT architectures for the CivilComments
Discussion.
The CivilComments dataset exhibits both an accuracy on a line and an accuracy on the inverse line phenomenon across all test environments in the Leave-OOe-DomaannOO procedure. The spurious correlation, which involves the presence absence of 8 different demographic identities (e.g., male, white, Christian, Muslim, etc.), is rather strongly correlated with the toxicity label. Empirically, the correlation coefficient (R) values range from low as -0.47 to as high 0.43. There appear to be many ID/OOD splits that can be derived from this dataset for benchmarking domain generalization.
Table 27: CivilComments ID vs. OOD properties.
OOD
slope
intercept
Pearson R
p-value
standard error
Env 0 acc
0.36
-0.29
0.28
0.00
0.04
Env 1 acc
-0.49
0.16
-0.47
0.00
0.03
Env 2 acc
0.39
-0.17
0.28
0.00
0.04
Env 3 acc
-0.49
0.12
-0.42
0.00
0.03
Env 4 acc
0.58
0.20
0.43
0.00
0.04
Env 5 acc
-0.54
0.00
-0.43
0.00
0.04
Env 6 acc
0.46
-0.04
0.30
0.00
0.05
Env 7 acc
-0.50
0.16
-0.43
0.00
0.03
Env 8 acc
0.49
0.03
0.36
0.00
0.04
Env 9 acc
-0.49
0.06
-0.41
0.00
0.03
Env 10 acc
0.53
0.17
0.34
0.00
0.05
Env 11 acc -0.57
-0.09
-0.46
0.00
0.03
Env 12 acc
0.46
0.05
0.30
0.00
0.05
Env 13 acc
-0.54
-0.04
-0.45
0.00
0.03
Env 14 acc
0.41
-0.10
0.29
0.00
0.04
Env 15 acc
-0.52
0.06
-0.43
0.00
0.04
74

---

Published in Transactions on Machine Learning Research (08/2025)
70
R=0.28
99
60
50
Test Env 0 ACC
40
Test Env 1 Ace
80
70
60
20
R=-0.47
30
20
10
20
30
40
50
60
70
1
10
20
30
40
70
90
99
Average Train Envs Acc
Average Train Envs Acc
R=0.28
60
90
Test Env ACC
50
40
Test Env 3 Acc
80
70
30
60
20
40
10
20
R=-0.42
1
10
20
30
40
50
60
70
79
1
10
20
30
4050 60 70 80
90
99
Average Train Envs Acc
Average Train Envs Acc
98
R=0.43
90
90
80
Env 5 Acc
80
Test Env
40
30
20
10
10
20
30
40
70 80
90
98
1
10
20
30
40
60
70
80
90
99
Average Train Envs Acc
Average Train Envs Acc
R=0.30
99
90
70
Test Env 7 Acc
Test Env 6
50
40
60
30
20
30
10
R=-0.43
40
70
10
30 40:0000
90
Average Train Envs Acc
Average Train Envs
Figure 31: CivilComments: Average train Env Accuracy Test Env Accuracy.
75

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Published in Transactions on Machine Learning Research (08/2025)
88
R=0.36
99
Env Acc
60
Test Env 9 Ace
80
40
60
70
10
20
R=-0,41
10
20
30
40
50
60
70
80
1
10
20
30
40
70
90
Average Train Envs Acc
Average Train Envs Acc
99
R=0.34
90
90
Test Env 11 Acc
80
80
70
60
40
30
30
Test
20
20
R=-0.46
1
10
20
30
4050
60 70 80
90
99
1
10
20
30
4050 60
70 80
90
99
Average Train Envs Acc
Average Train Envs Acc
90
0
R=0.30
80
90
60
Test Env 13 Acc
80
Test Env 12
40
60
40
30
10
20
30
40
50
60
70
80
90
1
10
20
30
505006
70
80
90
99
Average Train Envs Acc
Average Train Envs Acc
80
R=0.29
99
70
60
Test Env 14 ACC
50
Test Env 15 Acc
40
30
60
20
30
10
R=-0.43
20
60
10 30 40
90
Average Train Envs Acc
Average Train Envs
Figure 32: CivilComments: Average train Env Accuracy
Test Env Accuracy.
76

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Published in Transactions on Machine Learning Research (08/2025)
99
99
Test Bri A.c
90
Nest
10
10 20 30:40 01440066700 70
99
30.00000070 80 90
99
Envs Accs
Acc. R=.0.8
R=0.97
R.-0.85
0.-0.82
Eny R=-0.72
Env Acc.
Eny Acc. Rm1.0
Env 7 Acc. Rm-0. 81
Enw 12 Acc. R=o 90
Env 2 Acc. R.
Eny 7 Acc. R=1.0
Env 12 Acc. 0.70
Eri Acc. Re-0.81 81
Env B Acc. R=0.98
Erv 13 Acc. Rm-0.84
Erv 3 Acc. R-1.0
Env 8 Acc. Rm-0.76
Env 13 Acc. R-0.99
R=0.96
Rm-0.03
R- 0.72
Eny Acc. R=2.0
Env Acc. Rw.0.79
Eri Acc. Rm-0.8t 84
Env 10 Acc. Rw0.95
Erw 15 Acc. Rm-0.82
Env 5 Aoc. Rw0.99
Env 10 Acc. Re--0 1o-60.69
Env 15 Acc. R=1.0
99-
%
Iest 86
Test Acc
10
10 20 30 40566770 90
90
99
20 3040506070 70 80
90
99
Train Envs
Accs
Acc. R=0.99
R=0.99
R-
Env R-
Acc. 0.70
Env 11 Acc. R=0.99
Eny Acc. R.00774
Erv Rm-0.74
Erv 12 Acc. R-1.0
Env 1 Acc. R=1.0
Eny 7 Act. R=100
Env 12 Acc. R=-0.78
Eni 3 Acc. R- 0.75
Env Acc. R=1.0
Erv 13 Acc. R- 0.79
Env 2 Aoc. R- 0.83
Env B Acc. R 0.B
Env 13 Acc. R=0.99
Acc. R-0.99
R--0.78
R-1.0
0:70.6
Eny R=1.0
Env Acc. 8.:0.93
Env 5 Acc. R.0
Env 10 Acc. R-0.98
Erw 15 Acc. R- 0.77
Env Acc. R=0.99
Env 10 Acc. 0.74
Env 15 Acc. R=1.0
99
Test Env 4 fcc
9
Env Acc
90
10
30 00440560777 70 Eriss
99
Train Enus Aces
Env
Rm-0.8
Acc. R-
Eny R- 0.79
Env 11 Acc. R=1.0
Env Act. 0.7
Env?? 7 Acc. R=-0.71
Eny 12 Acc. R=0.99
Env 1 Aoc. R-0.99
Eny 7 Acc. R-0.99
Env 12 Acc. R-.0.8
Env Acc. Reo,0e
Eriv 8 Acc. Reo 99
Env 13
Acc. Rm-0.78
0.844
R.00.81
Env 13 Acc. R=1.0
Erv Acc. Rm00.71
9 Acc. R--0.75
Fnv 14 R-O 98
Env Aoc. Ro0.00
Env Acc. R=1.0
Env Acc. Rm-0.84
Env S Acc. R=-0.78
Env 10 Acc. R=110
Env 15 Acc. Ro-0.74
Env Aoc. Rw.0.78
Eni 10 Acc. 0-76
Env 15 Acc. R=1.0
99
Test Env 6 Acc
A25
20
30:40 Envs
306070
90
99
Train Envs Accs
Env Acc. R-0.95
Eri'3 8 Acc. Rw.0.78
Env 11 R--0.8
Env O Acc, Rm-o 86
Eny 5 Acc. R=0.99
Env 11 Acc. Rm
Env Acc. R 0.7
Env 7 Acc. R.-0.7
Eny 12 Acc. R=0.99
Env 1 Aoc. R=1.0
Env 6 Acc. R=-0.76
Env 12 Acc. R=-0.78
Acc. R-0.00
8
Acc. R-0.99
Env R-70.77
R.80.83
R=-0.79
Env Act. Env 14 Acc. Rm-0.83
Enw 3 Acc. R- 0.71
Env 9 Acc. R=-0.75
Env 14 Acc. R=0.98
Env 3 Acc. Rw1.0
Env Acc. R-1
Env Acc: R=1.0
Env Acc. 1=0
Cnv 15 0.70.33
Env Acc. 0.75.7
Env Acc. 0.73
Env 15 Acc. R=1.0
Figure 33: CivilComments: Train Env Accuracy vs. Test Env Accuracy.
77

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Published in Transactions on Machine Learning Research (08/2025)
8
Brw Acc
Test Env
10
10 20 30 4050000770 70 50 90
99
20 30.00500070 80 90
99
Train Envs Accs
Envs Accs
Env Act. R=0,95
Enw S Acc. R=
0.79
Cnv 11 R=-0.81
R=-0.93
Eny
Ent 11 Acc.
Env Acc. Rm-0.73
Env 6 Acc. R=1.0
Eny 12 Acc. Rm1.0
Env 1 Acc. R=1.0
Eny Acc. R.05
Env 12 Acc. R..0.97
Env 2 Acc. R=0.99
Env 7 Acc. 0.74
Env 13 Acc. 0.79.79
Erv Aoc. Rm.0991
Env Acc. R=1.0
Env 13 Acc. R=1.0
Acc. Rm-0,74
9
Acc. Rw1.0
Eny R--0.88
Env Acc. Rw.0.31
Erv 4 Arc. R=1.0
Eni 10 Acc. Re1.0
Env 15 Arc. Rw-0.76
Env 4 Acc. Rm-0.84
Env 10 Acc. Re-00.83
Env 15 Acc. R-
Test Env Acc
8 $
Test Env 11 Acc
<0
30:00560070
10 30 00.40050007
90
Envs
Train Envs Accs
Acc. R=0.95
5 0.82
Env R=-0.83
Acc.
Env Acc. R=1.0
10 Acc. 0.77
Enw 1 Acc. R.70.75
Env 6 Acc. R-0 39
Env 12
Acc. R=0.99
Env Acc. R-0 98
Env Acc. Rv-0. 8
Erw 12 Acc. R- 0.81
Env Acc. R=0.97
7 R- 0.75
Env 13
R.-0.82
R-
Env R.0.99
Enw Acc. R=1.0
Env 3 Act. 0.70.75
Env 8 Acc. R=0 99
Eny 14 Acc. R=0.97
Eny 3 Acc. R=0 99
Env Acc. Re=-0. 92
Enw 14 Acc. R= 84
Env 4 Acc. R-1.0
Erv 9 Acc. 0.79
Env 15 Acc. Re. 0.78
Eny 4 Acc. 0.79
Env Acc. R=1.0
Enw 15 Acc. Rw0.99
99
Test Env 12
10 3366600
Tost Env 13
20
10
5060 Train Ervs Accs
Train Erivs Aces
Env Acc. R-O.97
Eriv S Acc: Rt-0079
Env 10 Arc. R=1.0
0 Arc. Rm-O
Env Acc. R=1.0
Frw 10 Acc. Rm-o0 78
Env Acc. R-0.72
Erw 6 Acc. R=1.0
Env 13 Acc. R=-0.81
Eny Acc. R-0 99
Env Acc. Rm-0 B1
Erw 11 Acc. R-1.0
Rm-0.73
Rm-0.79
Rm-0
Enw 12
Env 3 Acc. R-00.73
Eri 8 Acc. Rw1.0
Env 14
Acc. R-0.90
Env 3 Acc. R-0 99
Env Acc. Rv--O. 84
Enw 14 Acc. Re-O. 86
Acc. R=1.0
R-.0.76
Env R.70.75
Env Rw110
Enw 15 Acc. R=1.0
99-
Env 15 46
30
30 8
80
Test Env 14
10-
1
10 20 30 5060 70 80 90
99
10 20 30 4050060 80 90
Train Erivs Aces
Train Envs Accs
Env O Acc. Reo.gg 99
Env S Acc: Rm-0079
Env 10 Acc. Rm0.99 99
Env 0 Acc. Rm-0 87
Env Acc, Rmn.
Erw 10 Acc. Rm-0 74
Env Act. R= 0.73
6 Acc. R=0.99
Env 11
0.8
R=1.0
Env Acc. R= 0.77
Enw 11 Acc. R=0.99
Enw 2 Acc. Rm1.0
Env'? 7 Acc. Rm-0.74
Env 12 Acc. Rm1.0
Eny 2 Acc. Rm-0
Env Acc, R==10
Enw 12 Acc. Rm- 79
Env Acc. R-00.7
Erw 8 Acc. R- 1.0
Env 13 Acc. R=-0.78
Eny 3 Acc. R=1.0
Env Acc. Rm-O 81
Erw 13 Acc. R-0. -9..99
Acc. R=0.99
R=-0.77
Env R=-0.76
R=
Env R=110
Acc. R=
Figure 34: CivilComments: Train Env Accuracy Test Env Accuracy.
78

---

Published in Transactions on Machine Learning Research (08/2025)
Acknowledgements
OS was partly supported by the UICC Beckman Institute Graduate Rescarch Fellowship, NSF-NRT 1735252, GEM Associate Fellowship, and the Alfred P. Sloan MPhD Program. SK acknowledges support from NSF 2046795 and 2205329, the MacArthur Foundation, Stanford HAT, and Google Inc. We thank A. Anas Chentouf, Tyler LaBonte, Vivian Nastl, Haoran Zhang, and Yibo Zhang for their comments on an earlier draft. We also thank the TMLR reviewers and the action editor for their valuable feedback and suggestions, which helped improve this work.
79


Reviews:
[{"review_id": "Nm2gurTB0U", "reviewer": "Reviewer_svrV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths:\n\n(+) This work focuses on a fundamental challenge in evaluating the OOD generalization benchmarks.\n\n(+) The authors provide detailed theoretical discussion on the influence of spurious features, and how it's related to ID-OOD performance correlations.\n\n(+) The experiments are quite extensive and comprehensive.\n\n## Weakness\n\n(-) Although the discussion is meaningful, it does not follow the real-world data generation process, limiting the impact.\n\n(-) It's also worth discussing the correlations in terms of other metrics.\n\n(-) A number of related works have also discussed this issue.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work investigates the limitations of domain generalization benchmarks in evaluating model robustness to spurious correlations under distribution shifts. The authors argue that many existing benchmarks are misspecified, as they fail to include shifts in spurious correlations that meaningfully degrade out-of-distribution (OOD) generalization. To address this, they propose theoretical conditions for well-specified benchmarks, showing that a strong positive correlation between in-distribution (ID) and OOD accuracy—known as \"accuracy on the line\"—indicates that a benchmark is unsuitable for assessing robustness to spurious correlations. Through extensive analysis of over 40 ID/OOD splits across 12 popular benchmarks, they demonstrate that most existing benchmarks exhibit accuracy on the line, limiting their utility for domain generalization evaluation. The findings emphasize the need for rethinking benchmark design to better assess robustness to spurious correlations.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Although the discussion is meaningful, it does not follow the real-world data generation process, limiting the impact. More specifically, if the assumption and the data model on the spurious features does not reflect the trends in real-world datasets such as those from DomainBed and Wilds, perhaps we need to reconsider better data models instead of the benchmarks. Otherwise, the gap between the methodology development in OOD generalization and the real-world practice will gradually lose the relevance.\n\n2. It's also worth discussing the correlations in terms of other metrics. For example, mean accuracy and worst case accuracy will reflect different trends. It's also encouraged to consider other metrics in the Wilds benchmark.\n\n3. A number of related works have also discussed this issue:\n- Rich feature construction for the optimization generalization dilemma, ICML'22.\n- Pareto invariant risk minimization, ICLR'23.\n- Understanding and Improving Feature Learning for Out-of-Distribution Generalization, NeurIPS'23.\n- Spurious Feature Diversification Improves Out-of-distribution Generalization, ICLR'24.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "xlFOYYq2rl", "reviewer": "Reviewer_jq8n", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n\n+ The theoretical framework is solid.\n\n+ The theoretical framework developed in the paper offers valuable insights.\n\n+ The paper proposes a new framework for assessing DG benchmarks and provides guidelines for constructing more effective evaluation protocols for domain generalization tasks.\n\n+ Introduces the concept of well-specified benchmarks.\n\nWeakness:\n\n-  One of the major concerns is that, the spurious correlations follow the Gaussian assumption. What if the correlation does not follow this assumption, which can be common in many real-world scenarios? More discussion and analysis on this aspect is needed. \n\n- Does the property and conclusion of this paper hold on some larger-scale domain generalization settings, such as between ImageNet and ImageNet-R, ImageNet-A, ImageNet-C, or other larger-scale domain generalization dataset?\n\n- More in-depth discussion on how the findings can influence the development of new DG algorithms or the improvement of existing ones is needed.\n\n- More feature space visualization on the different types of features from a model on ID and ODD scenarios is needed, such as t-SNE, to help the understanding of the relationship.\n\n- An exploration on alternative evaluation metrics may provide a better understanding of model robustness to spurious correlations. Metrics such as area under the ROC curve (AUC), average worst-case performance, or robustness to adversarial perturbations could be used in conjunction to offer a more nuanced assessment.\n\n- The paper does a good job of critiquing existing benchmarks, but it does not provide a detailed comparison with other recent domain generalization methods that specifically address spurious correlation removal. A more comprehensive comparison would position the paper’s contributions within the broader DG research landscape.\n\n- The paper does not include an ablation study to assess how the accuracy on the line phenomenon varies under different configurations of domain splits and model types. Such an experiment would help validate the framework's robustness and further illustrate its practical utility.\n\n- The paper introduces the concept of well-specified benchmarks, but it does not provide sufficient guidelines for practitioners to easily create or adapt their own datasets according to this framework. A more detailed toolkit or practical advice on how to curate well-specified benchmarks would be valuable for future research.\n\n- Is table 3 necessary in the main text? Or can be put into the supplementary material?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper examines the limitations of current domain generalization (DG) benchmarks, specifically on a phenomenon where in-distribution (ID) accuracy strongly correlates with out-of-distribution (OOD) accuracy. \n\nThe authors argue that these benchmarks may be misspecified for evaluating the true robustness of models to spurious correlations, which are statistical associations that do not generalize across domains. \n\nThrough both theoretical analysis and empirical experiments, the paper introduces the concept of well-specified benchmarks and shows that many popular DG benchmarks fail to capture meaningful distribution shifts. \n\nThe results suggest that for robust evaluation, benchmarks should not show a strong positive correlation between ID and OOD accuracy.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "-  What if the correlation does not follow this assumption, which can be common in many real-world scenarios? More discussion and analysis on this aspect is needed. \n\n- Does the property and conclusion of this paper hold on some larger-scale domain generalization settings, such as between ImageNet and ImageNet-R, ImageNet-A, ImageNet-C, or other larger-scale domain generalization dataset?\n\n- More in-depth discussion on how the findings can influence the development of new DG algorithms or the improvement of existing ones is needed.\n\n- More feature space visualization on the different types of features from a model on ID and ODD scenarios is needed, such as t-SNE, to help the understanding of the relationship.\n\n- An exploration on alternative evaluation metrics may provide a better understanding of model robustness to spurious correlations. \n\n- Please provide a detailed comparison with other recent domain generalization methods that specifically address spurious correlation removal. \n\n- Please include an ablation study to assess how the accuracy on the line phenomenon varies under different configurations of domain splits and model types. \n\n- The paper introduces the concept of well-specified benchmarks, but it does not provide sufficient guidelines for practitioners to easily create or adapt their own datasets according to this framework. A more detailed toolkit or practical advice on how to curate well-specified benchmarks would be valuable for future research.\n\n- Is table 3 necessary in the main text? Or can be put into the supplementary material?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "TPG5lABWoc", "reviewer": "Reviewer_ULfo", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper contains theoretical guarantees and experiments.\n- The problem is well-motivated. Domain generalization and the accurate evaluation of models are crucial for real world application.\n- The paper is clear about the limitations of the proposed test and recommends a holistic and well-hedged approach for real world problems.\n\nWeaknesses:\n- The theoretical results are necessary, but not sufficient. Ideally, they should be sufficient as it allows conclusions to be drawn with more confidence. Currently, the results allow us to reject the hypothesis that the dataset is well-specified if the correlation between the OOD and ID accuracies is sufficiently high. However, this implies that some datasets may be prematurely discarded.\n- It is unclear in practice how many and what type of models are needed to be evaluated to obtain the correlation between the OOD and ID accuracies. Are the ones used in the experiments sufficient? \n- The theory assumes that the domain-general and the spurious features are disentangled and combine linearly to get the target variable. I feel that this disentanglement is a strong assumption, and the authors do not go into detail why it is reasonable. The linearity assumption seems reasonable given results on the neural tangent kernel, for example.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper considers the problem of evaluation in domain generalization. Motivated by the observation that ERM achieves the best OOD accuracy in many benchmarks, it studies when the best domain generalizable model achieves the best OOD accuracy and the behavior of the OOD accuracy relative to the in-distribution (ID) accuracy. It shows that when the OOD spurious correlations are sufficiently misaligned with the ID spurious correlations, the best domain generalizable model achieves the best OOD accuracy and the relationship between the OOD accuracy and the ID accuracy is non-linear with high probability. Therefore, the paper proposes to use the correlation between the OOD and ID accuracies as a test for whether a domain generalization benchmark is well-specified. Experiments on some common neural networks and datasets indicate that in general, semisynthetic (like colored MNIST) and subpopulation shift datasets are well-specified.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Critical:\n- Please provide some guidance on how many and what type of models need to be evaluated to carry out the test on the correlation between the OOD and ID accuracies.\n- Can you expand on why the assumption on the disentanglement of the domain-general and spurious features is reasonable? (Last full paragraph of page 2.)\n\nMinor:\n- What are the obstacles to obtaining sufficient theoretical guarantees?", "broader_impact_concerns": "No broader impact statement is necessary.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)