Understanding Domain Generalization: A Noise Robustness Perspective
Authors: Rui Qiao
Bryan Kian Hsiang Low
Venue: ICLR 2024 poster (2024)
Keywords: out-of-distribution generalization
distribution shifts
spurious correlation
noise robustness
Primary area: general machine learning (i.e., none of the above)
TL;DR: Label noise exacerbates the effect of spurious correlations for ERM. Invariance learning algorithms with label-noise robustness may improve the situation under certain circumstances.

Abstract: Despite the rapid development of machine learning algorithms for domain generalization (DG), there is no clear empirical evidence that the existing DG algorithms outperform the classic empirical risk minimization (ERM) across standard benchmarks. To better understand this phenomenon, we investigate whether there are benefits of DG algorithms over ERM through the lens of label noise.
Specifically, our finite-sample analysis reveals that label noise exacerbates the effect of spurious correlations for ERM, undermining generalization. 
Conversely, we illustrate that DG algorithms exhibit implicit label-noise robustness during finite-sample training even when spurious correlation is present.
Such desirable property helps mitigate spurious correlations and improve generalization in synthetic experiments. 
However, additional comprehensive experiments on real-world benchmark datasets indicate that label-noise robustness does not necessarily translate to better performance compared to ERM. 
We conjecture that the failure mode of ERM arising from spurious correlations may be less pronounced in practice. Our code is available at https://github.com/qiaoruiyt/NoiseRobustDG

Published as a conference paper at ICLR 2024
UNDERSTANDING DOMAIN GENERALIZATION: A NOISE ROBUSTNESS PERSPECTIVE
Rui Qiao and Bryan Kian Hsiang Low
Department of Computer Science, National University of Singapore rui qi acêu. nus edu, 1 lowkhe comp. nus edu. sg
ABSTRACT
Despite the rapid development of machine learning algorithms for domain general- ization (DG), there is no clear empirical evidence that the existing DG algorithms outperform the classic empirical risk minimization (FRM) across standard bench- marks. To better understand this phenomenon, we investigate whether there are benefits of DG algorithms over ERM through the lens of label noise. Specifically, our sample analysis reveals that label noise exacerbates the effect of spuri- ous correlations for ERM, undermining generalization. Conversely, we illustrate that DG algorithms exhibit implicit label-noise robustness during training even when spurious correlation is present. Such desirable property helps mitigate spurious correlations and improve generalization in synthetic experiments. However, additional comprehensive experiments on real-world benchmark datasets indicate that label-noise robustness does not necessarily translate to better perfor- mance compared to ERM. We conjecture that the failure mode of ERM arising from spurious correlations may less pronounced in practice. Our code is available com http::///tttto.oooooooooooooo/aa
INTRODUCTION
A common assumption in machine learning is that the training and the test data are independent and identically distributed (i.id.) samples. In practice, the unseen test data are often sampled from distributions that are different from the one for training. For example, a camel may appear on the grassland during the test time instead of the desert which is their usual habitat (Rosenfeld al., 2020). However, models such as overparameterized deep neural networks are prone to rely on spurious correlations that are only applicable to the train distribution and fail to generalize under the distribution shifts of test data (Hashimoto al. 2018 Sagawa et al. 2019 Arjovssy al. 2019). It is crucial to ensure the generalizability of ML models, especially in safety-critical applications such autonomous driving and medical diagnosis (Ahmad et al. 2018 Yurtsever al. 2020).
Numerous studies have attempted to improve the generalization of deep learning models regarding distribution shifts from various angles (Zhang al. 2017 2022 Nam al., 2020; Koh et al. 2021 Liu et al., 2021a: Yao et al.] 2022] Gao et al. 2023). ). An important line of work addresses the issue based on having the training data partitioned according to their respective domains/envirromentss For example, cameras are placed around different environments (near water or land) to collect sample photos of birds. Such additional environment information sparks the rapid development of domain generalization (DG) algorithms (Arjovsky et al. 2019 Sagawa et al. 2019; Krueger et al. (021). The common goal of these approaches is discover the invariant representations that are optimal in loss functions for all environments to improve generalization by discouraging the learning of spurious correlations that only work for certain subsets of data. However, Gulrajani & Lopez-Paz (2020) and Ye et al. (2022) have shown that no DG algorithms clearly outperform the standard empirical risk minimization (ERM) (Vapnik 1999). Subsequently, Kirichenko et al. (2022) and Rosenfeld et al. (202)) have argued that ERM may have learned good enough representations. These studies call for understanding whether, when, and why DG algorithms may indeed outperform ERM.
In this work, investigate under what conditions DG algorithms superior to ERM. We demon- strate that the success of some DG algorithms can be ascribed to their robustness to label noise under subpopulation shifts. Theoretically, we prove that when using overparameterized models trained with

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finite samples under label noise, ERM is more prone to converging to suboptimal solutions that mainly exploit spurious correlations. The analysis is supported by experiments on synthetic datasets. In contrast, the implicit noise robustness of some DG algorithms provides extra layer of performance guarantee. We trace the origin of the label-noise robustness by analyzing the optimization process of few exemplary DG algorithms, including IRM (Arjovsky et al. 2019), V-REx (Krueger et al. 2021) and GroupDRO (Sagawa et al., 2020), while algorithms including ERM and Mixup (Zhang et al. 2017) without explicit modification the objective function, do not offer such benefit. Empirically, our findings suggest that the noise robustness of DG algorithms can be beneficial in certain synthetic cireumstances, where label noise is non-negligible and spurious correlation severe. However, in general cases with noisy real-world data and pretrained models, there is still no clear evidence that DG algorithms yield better performance at the moment.
Why should we care about label noise? It can sometimes be argued that label noise is unrealistic in practice, especially when the amount of injected label noise all but small. However, learning a fully accurate decision boundary from the data may not always be possible (c.g.. some critical information is missing from the input). Such kinds of inaccuracy can be alternatively regarded as manifestations of label noise. Furthermore, common setup for analyzing the failure mode of ERM assumes that the classifier utilizing only the invariant features is only partially predictive (having <100% accuracy) of the labels (Arjovsky al.] 2019. Saaaaaa al. 20200 (Shi et al., 2021). These setups are subsumed by our setting where there initially exists a fully predictive invariant classifier for the noise-free data distribution. and some degree of label noise is added afterward. More broadly, we analyze the failure mode of ERM without assuming the spurious features are generated with less variance (hence less noisy) than the invariant features. By looking closer at domain generalization through the lens of noise robustness, we obtain unique understanding of when and why ERM and DG algorithms work.
As our main contribution, we propose an inclusive theoretical and empirical framework that analyzes the domain generalization performance of algorithms under the effect of label noise:
1. We theoretically demonstrate that when trained with ERM on finite samples, the tendency of learning spurious correlations rather than invariant features for overparameterized models is jointly determined by both the degrees of spurious correlation and label noise. (Section 44)
We show that several DG algorithms possess the noise-robust property and enable the model to learn invariance rather than spurious correlations despite using data with noisy labels. (Section 5)
3. We perform extensive experiments compare DG algorithms across synthetic and real-life datasets injected with label noise but unfortunately find no clear evidence that noise robustness necessarily leads to better performance in general. To address this, we discuss the difficulties of satisfying the theoretical conditions and other potential mitigators in practice. (Section 6
RELATED WORK
Understanding Domain Generalization. It has been demonstrated that ERM can fail in various scenarios. When spurious correlation is present, ERM tends to rely on spurious features that do not generalize well to other domains because of: model overparameterization (Sagawa et al. 2020),
geometric failure from the max-margin principle (Nagarajan et al., 2020; Chaudhuri al., 2023). feature noise (Khani& & Liang 2020), gradient starvation (Pezeshki et al. 2021), etc. Those setups assume that the invariant features are either partially or fully predictive of the labels. By incorporating label noise, our analysis applies to a broader range of settings. We also explicitly quantify the effect of spurious correlation along with label noise and provide more intuitive understanding.
Algorithms for Domain Generalization. Myriads of algorithms and strategies have been proposed from different angles of the training pipeline, including 1) new training objectives by considering
the domain/ /grop information (Arjovsky et al., 22199; Sagawa et al. 22099 Krueger et al. 2021 Liu
jet al. 2021bb Zhang et al., Ahujaa et al. |0_2202 | Zhou et al. [2022); 2) optimization by aligning
the gradients (Koyama & Yamaguchi 2020 Parascandolo et al. 2020 (Shi et al., 2021 Rame et al.
2022); 3) efficient data augmentation (Zhang et al. 2017 Verma et al. 2019 Li et al.
202 Yao et al. 2022 Gao et al. 2023); 4) strategically curated training scheme (Izmallov et al. 2018 Nam
ee al. 2020 Cha et al. 2021 Liu et al. 2021al Zhang et al. 2022). However, multiple works have
benchmarked that ERM and its simple variants can already achieve competitive performance across
datasets (Gulrajani& & Lopez-Paz 2020; [drissi et al. 2022; Khani & Liang] 2021 Ye et al.) 2022
Chen et al. 2022 Yang et al. 2023 Liang et al. 2023). The quality of features learned by ERM
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has also been assessed to be probably good enough (Kirichenko et al. 2022 Rosenfeld al. 2022
Izmailoy et al.) 2022), challenging the practicality of various DG algorithms. Our work complements these studies with extensive experiments and discussions about the gap between theory and practice.
Algorithmic Fairness aims to remove the predictive disparity from groups (Dwork et al. 2012 Hardt et al. 2016) and its objectives related to DG (Creager al. 2 |Wang al. (2021) analyze the
harm of grouppdepennent label noise to models trained with fairnsss algorithms, while we show that
uniform label noise hurts ERM but can be resisted by some DG algorithms.
DOMAIN GENERALIZAAIOO WITH LABEL NOISE
Let (X)))) denote the space of input and label, respectively. Let & represent the set of environ- ments. Assuming that the environment information is known, denote a noise-free dataset by
D = {x(1), y()) (()))e())))))))))))))))))) e(")).... ...... (x((N,, y y(M) ( )( where each data point (x((), , e X xxxxx is sampled i..d. from a data distribution P. For clarity, we will slightly abuse the notation by omitting e(9) from the data point when we only need (x((), ())) Each environment e E has its subset of data Dr = Unnto=ad -e (x:(() y()) } with the corresponding distribution Pe. Let f E F be the classifier in the hypothesis space F and e be the loss function. The risk for the environment is defined as:
Re(s) = ((.w)), ((e_)),,,(((((xx,.)) vl)
R.( = (1/) ((.,)) {(f()),))
Given the training environments Exr, the environment- ERM estimator is:
RERM = Emee, R.(S).
The classic environment-agnosti ERM can be viewed as just having one environment in the training set. On the other hand, the objective of DG is to minimize the out-of-distribun (OOD) risk:
()) = max.ge Re(s).
There are two major types of distribution shifts encountered DG: domain shifts and subpopulation shifts. For domain shifts, the test environments generally have unseen populations with different compositions and features compared to the training environments. For example, cameras placed at different locations capture images for different populations with different backgrounds. Usually, there is no association between environments except that it is common to assume the existence of some comain-invariann feature that generalizes equally well to all training and test domains.
For subpopulation shifts, the training and the test environments share the same composition in terms of subpopulations, but the proportion of the subpopulations may vary. The subpopulations are also referred to as groups, denoted as G. Note that the notion of environment is different from group. In general, group information, often associated with specific spurious features/attributes, challenging to obtain than environment information in practice. Let E G be an element of the groups and let 2, be its associated data distribution. For subpopulation shifts, we can rewrite the data distribution for each environment e as Pa En 2g. where T [0. 1] is the mixing coefficient and satisfies Eg o 1. Thus, the OOD risk in Equation I can be rewritten as:
more
R subpop (f) max R₂)
Therefore, may evaluate the robustness to subpopulation shifts using the worst-group (WG) error in the test set. In this work, our theoretical results focus on subpopulation shifts, but we provide extensive empirical results for both types of distribution shifts.
Let the -abel-independent noise level be n, which is then applied to the noise-fre free dataset and generate the noisy dataset 5 = (1x _))))e())))))))))) e())),.... (z(()), g(N), c(N))), where y(()) is the result of randomly perturbing () to another class with probability n,. Let P. be the corresponding noisy data distribution. Define the training risk of the classifier f w.r.t the noisy data distribution as:
RT()) = (a ( (())))))))
R²( (1/N) E (x.D)ED
((((x) ((())))
4 UNDERSTANDING THE EFFECT OF LABEL NOISE
Our analysis primarily focuses on subpopulation shifts. One common issue for subpopulation shifts is the strong spurious correlations between certain input features and the labels in the training
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set, meaning that such spurious features can accurately predict the labels for the majority of the training data. This causes the trained classifier to rely heavily on the spurious features that do not generalize, especially to the minority groups that have the inverse of such correlation. Consequently, it is challenging to ensure the OOD performance when the test set is dominated by minority groups.
To obtain more intuitive theoretical results, we opt for the setup of binary linear classification with overparameterization without environment information (Sagawa et al., 2020; Nagarajan et al., (020). For training set yX sampled from P. we have X E Inxxa d > for overparameterization. and y E {0, 13". Consider that there exists a fine-grained and disjoint partition of the input features for each instance x x Xouil, where Xnv are invariant features that generalize across all environments E, Xspu E 12 are spurious features that only have a high spurious correlation ? with label y in the training environments Exp, and Xuui E Rd are nuisance features sampled from a high- -igeenimensinna isotropic Gaussian distribution Xmui ~ -N(Odmm: o "ui duu- and irrelevant to the task, but nonetheless provide linear separability for both the noise-free and the noisy training sets D and 1 due to overparameterization of the model and underspecification of finite samples (D' Amour et al., 2022). We use E (0.5, 1) exclusively for the correlation between spurious features and the label. The cardinalities of the features d = dinv + dep + dnui and dmai >> dinv + dspu. This also resembles practical situations such as image classification where for the high-dimensional inputs, there are only a few invariant and spurious features that work for the classification task, but deep neural networks have high capacity and hence the ability to memorize using the nuisance features. Let the overparameterized linear classifier [(x) = w x, where the weights w E IRd can also be decomposed into [Winv; Wspu: Wnui] w.r.t. (iivv Xuu; Xnui We drop the bias term b for simplicity. Assumption 4.1 (Linear Separability for Invariant Features). The noise-free data distribution P is linearly separable using just the invariant features Xinv
This assumption ensures that the invariant features sufficiently expressive and have accuracy that is at least as good as the spurious features for any environments or groups in a noise-free setting. When combined with label noise level 7) [0, 0.5), this also accommodates the scenario with imperfect invariant classifiers. Thus, the assumption is in fact more inclusive than it is restrictive.
4.1 FAILURE-MODE ANALYSIS FOR ERM DUE TO LABEL NOISE
We first provide an analysis of the failure mode for ERM given a finite and noisy dataset without any environment information. Let (ime) be the optimal classifier that only uses the invariant and nuisance features (i.e., the weights for spurious features wspu
(inv) = O, but not necessarily for the invariant component w (((ir) and nuisance component w (im)), Denote the f--norm for the invariant component
(inv)
as ( |]. Similarly, let (s")) and [ || be the optimal classifier that only uses the spurious and
nuisance features (i.e., w w((sp)) = 0) and its associated nn-nom.. We assume that || inv || 2 || (5po) ||- Since norms are usually the complexity measure of the models, this assumption conforms to the practical cases that invariant correlations are usually more complex and harder to learn compared
[i()))
to spurious correlations (recall the cow-camel example in Section []] When ERM is the objective function, both types of classifiers w(in) and ((pu) could reach the global optimum on the training set by learning the weights corresponding to the invariant/spurious features and memorizing all other noisy samples (because both classifiers achieve near O loss and gradient).
has been demonstrated that a classifier optimized over ERM with gradient descent (GD) has the implicit bias of converging to a solution with minimum fg-norm while minimizing the training objective (Zhang et al., 202 la), i.c., the min-norm bias. More details about this on logistic regression can be found in Appendix it suffices to find a classifier that achieves near 0 loss with less norm than Win- We then analyze which rype of classifier has smaller norm and is thus preferred by GD under the noisy and overparameterized linear classification setting. Due to the simplicity bias of deep learning (Geirhos) al. 2020 Shah et al. 2020) and abundant empirical observations, we assume that during training, the classifier first reaches small-norm solution that only uses spurious features Xapu- Thereafter,
To show that the model does not rely only the invariant features,
to further minimize the noisy empirical risk ((),), the classifier either (1) memorizes the noisy data, or (2) gradually picks up the invariant features to correctly classify the remaining noise-free non-spurious data (Arpit et al. 2017 Nagarajan et al. 2020) and potentially become w (in)) If w (inv) has larger norm, is harder for the model to escape the spurious solution using option (2).
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Figure Simulation on synthetic data trained with overparameterized logistic regression. The dotted lines are the means and the shaded regions are for the standard error from 5 independent runs. Figure Ia] shows that the minority-group (worst-group) error increases much more than the majority-group error when label noise is injected. Figure 1b indicates that gathering more data effectively reduces all test errors despite the presence label noise. Figure IC visualizes the learned decision boundaries for the same training set under different 7/, where markers "o" and "x" are for the majority and the minority groups respectively. All data points are colored by the true labels. By adding more label noise, the classifier becomes more skewed towards using the spurious features. Figure ld shows that as more noise is present, (xpu) indeed tends to have smaller norm than w (im)) even though it is bigger when 1j
Theorem 4.2. Under Assumption 4.1 if [w (inx) high probability, [| [| where C is a constant for memorization cost.
||2
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|| 2 n(1 - 7) Cn) then with
For readability, we use C' to represent the average memorization cost. The proof is available in Appendix C.I The theorem describes the condition required for the squared 2n-norm between the invariant component w/( of (lnv) and the spurious component w Wapa of w(spu), such that the fn-norm of the desirable invariant classifier w-(i))) will be larger than the undesirable spurious classifier (ssu), It implies that having a more severe spurious correlation 7 and higher noise level 77 both make the condition in Theorem 4.2 casier satisfy. By the min-norm bias of GD, ((m)) having lower norm than (inn) results in the overparameterized linear classifier to prefer exploiting the spurious correlation appearing in the training set and pay less attention to the invariant features, which hurts the generalization performance on minority groups in the test set. Besides, the theorem has intuitive interpretations of different factors. When the spurious correlation y - 1, the spurious features become invariant for the training environments, and the gap condition in Theorem 4.2 is satisfied (||w
(inv) ||a - || [ s (spu) || Z 0) so that this "newborn" invariant features with less norm would be favored. Furthermore, when the noise level + 0.5, there will be no useful features under this binary setting, and there is no difference between spurious features and invariant features anymore. As the gap condition is also satisfied, the spurious dummy classifier with the least norm is preferred. Lastly, when n grows, the gap condition is enlarged and becomes harder to satisfy. This means that having more data could gradually resolve the harm from both spurious correlation and label noise.
To further illustrate the result, we study the effect of label noise on toy data trained with overparam- eterized logistic regression under the same setting. For each class in {0, 1). the data that can be classified correctly by only Xapa forms majority group, and those that cannot belong to the minority group. We let there be i-dimensional latent variable that controls the generation for invariant and spurious features respectively. We set duvv dupu 5, y 0.99, n 1000, d 3000. The setup modified from Sagawa al. (2000) and more details are available in Appendix
As shown in Figure Ia despite ERM generalizing to minority groups in the test set almost perfectly when it is trained with high spurious correlation on noise-free data, gradually adding noise degrades the minority (worst-case) performance significantly. Moreover, label noise increases the classifier's dependence on the spurious features, as the decision boundary becomes more tilted in Figure Ic] This complements the previous analyses that attribute the failure mode of ERM to the geometric skew (Nagarajan et al., 2020) of the max max-margin classifier in the noise-fre free setup. Our study shows that adding label noise also intensifies the skew. Since the classification error for the minority group goes up gradually, one interpretation is that adding more noise gradually reduces the number of effective data points from the minority groups for invariance learning. Consequently, sampling more data
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could help mitigate the effect of label noise and reduce test error the minority groups, and verify that by adding more data shown in Figure [b Thus, data collection and augmentation are still the keys to improving domain generalization. In addition, we train the classifiers w(in)) and (spu) and plot the related f2-norms w.r.t. increasing noise level in Figure Id As the noise level goes up. ((s")) indeed tends to have smaller norm than w(im)) At the same time, the change in norm gap (inv) is relatively small. Moreover, the classifier w that uses all features also gets closer to the spurious classifier w( It also hints that fz-regularization may not help with ERM for better robustness to subpopulation shifts when the label noise is present.
between W (spu) and w/m)))
5 THE IMPLICIT NOISE ROBUSTNESS OF DG ALGORITTMS
We analyze few exemplary baseline algorithms used for DG on their noise robustness. In particular, invariance learning (IL) algorithms such IRM and V-REx have better label-noise robustness by providing more efficient cross-domain regularization. We show the analysis for IRM and leave the rest in Appendix [ Let the classifier be f = w o b where w is linear and O is the features/logits extracted from the imput x. The practical surrogate for IRM objective on the training set is:
RIRMvl Eeec, (w o 4.) + >/ who===1. Re (w &e))l
where À the regularization hyperparameter. In particular, IRM aims to learn representations that are local optima for all environments in addition to minimizing the sum of all risks. Suppose that these representations are invariant. Intuitively, since the noisy data need to be memorized independently. each of such costly memorization is non- oon-inariannt across domains and induces extra penalties on the regularization terms for domain invariance. For binary classification trained with the logistic loss (equivalent to the cross-entropy loss), the gradient for IRMv1 can be written as:
V & (1 2X (6)) + tha (0) - (()) - y]) VoR. V
where @ is the predictive logit for instance s. a is the sigmoid function, and VoR. is the ERM gradient w.r.t. the model parameters. As we have rewritten the IRM gradient as ERM gradient multiplied by a coefficient a()), we can compare the two optimization trajectories directly. We plot the coefficient function a w.r.t. logit ò in Figure 2 Since a is symmetric w.r.t. y 0 and y = 1, we only need to analyze for y = 1. During training, e grows for y 1 with ERM gradient. However, when & is larger than a certain threshold (inversely proportional to the regularization strength A), a(()) becomes negative, which reverses the gradient direction and does the opposite of minimizing the ERM loss by decreasing ø. This behavior with strong regularization generally prevents the model from predicting training instances with high probabilities and memorizing the noisy samples with overvonfidence. When the regularization for IRM is sufficiently strong (A > 100 in practice), the loss for a single sample would oscillate back and forth instead of saturating. This is part of the reason that focus alyzing the gradients of the algorithms trained on finite samples instead of performing conver- gence analysis. The gradient for V-REx has similar property and we leave the analysis in Ap- pendix On the other hand, the objectives for ERM. Mixup, and GroupDRO do not explicitly penalize memorization. As long as the model capacity allows, memorizing noisy data always leads to lower training loss and is encouraged by gradient descent. Thus, they are likely to have poorer noise robustness and worse general- ization. However, since GroupDRO prioritizes optimizing the worst group, it generally slows down noise memorization in the short horizon. We will verify these claims in Section 6
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6 Is NOISE ROBUSTNESS NECESSARY IN PRACTICE?
We empirically investigate how label noise affects OOD generalization across various simulated and real-life benchmarks. We adopt the Domainbed framework (Gulrajani & Lopez-Paz 2020 with

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Figure 3: Simulation on CMNIST dataset. As the noise level r increases, both IRM and V-REx exhibit better generalization and noise robustness compared to approaches with ERM objectives.
standard configurations and add varying levels of label noise to the training environments of the datasets. We assume the availability of small validation set from the test distribution for model selection. For a fair and comprehensive comparison, Domainbed implements 3 trials of 20 sets of hyperparameters, which may not be sufficient to reproduce the best results for all algorithms. We evaluate the robustness to label noise of the DG algorithms based on the average OOD accuracy on the test set of the pest-performing models (selected by the validation set) across trials.
6.1 DATASETS
Subpopulation shifts (Synthetic). CMNIST (ColoredMNISTT) (Arjovsky et al. 2019) is a synthetic binary classification dataset based on MNIST (LeCun 1998) with class label {0, Ij. There are four groups: green 1s (g1). red Is (g2). green Os (gs). and red Os (g1). The color of digit has a high spurious correlation y with the label in the training set, where most of the instances are from Φ1 and 94. which can be viewed as the majority groups. There are two training environments 61, 62 with different degrees of spurious correlations 71 0.9, 72 0.8, meaning that using only color achieves 85% training accuracy. However, the minority groups G2, 93 dominates the test set with "festt 0.1.
Subpopulation shifts (Real-world). Waterbirds (Wah al., 2011 Sagawa et al. 2019) is binary bird-type classification dataset that has the label spuriously correlated with the image background. There are four groups: waterbirds on water (91), waterbirds on land (G2). landbirds water (9g), and landbirds on land (G.). The majority groups are G1 and G1. and the minority groups are G2, G3. Similarly, CelebA (Liu et al. 2015) Sagawa al.) 2019) is a binary hair color prediction dataset and blondness is much less correlated with the gender male than female, so blond male is the minority group and the rest are the majority. Based on these two datasets, we consider two types of simulated environments: (1) Waterbirds and CelebA: Directly treat the 4 groups as 4 environments, which similar to the common setups, but we allow all algorithms including ERM to access the group information. However, this setup can be unfeasible when the group information is not available. (2) Waterbirds+ and CelebA+: To simulate a more realistic setting, we create two derivative datasets based Waterbirds and CclebA. There are two environments e c2. Each environment has half the data from each majority group. Then for each minority group, 1/3 of the data is allocated to e and the rest 2/3 is allocated 62. This resembles the CMNIST setup so that e2 has twice the amount of the minority data as éj. This setting is less restrictive and the difference in the degree of spurious correlations represents type of subpopulation shift. We also extend the result to the language modality using CovilComments (Borkan et al. 2019) in Appendix F
Domain shifts (Real-world). Based on Domainbed (Gulrajani & Lopez-Pate 2020), we use PACS, VLCS, OfficeHome, and TerraIncognita Each dataset has multiple environments. Each environ- ment has specific populations that have unique features that are different from the ones other environments. This is distinct from subpopulation shifts where only the composition of the population changes across environments. We leave the full descriptions of all datasets in Appendix E.
6.2 EXPERIMENTS ON SYNTHETIC CMNIST DATA
For CMNIST, we train simple Convolutional Neural Network (CNN) models for 5k steps. Only models at the last epoch are compared for validation performance (i.e., no early stopping).
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As shown in Figure 3a when the noise level 17 is close O, ERM performs competitively even without using the environment label information compared to other DG algorithms. Since on average y 0.85 in the training set, the degree of spurious correlation is not severe enough to make ERM fail in the noise-free case. However, when the noise level goes up. II. algorithms (IRM, V-REx) significantly outperform ERM-based algorithms (ERM, Mixup). The performance gap is enlarged when the noise level 77 gets higher because the accuracy of ERM-based algorithms deteriorates much faster. On the other hand, GroupDRO is slightly behind IRM and V-REx but has clear advantage over ERM. Moreover, we track the accuracy of the noisy data as a surrogate of noise memorization in Figure Sa which shows that DG algorithms naturally are more robust to label noise memorization compared to vanilla ERM-based approaches. Furthermore, strong negative correlation can be observed between noise memorization and the test error.
To further examine noise memorization, we "overfit" CMNIST dataset with 25% label noise by extended training for 20k steps in Figure(3b) II. algorithms such as IRM, V-REx with strong invariance regularization do not memorize the noisy data severely, despite the complete memorization still being a valid local minimum (all environments have O and equal loss). On the other hand, ERM, Mixup, and GroupDRO suffer from noise memorization in the long horizon. The phenomenon of GroupDRO having contrasting results when trained with different steps aligns with the previous conclusion that GRoupDRO works well with strong regularization such early stopping (Sagawa al. 2019).
6.3 EXPERIMENTS ON WORLD DATA
We add 10% and 25% label noise to the benchmark datasets and compare the out-of-distribution performance for various algorithms. For all datasets, we use ResNet-50 (He et al., 20T6) pretrained on ImageNet (Deng et al. 2009). For the real-world subpopulation-shitf datasets, worst-group (WG) accuracy of the model on the test set selected according to the validation WG performance. For the domain-shift datasets, we perform single-domain cross-test experiments by setting each domain as the test set and report the average accuracy of the model selected using 20% of the test data. Standard data augmentation is performed. More details available in Appendix E
report the
Dataset
Waterbirds+
Waterbirds
CelebA+
CelebA
Alg/ /g/n
0.1
0.1
0.1
0
0.1
FRM
81.5+to.4
64.6+to.s
86.7 86.7+1.0
61.4+1.3
71.7+
65.4+1.9
88.1+1.2
63.1
Mixup
79.8+to.9
61.8+220
88.0
56.9
71.1
64.6
86.9
61.7
DRoupDRO
82.6.5.44
65. 65.61p.s
84.91...7
61.9
71.912..
61.721.7
87. 87.6
63.0
IRM
78.51:0.
63.3
87.0
58.81o..
75.4
62.4
88.0 88.020.4
59.31117
V-REx
78.3
66.6-to.9
87.4
59.2+1..
73.9
69.9-b2.
89.3
60.2
Table I: Worst-group accuracy (%) for subpopulation shifts.
Dataset
PACS
VLCS
OfficeHome
TerraIncognita
Alg/n) /)
0.1
0.25
0.1
0.25
0.1
0.25
0.1
0.25
ERM
82.0
74.5:bo.6
75.0
71.9
62.2.b..1
54.9:bb.3
52. 52.1
48.5:
Mixup
83.6
75.2
75.5
71.9
63.9
57.5
53.2
51.3
GroupDRO
82.4+oo.3
74.7+h0.4
75.1
71.2
61.3
54.1
51.8
50.6
IRM
80.4+1..
71.0
74.6
70.3
61.2
53.9th1.
48.3
44.4
VREx
81.4+o..2
73.5+t..7
75.0+t..1
71.8
60.6
53.0+ho.9
48.3
48.9+o.4
Table 2: Cross-test accuracy (%) for domain shifts.
For real-world subpopulation-shift datasets in Table 1 the ranking of different algorithms varies across different datasets and noise levels. Despite being prone to spurious correlation and label noise, ERM has decent performance given the environments. The superiority of IL algorithms on CMNIST does not translate into real-world datasets. For real-world domain-shift datasets, adding label noise degrades the performance of all tested algorithms all other algorithms consistently, followed closely by ERM. This indicates the effectiveness of data augmentation for OOD generalization under label noise. Despite being noise-robust, IRM and
similar scale. Surprisingly. Mixup outperforms
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VREx rarely catch up with ERM. Thus, there is no clear evidence that the noise robustness property necessarily leads to better performance in weal-world subpopulation-she shift and diomain-shil datasets. In practice, the choice of DG algorithms depends heavily the dataset and hyperparameter search. An effective model selection strategy could be more important than an algorithm.
For Waterbirds and CelebA, our GroupDRO results are much lower than the reproducible baseline (Sagawa et al. 2019), showing that the current hyperparameter search space can be far from complete. However, by utilizing the group labels, our results for ERM, IRM, and V-REx are already significantly higher than the reported scores from prior works (Yao et al., 2022), setting new baseline records. Given that ERM and DG algorithms perform similarly under the current setting, it may be possible that ERM can be improved further by better model selection and broader hyperparameter search.
7 DISCUSSION
Why does ERM objective perform competitively on real-world subpopulation-shif datasets? (1) The real-world training is usually coupled with model pretraining and data augmentation. With higher-quality representations and more augmented data, the learning requires much fewer training steps. which prevents noise memorization. We observe that even with strong spurious correlation > 0.95, noise memorization is less severe for all algorithms including ERM on Waterbirds(+)) dataset in Appendix H (2) The failure condition based on spurious correlation and label noise may not be satisfied in practice. Without knowing the group labels. the Waterbirds+ and CelebA+ datasets both exhibit strong spurious correlations between some input features and the labels. However, it is unclear how much more difficult it is to learn the invariant features compared to the spurious ones, which relates to the norm difference between the spurious and invariant classifiers. For example, in CelebA, the label hair blondness is spuriously correlated with the feature gender, but the gender feature could be more difficult to learn than the hair blondness, unlike CMNIST where the spurious feature color is real- world
more straightforward for the model to discover. Thus, the issue of spurious correlation datasets may be less severe. (3) The invariance learning condition may not be satisfied. IL algorithms require training environments to have sufficient distributional differences for the subpopulations to distinguish invariant features. This condition might be true for CMNIST (?1 72 = 0.1), ), but not necessarily for Waterbirds+ and CelebA+ with a less-than-255 difference in 7-
Does spurious correlation affect domain shifts? In domain shifts, it is common to assume there are coomain-speciff features that have spurious correlations with the labels. However, these spurious correlations picked up from the training environments are unlikely to be present in new domains. If this correlation is not so strong, we expect some level of meaningful representations can be learned by ERM and yield reasonable performance for test distributions. Moreover, when there are multiple training domains available that come with non-trivial spurious correlations, all the domain-speciff spurious correlations combined may result in even more complexity and become a worse burden for the classifier to learn compared to the invariant features. We hypothesize that adding data from more domains is likely to encourage invariant feature learning even for just ERM.
8 CONCLUSION AND FUTURE WORK
Label noise constitutes another failure mode for ERM by exacerbating the effect of spurious correla- tions. Invariant learning algorithms exhibit a clear advantage of noise robustness over ERM-based algorithms in certain synthetic circumstances. Unfortunately, such desirable property does not trans- late to better generalization in real-world situations. ERM objective with effective data augmentation still constructs simple yet competitive baseline. We believe that more theoretical and empirical analyses are needed to understand when and why ERM and DG algorithms work and fail.
In practice, it is uncertain whether the invariant and spurious features can be perfectly extracted. Subsequently, the invariant features may violate the linear separability. As such, our theoretical analysis may not be directly applicable in the real world. A more general understanding can be obtained by removing these assumptions. Furthermore, though the spurious correlations may be non-trivial to learn for some datasets, certain types of adversarial attacks may create similar conditions as in corrupted CMNIST, where invariance learning might provide stronger resilience. The implicit noise robustness of II. algorithms could also be beneficial to learning with noisy labels.

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9 ACKNOWLEDGEMENT
We would like to thank the anonymous reviewers and AC for the constructive and helpful feedback. We thank Zheyuan Hu for the helpful discussions. This research/project supported by the National Research Foundation, Singapore under its Al Singapore Programme (AISG Award No: AISG2- PhD/2021-08-017(TT)
10 REPRODUCIBILITY STATEMENT
For our theoretical analysis, we have stated the assumptions in Section 4 and given the proofs in Appendix C For our empirical results, we have documented the complete experimental setup in Appendix We have also submitted our code with the necessary instructions to reproduce our results.
REFERENCES
Muhammad Aurangzeb Ahmad, Carly Fekert, and Ankur Teredesai. Interpretable machine learning in healthcare. In Proceedings of the 2018 ACM International Conference on Bioinformatics, Computational Biology, and Health Informatics, pp. 559-560, 2018.
Kartik Ahuja, Ethan Caballero, Dinghuai Zhang, Jean-Cheee Christophe Gagnon Audet, Yoshua Bengio, loannis Mitliagkas, and Irina Rish. Invariance principle meets information bottleneck for out-of- distribution generalization. In Advances in Neural Information Processing Systems, pp. 3438-3450, 2021.
Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv: 1rXiv:/907.02889) 2019.
Devansh Arpit, Stanislaw Jastrzębski, Nicolas Ballas, David Krueger, Emmanuel Bengio. Maxinder S. Kanwal, Tegan Maharaj, Asja Fischer, Aaron Courville, Yoshua Bengio, and Simon Lacoste-Julien. A closer look at memorization in deep networks. In International Conference Machine Learning, pp. 233-242, 2017.
Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. In Proceedings of the European Conference on Computer Vision, pp. 456-473, 2018
Daniel Borkan, Lucas Dixon, Jeffrey Sorensen, Nithum Thain, and Lucy Vasserman. Nuanced metries for measuring unintended bias with real data for text classification. In Companion proceedings of the 2019 world wide web conference. pp. 190-500, 2019.
Junbum Cha, Sanghyuk Chun, Kyungjae Lee, Han-Cheol Cho, Seunghyun Park, Yunsung Lee, and Sungrae Park. Swad: Domain generalization by seeking flat minima. In Advances in Neural Information Processing Systems, pp. 22405-22418, 2021.
Kamalika Chaudhuri, Kartik Ahuja, Martin Arjovsky, and David Lopez-Paz. Why does throwing away data improve worst-group error? International Conference on Machine Learning. 2023.
Yimeng Chen, Ruibin Xiong, Zhi-Ming Ma. and Yanyan Lan. When does group invariant learning survive spurious correlations? In Advances in Neural Information Processing Systems, pp. 7038- 7051, 2022.
Elliot Creager, Jörn-Henrik Jacobsen, and Richard Zemel. Environment inference for invariant learning. In International Conference Machine Learning, pp. 2189-2200, 2021.
Alexander D' Amour, Katherine Heller, Dan Moldovan, Ben Adlam, Babak Alipanahi, Alex Beutel, Christina Chen, Jonathan Deaton, Jacob Fisenstein, Matthew D Hoffman, et al. Underspecification presents challenges for credibility in modern machine learning. The Journal of Machine Learning Research, 23(1): (23):10237-10297, 2022.
Jia Deng, Wei Dong. Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 248-255, 2009.
10

---

Published as a conference paper at ICLR 2024
Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. Fairesss through awareness. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, pp. 214-226, 2012.
Chen Fang, Ye Xu, and Daniel N Rockmore. Unbiased metric learning: On the utilization of multiple datasets and web images for softening bias. In Proceedings of the IEEE International Conference Computer Vision, pp. 1657-1664, 2013.
Irena Gao, Shiori Sagawa, Pang Wei Koh, Tatsunori Hashimoto, and Percy Liang. Out-of-domain robustness via targeted augmentations. arXiv preprint arXiv:2302.11861, 2023.
Robert Geirhos, fom-Henrik Jacobsen, Claudio Michaelis, Richard Zemel, Wieland Brendel, Matthias Bethge, and Felix A Wichmann. Shortcut learning in deep neural networks. In Nature Machine Intelligence, volume 2, pp. 663-673. Nature Publishing Group UK London, 2020.
Ishaan Gulrajani and David Lopez-Paz. In search of lost domain generalization. arXiv preprint arXiv:2007.01434, 2020.
Moritz Hardt, Eric Price, and Nati Srebro. Equality of opportunity in supervised learning. In Advances in Neural Information Processing Systems, 2016.
Tatsunori Hashimoto, Megha Srivastava, Hongseok Namkoong, and Percy Liang. Fairness without demographics in repeated loss minimization. In International Conference on Machine Learning, pp. 1929-1938, 2018.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Putlern Recognition, pp. 770-778, 2016.
Badr Youbi Idrissi, Martin Arjovsky, Mohammad Pezeshki, and David Lopez-Paz. Simple data balancing achieves competitive worst-group-aceurad In Conference on Causal Learning and Reasoning, pp. 336-333663333 1566-351,22 2022.
Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson, Av- eraging weights leads to wider optima and better generalization. arXiv preprint 1rXiv:1803.05407, 2018.
Pavel Izmailov, Polina Kirichenko, Nate Gruver, and Andrew G Wilson. On feature learning in the presence of spurious correlations. In Advances in Neural Information Processing Systems, pp. 38516-38532 2022.
Ziwei Ji and Matus Telgarsky. Risk and parameter convergence of logistic regression. arXiv preprint arXiv:1803.07300 2018
Fereshte Khani and Percy Liang. Feature noise induces loss discrepancy across groups. In Interna- tional Conference on Machine Learning, pp. 5209-5219, 2020.
Fereshte Khani and Percy Liang. Removing spurious features can hurt accuracy and affect groups disproportionaaelyy In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, pp. 196-205. 2021.
Polina Kirichenko, Pavel Izmailov, and Andrew Gordon Wilson. Last layer re-training is sufficient for robustness to spurious correlations. arXiv preprint wrXiv:2204.02937, 2022.
Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Bal- subramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al. Wilds: A benchmark of in-the-wild distribution shifts. In International Conference on Machine Learning, pp. 5637-5664, 2021.
Masanori Koyama and Shoichiro Yamaguchi. Out-of-distributioon generalization with maximal invariant predictor. 2020.
David Krueger, Ethan Caballero, Joemn-Henrkk Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi Le Priol, and Aaron Courville. Out-of-distributtin generalization via risk extrapola- tion (rex). In International Conference on Machine Learning, pp- 5815-5826, 2021.

---

Published as a conference paper at ICLR 2024
Yann LeCun. The mnist database of handwritten digits. http://yann. lecun. com/exdb/nmist// 1998.
Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M Hospedales. Deeper, broader and artier domain generalization. In Proceedings of the IEEE International Conference on Computer Vision, pp.
5542-5550, 2017.
Pan Li, Da Li, Wei Li, Shaogang Gong, Yanwei Fu, and Timothy M Hospedales. A simple feature the IEEE/CVF International Conference
augmentation for domain generalization. In Proceedings the Computer Vision, pp. 8886-8895, 2021.
Weixin Liang, Yining Mao, Yongchan Kwon, Xinyu Yang, and James Zou. Accuracy the curve:
On the nonlinear correlation of ml performance between data subpopulations. In International Conference Machine Learning, pp. 20706-20724, 2023.
Evan Z Liu, Behzad Haghgoo, Annie S Chen, Aditi Raghunathan, Pang Wei Koh, Shiori Sagawa, Percy Liang, and Chelsea Finn. Just train twice: Improving group robustness without training group information. In International Conference Machine Learning, pp. 6781-6792, 2021a.
Jiashuo Liu, Zheyuan Hu, Peng Cui, Bo Li, and Zheyan Shen. Heterogeneous risk minimization. In International Conference on Machine Learning, pp. 6804-6814, 2021b.
Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes the wild. In Proceedings of the IEEE International Conference on Computer Vision, pp. 3730-3738, 2015.
Vaishnavh Nagarajan, Anders Andreassen, and Behnam Neyshabur. Understanding the failure modes of ofi-t--distribution generalization. arXiv preprint arXiv:2010.15775, 2020.
Junhyun Nam, Hyuntak Cha, Sungsoo Ahn, Jacho Lee, and Jinwoo Shin. Learning from failure: De-biasing classifier from biased classifier. Advances in Neural Information Processing Systems, 23:20673-20684, 2020.
Andrew Y Ng. Feature selection, 11 vs. 12 regularization, and rotational invariance. In International Conference Machine Learning, pp. 78, 2004.
Giambattista Parascandolo, Alexander Neitz, Antonio Orvieto, Luigi Gresele, and Bernhard Schölkopf. Learning explanations that are hard to vary. arXiv preprint arXiv:2009.00 00000, 2020.
Mohammad Pezeshki, Oumar Kaba, Yoshua Bengio, Aaron C Courville, Doina Precup, and Guil- laume Lajoie. Gradient starvation: A learning proclivity in neural networks. In Advances in Neural Information Processing Systems. pp.
2021.
Alexandre Rame, Corentin Dancette, and Matthicu Cord. Fishr: Invariant gradient variances for out- of-distribution generalization. In International Conference on Machine Learning, pp. 18347-18377, 2022.
Elan Rosenfeld, Pradcep Ravikumar, and Andrej Risteski. The risks of invariant risk minimization. arXiv preprint arXiv:2010005761, 2020.
Elan Rosenfeld, Pradeep Ravikumar, and Andrej Risteski. Domain-adjuster regression Erm may already learn features sufficient for out-of-distribution generalization. arXiv preprint 20Xi::2202.068866, 2022.
Saharon Rosset, Ji Zhu, and Trevor Hastie. Margin maximizing loss functions. In Advances in Neural Information Processing Systems, 2003.
Shiori Sagawa, Pang Wei Koh, Tatsunori Hashimoto, and Percy Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. arXiv preprint
2019.
Shiori Sagawa, Aditi Raghunathan, Pang Wei Koh, and Percy Liang. An investigation of why
overparameterrizatoon exacerbates spurious correlations. ArXiv, abs/2005.06 00445, 2020.
Harshay Shah, Kaustav Tamuly, Aditi Raghunathan, Prateek Jain, and Praneeth Netrapalli. The pitfalls of simplicity bias neural networks. In Advances in Neural Information Processing Systems, pp. 9573-9585, 2020.
12

---

Published as a conference paper at ICLR 2024
Yuge Shi, Jeffrey Seely, Philip HS Torr, N Siddharth, Awni Hannun, Nicolas Usunier, and Gabriel Synnaeve. Gradient matching for domain generalization. arXiv preprint arXiv:2104.09937, 2021.
Daniel Soudry, Elad Hoffer, Mor Shpigel Nacson, Suriya Gunasekar, and Nathan Srebro. The implicit bias of gradient descent on separable data. The Journal of Machine Learning Research, 19(1): 2822-2878, 2018.
Baochen Sun and Kate Saenko. Deep coral: Correlation alignment for deep domain adaptation. In ECCV Workshops, 2016.
Vladimir Vapnik. The nature of statistical learning theory. Springer science & business media, 1999.
Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5018-50100 8-5027, 2017.
Vikas Verma, Alex Lamb, Christopher Beckham, Amir Najafi, Ioannis Mitliagkas, David Lopez-Paz, and Yoshua Bengio. Manifold mixup: Better representations by interpolating hidden states. In International Conference on Machine Learning. pp. 6438-6447 2019.
C. Wah, S.Branson, P. Welinder, P. Perona, and S. Belongie. The caltech-ucsd birds-200-2011 dataset. California Institute of Technology, 2011.
Jialu Wang. Yang Liu, and Caleb Levy. Fair classification with group-dependent label noise. In Proceedings of the 2021 ACM conference on fairness, accountability, and transparency, pp. 526-536, 2021.
Yuzhe Yang, Haoran Zhang, Dina Katabi, and Marzyeh Ghassemi. Change is hard: A closer look at subpopulation shift. In International Conference Machine Learning, 2023.
Huaxiu Yao, Yu Wang. Sai Li, Linjun Zhang, Weixin Liang, James Zou, and Chelsea Finn. Improving out-of-distribution robustness via selective augmentation. In International Conference on Machine Learning, pp. 25407-25437, 2022.
Nanyang Ye, Kaican Li, Haoyue Bai, Runpeng Yu, Lanqing Hong, Fengwei Zhou, Zhenguo Li, and Jun Zhu. Ood-bench: Quantifying and understanding two dimensions of out-of-distribute generalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 7947-7958, 2022.
Ekim Yurtsever, Jacob Lambert, Alexander Carballo, and Kazuya Takeda. A survey of autonomous driving: Common practices and emerging technologies. IEEE access, 8:58443-58469, 2020.
Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning (still) requires rethinking generalization. Communications of the ACM, 64(3):107-115, 2021a.
Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv: 1710.0949 2017.
Michael Zhang, Nimit S Sohoni, Hongyang R Zhang, Chelsea Finn, and Christopher Ré. Correct-n- contrast: A contrastive approach for improving robustness spurious correlations. arXiv preprint arXiv:2203.01517. 2022.
Xingxuan Zhang, Peng Cui, Renzhe Xu, Linjun Zhou, Yue He, and Zheyan Shen. Deep stable learning for dfist---iittibution generalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5372-5382, 2021b.
Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Antonio Torralba. Places: A 10 million image database scene recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 40(6):1452-) 1464, 2017.
Xiao Zhou, Yong Lin, Renjie Pi. Weizhong Zhang, Renzhe Xu, Peng Cui, and Tong Zhang, Model agnostic sample reweighting for out-of-dit nat-of-distribution learning. In International Conference on Machine Learning, pp. 27203-27221, 2022.
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A DG ALGORITHMS
In this work, we will analyze some representative algorithms including Invariance Learning (IL) (Arjovsky et al. 2019
Krueger et al. (222)) and Distributionally Robust Optimization (DRO) (Sagawal w o 4, where w is linear and & is the features extracted from the input x. Let Exp be the training environments. For IL, two popular risk objectives are:
et al. 2019). Let f
RIRM Ecee, R. o Φ) V-REX Eeee, Re(f) + eVeEE. (Ree....R.(()
s.t. aig Re +), Ve Etr
where V[Re (f) the variance of risk across environments and À the hyperparameter. For DRO, one of the risk objectives is the worst-case loss over all training environments:
R = maxeeE.. (J)-
(2)
B NOISE ROBUSTNESS ANALYSIS
B.I GRADIENT ANALYSIS FOR IRM ALGORITHM
We show this by analyzing the gradient of IRM for finite samples instead of doing convergence analysis. Recall that the IRM objective function is:
R
IRM
Σ
Re +)
s.t. w min R (1 L), Ve E Eir
CEEtr
The goal of IRM is to obtain classifiers that are simultancously optimal across all training environ- ments, but this induces bilevel optimization problem that is hard optimize. As zero gradient is necessary for reaching the local minimum, IRM has the practical surrogate that minimizes the norm of the gradient encourage reaching the local minimum for all training environments. Since goal to understand how the DG algorithms work in practice, we only analyze the practical surrogate of IRM, which the de facto implementation across standard benchmarks. The practical surrogate called IRMv1 has the form:
R IRM
Σ eEEir
Re(w 4e) A/V w/ w- 1.0 (whb
(3)
For cach environment
Σ R.(w o p,) Re o b2) n
1
(4)
The gradient w.r.t. 1.0 for environment is scalar that can be rewritten as:
RR
ar | 10|w=10 Re
)T
aΦ
1
JΦ. +
For simplicity, focus on IRMvl and consider the binary classification case with logistic regression.
The input x E Rd and the label {0. 1}. We analyze with logistic regression due to its simple form and most of the analysis can be done with scalars. Moreover, it is widely known that logistic regression is equivalent to training with cross-entropy loss, so there is no loss of generality. Logistic regression has the form:
ply 1|x) (0Tx)) where & are the weights of the linear classifier, and a is the sigmoid function:
(xx) =-x
à= I, which corresponds to /b in previous formulations. The IRM-ready logistic regression simply:
ply 1 1/=) a (wo))
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-20
0.
0.5
20
20
-15
-10
-05
0.0
(a) y 1, 10
(h) y 0, A
10
(c) y 1, varying À
(d) 0, varying X
Figure 4: IRM gradient coefficient w.r.t. regularization strength
where w = 1 and we will drop it in the following analysis unless it is needed in the term. The IRM-ready logistic loss for an environment:
L y log o (wo) - (1 - y) log(1 (1 - o (wo)).
The partial gradient w.r.t. linear weights A:
aL(0)
(o(Ø) - y)x.
00
The partial gradient for w used by IRMv1:
V www== ((0) (o( y)
Let Le be the logistic loss for environment e. The IRMvl objective is:
L ZIRM
Le()) A/| /w== L (0)
For simplicity, we just analyze the behavior of IRMv1 for a single environment, so we abuse the notation and the subscript can be dropped temporarily. The gradient for IRMv1 in single environment for one sample (xz, y) is:
ØLRM
(l)
dAA V (0]]=
00
00
00
= (a y) + 2x((()) (0) + - - y|æ
(a(Ø) y)=(1 + 2xo(o(0) (10))) o()) y))
(1+ + ( (0) -o()) - y1) (l)
00
(d) (L)
09
We have re-expressed the gradient of IRM in terms of the gradient of the original logistic function multiplied by new coefficient function This new coefficient (σ)) for the gradient has interest- ing property that restricts the predictive probability to non-extreme values when the regularization strength À sufficiently large. We plot the coefficient function w.r.t. Φ in Figure 4 The function is symmetric with respect to label Without loss of generality, we only look at the case for 1. Recall that for logistic regression, we predict positive if 6 > To minimize the logistic loss for an instance (z,, 3:) where y: = 0. ø would keep growing because it increases p(y)(z,). The coefficient needs to stay positive for gradient descent. However, as ø increases, a becomes negative before @ is large enough, which blocks the logistic loss to continue decreasing and severe memorization. Moreover, a larger leads to stronger resistance during the optimization process as shown in Figure The same rationale applies to the 0 case due to the symmetry. In practice, À (usually > 100) tends to be very large in order to work well on datasets like CMNIST. Thus, for IRM, data points may be oscillating between gradient descent and gradient ascent (w.nt. the ERM case), and do observe that IRM loss typically does not converge on the best persforrfor models. We hypothesize that only classification patterns that work for most of the data could cross the hill and get preserved.
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B.2 GRADIENT ANALYSIS FOR V-REX ALGORITHM
Recall that the V-REx loss function is:
V-REX
Σ
Ro(s) + AVEEE, (R((I,
[Etr]
eEEtr
A m
1 1 Σ R.(f).
m
(f) RR)² -R)²
m
where R. = (1/m) 1/mm)ree, R. is the average loss of all training environments. For specific environment i Etr, can rewrite the loss as:
m
m Re(
R
V-REx
-R)²
m
m
+R(()+ 1 Σ Re(l). m e=1,efi
(R,(f)- - R)²
(
m
e=
The gradient incurred from R, can derived as follows:
Ö
V-R
ÖR,
00
OR;
00
2A
(R. R1(1 - Σ (Re RR(
1
2A
1 8R,
+ 00
e=1,e
2A
(R+(J) R) + 21
1 0R:
(Rel)) R)(-
00
2A
(R:(f) - R) + 1 0R,
00
(2(R((f) - R) + 1] OR;
00
The overall gradient for the network parameters can be expressed as:
V-REx
V-REx
Σ
a0
a0
OR.
(2(R.()) - R) + 1]
00
In practice, the strength of regularization for invariance learning tends to be very large (A > 100) for the better-performing models. For environments with below-average risk where R. < R. The coefficient will become negative, and gradient ascent instead of descent performed for this batch of data points. We can view such gradient ascent as an "unlcarning" process. We hypothesize that since memorization is gradual process and only decreases the risk for its own environment, making it below average, its effect can be purged later on by gradient ascent just as other spurious features. As result, only the invariant features that useful to all environments preserved.
C PROOFS
C.1 PROOF OF THEOREM 4.2
We first look at the memorization cost for overfitting the noisy data points.
Lemma C.1. For an overparameterized linear classification problem with label noise, with high probability, the squared norm of the classifier |w| will be increased by 3 d on average for
memorizing a single sample (mislabeled or non-spuriously correlated) in the training set, where 2 is constant.
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Proof. First, by the definition of fg-norm:
| 1| = || 1| + [[ + |miil
(5)
Since Wiw and Wauu can both result in relatively low classification error, memorization will mostly incur larger norm the term |/Wauill?.
Recall that Xuui ~ (0, mmi) and daui >> n. Vectors sampled from high-dimensiona Gaussian
spheres are orthogonal to each other with high probability: Vi + j. X 1 x To memorize a set of data points S using only the nuisance features, it is then sufficient to construct w (8) Eies By x mui where 3 and more importantly:
(8)
32
|
(6)
By Bernstein's 's inequality for the norm of subgaussian random variables:
2 S 2cxp (- cdnui min(t? (f,t) .t)
P
( | 12
B[|
E
nui
(1)
12
K 2() S 2 exp cdnui min .t)
P ||xX(1) (2
o mui
Ka1
where K max, [ ds and e is absolute constant. Thus, with high probability, l/ nui I|2 concentrates on 3 [S mui- and the average cost of memorizing data point is d nui In particular, the probabilistic bound becomes tighter as dnui increases, which corresponds to the model becoming more overparameterized.
We first compare the purely spurious classifier (spr)) and the purely invariant classifier (im)) under the effect of noise memorization. Let [a: b] be the concatenation of vector a and b vertically. Let L be the logistic loss. The classifiers can be defined mathematically as:
(spu)
arg min [w Wall su]
Ily [Wapus Wnui]
[x
w (inv)
arg
min [Wir Wall
N Σ ((y [Wanvi Wmul [x x ((il))
Theorem C.2. If
[w (inv)
inv 1|2
|w
(spu) sqi
1|a 2 n11 (1 21)) dnuio then [[vv > || ||.
Proof. With label noise level n, the number of data points that need to be memorized using the optimal set of invariant features is:
Ninv = my
The number of data points that need to be memorized using the 7-correlated spurious feature is: Tus n(1 7 (2 1)n)
Then approximately,
|| w (w.(ine) (inv)) 1|2
w (inv)
||2 + w (inv) ( 1|2
~ w inv
finv)
1|2 + nnn,3? d 2
nui
(spu)
||
w (spu)
||2 + || w (spa)
(s(pp)))
spa
nui
(spu)
112 n0(1 + (12)-))))8 1)1))33 (2nn))ddmuute dmui
spu
Then rewrite the gap as:
/ ( (imv)
||w (1mi)) 1
(inv)
||2 + n d nui (11
(l/w/spu))p 1|2 + rn(1 - 7 (27 1)m) o
inv
spu
w (inv)
2 w (spu)
nnn33² 3 d aui nn(1 7 (2 1 1)))) 32 d pui
spu
w (ine)
2|
w (spu) spu
2 n2( 7 + (2 32
w inv
(ime)
||2
|| spu
(spu)
2
n( 1)(2n ddmii nu
w
(inv)
|| spu
(spn) (smn) 2
n1( 7) 2 2
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Suppose that ||w/(m)) || // ||. then:
inv || || /w(m spu
(inv)
(xpu))
2 n11
7) 2r dnui nui 2
|| w || spa 11 2 n(1 (1 2x) dmui
[imx)
(spu)
By the above equation, it is easy to see that larger 7 or 17 both lead to smaller difference required between [[w (imv) inv 1 and [[w.grr)) to have [| 2 [| As our classifier is implicitly
regularized to prefer a decision boundary with smaller norm, the converged solution becomes more
likely to rely on spurious correlation, resulting in significantly worse generalization performance. Q
Remark C.3. It is possible that the classifier that only uses the spurious features may achieve a smaller norm by replacing the memorization of certain data points with invariant feature learning. This is likely to be true in practice. However, our objective here is to identify at least one suboptimal candidate classifier that is preferred by ERM to the invariant classifier. Moreover, the classifier with both spurious and partially invariant features will have an smaller norm than || resulting in the model preferring this model more than the purely spurious classifier and thus also not converging the purely invariant classifier.
C.2 THE INDUCTIVE BIAS TOWARDS SANIMUM-NORI SOLUTION
The weight norm is usually associated with the complexity of the model. Techniques such as l2- regularization are often used to improve the generalization of linear models by introducing the bias towards smaller norm solutions. Moreover, Zhang et al. (2021a) hypothesized the implicit regularization effect of stochastic gradient descent (SGD) during model training. They show the result for linear regression:
Lemma C.4. For linear regression, the model parameters optimized using gradient descent w.r.t. mean squared error converges the minimum E--orrm solution:
arg min
w|| s.t. y Xw.
For classification, we follow a similar rationale proposed by (Sagawa et al. 2020). Recall that as we have assumed overparameteriiztion, the training data is linearly separable. Consider the hard-margin SVM:
w" arg min |w] s.t. ( T x0) Vi.
In this case, the max-margin solution corresponds to the minimum-nowe solution. Thus, it suffices to show if the model converges to the max-margin solution. Souurry et al. (2018) and Ji Telgarsky (2018) demonstrated that logistic regression trained with gradient descent converges in direction towards the max-marga solution:
Theorem C.5 (Soudry al. (2018)). For any linearly separable data, gradient descent for logistic regression will behave as:
(1) w log t p(t),
(7)
where is the step. The residual grows at most as [|p(t)]| = Oloo log t), and so it converges towards the direction of the max-margin solution w":
(t))
lim +0
w (L)l
w
However, the global minimum of unregularized logistic regression not well-defined. To bridge this gap, look at fz-regularized logistic regression with regularization hyperparameter A:
1 N
w X arg min W
Σ L
x
w|]
(8)
where L is the logistic loss. As \ 0-, we can recover the unregularized logistic regression solution. Moreover, also reflects the max-margin/minnnonn solution, which has been proved by Rosset et al. (2003) in their Theorem 2.1:
lim
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Thus, we mimic unregularized logistic regression with À that is extremely small, that the norm finite. The linear model also converges in a finite amount of time when it is trained in practice.
These results demonstrate the minimum-nome norm inductive bias of (stochastic) gradient descent. This allows us to derive conclusions based on the comparison of norms between different types of classifiers such the only using invariant features spurious features.
C.3 STANDARD ERM STILL WORKS IN EXPECTATION
Proposition C.6. For a binary classification problem using softmax classifiers trained with cross- entropy loss function on infinite samples, the cabel-independent noise level rj < 1/2, there exists f" that is Bayes optimal and also a global optimum that satisfies first-order stationarity.
Proof. Recall that (z, 17) denotes a data point that is subject to label noise with ratio Since use
the original label for is y, then classification with labels 0, 1. For notational convenience, we can let be the one-hot vector for y. It has the same dimension 2 f(x). Let h be the logits for input æ in the final layer before passing through softmax, meaning that f(z) = softmax(h). The gradient of the model parameters for correctly labeled data points y):
represent the flipped label since its binary
Ve+ VE(((r),))
V
Σ qi log f());;
(cross-entropy loss)
V log fx)
(gg 1)
VS(x)
f(x),
Vsoftmax(h),
f(z)y
f(x), [f(x) - [())))
(f(x) - 4) TVh
The gradient for mislabeled data points ( y):
VE- = V((f(z), yy
f(z) V Vf(z)1- (f(x)- 9)) Thh
We can consider the two logits ho. hn separately. Without loss of generality, consider 1:
vet = (f(x) - 9) Vh
-f((x) - 1)Vh, + (f(x)) -
7 =(f(x) (1 (1-1()))) Vhh TVh
= - 0 + +s(xx) 1/
With noise level 7), in expectation:
E = (1 ve+ T/ -1-10 (l)=))(f(=),)))))))))))))))))>)) 1)Vh, + (f(z)) ( m((f)) 00th, + (f(z), 1)
(((m)-11+1n -(t)) + (f(z), -)
+1 f(x)
(f(x)) + f(x); 1)
Clearly, f(x); = 1 1 is the unique solution to setting the expected gradient as 0, if we do not constrain the value of Vh1. Vhg, reaching the first-order stationary point with convergence. Moreover, this is also a minimizer and the Bayes optimal classifier f as it achieves the Bayes optimal error rate
n.
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Fundamentally, the label noise level 1) is essentially the same as Bayes rate in expectation. This shows that logistic regression classifiers trained with ERM are naturally noise-robust in expectation.
Proposition C.7. Given the noise level 1j < 1/2, the Bayes optimal classifier remuins at least as accurate as the classifier using only --correlated features, i.e, Rg. (J*) S R2 (17).
Proof. The Bayes optimal classifier using all features for the noisy data has 0-1 loss:
R ( (f*) = Ro-1 (f*) +1
The optimal classifier using only a set of y-correlated features has the following 0-1 loss:
(1) (1- 77 + 7 1- 1 + (27 - 1)n
We use proof by contradiction. Suppose that R (f*) > R (f*). Recall that y E [0, 1]. then
Rihh (f*) R (f*)
Ro + > 7 (27 - 1)y1
(2 27)1
7 (o)(
1
Ro- (5*)
1) 2
(1->>00
27
1
>
(Ra-(f") =
2
As 1) E 00,). the last inequality cannot be satisfied so the reverse (("( & Ro. (f)) must be true.
According to Proposition C.7 at least in expectation, the global minimum is still obtained by the Bayes optimal classifier even under the presence of label noise. However, in practice with only finite number of samples, ERM (w/ or w/o regularization) could fail to learn generalizable classifier when the noise level is high because of the joint effort between the low-dimensional spuriously correlated features and noise in the overparameterized setting. With limited samples, the training error refuses to converge easily to the Bayes optimal error since it can be further reduced by the memorization of noisy samples. In general, increasing the number of data would gradually resolve the issues of both label noise and spurious correlation (Figure lb).
D SYNTHETIC EXPERIMENTS
Our synthetic experiments are based on the framework developed in in/Sagaw et al. (2020). Without loss
of generality, we let E [-1,+1} instead of {0. } for the clarify of describing the data-generating
process. Let there be a spurious attribute a E {1] with spurious correlation y [0.5, 1]. such that in the training set, we have p(a = y) 7. Subsequently, four groups are created: (g) ):
y 1, a 1; (92): y = - 1,a = 1; (ga): y = 1, a 1; (g4): y = 1, a = 1. Let there be n
training data points. The number of data in each group !61 [g4l 7n/2. 2, 192l = Igal (1 /
The data is sampled as follows:
Xinv NNy1,, 1 den.):
Xapu N(a1, 02 LT Xaui ~N ai /dmn//mm))) dmai);
In this case, we set dim- = depu and of = 2² so that the two features are expected to have the same complexity and noise level. For the main plots (Figure lb) lbb w.r.t. noise level 7), we set 7 0.99, 1000, dmmi 3000, dim- dape 5, of logistic regression with regularization À 10- For each configuration (by noise level n or number of data n)), we repeat the experiments 10 times with seeds from 0 - 9 to obtain the mean and standard error.
spu 0.25. We fit the model with
For the figure of the decision boundaries (Figure Ic). we set dum = dapu for easier visualization in 2D. We also set o = o = 0.1 and remove regularization, so that the plot is more perceptible than o. 0.25. Nonetheless, we also show the decision boundaries in Figure 5 for oi = o 0.25, dim dapu 1, A 10-1 which essentially has the pattern. The random seed is 0.
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Figure 5: Decision Boundaries for o o 0.25, dnm
dapa 1,1 = 10
Hinontyy Groups Hajorly Groups
D5
- Najority Groups
Ninerty Gmoup
Spurious Feature
0.2
0.00
0.05
0.10
0.15
0.00
ols
20a
124
30
10a
Noree Leoel
Kumber Thaining Datl
Invoriant Feature
(a) Label noise test
(b) Number of data
(c) Decision boundaries.
Figure 6: Simulations on randomly projected synthetic data trained with overparameterized logistic regression. The dotted lines the means and the shaded regions are for the standard error from independent runs. The result strongly resembles the case in Figure I showing that our result is also applicable to linearly transformed features.
D.1 COMPARISON TO OTHER SYNTHETIC EXPERIMENTS
Sagawa et al. (2020) proposed the concept of spurious-core information ratio (SCR): The SCR is the ratio of the variance between the invariant (core) features and the spurious features: / The higher the SCR, the more variations the invariant features. This will lead to noisier predictions for the invariant classifier (that only rely on the invariant features), subsequently resulting in the overparameterized models relying more on the spurious features. We do not assume that the SCR is high here and set of 2 spu- attaining more general setting where invariant features are at least as informative the spurious features in both noise-free and noisy settings.
D.2 TRAINING WITH LINEARLY PROJECTED FEATURES
So far, the invariant features Xinv and spurious features Xspu occupy independent dimensions. In this section, extend the synthetic setting such that invariant and spurious features are not perfectly disentangled into different dimensions. The only requirement is that they remain orthogonal to each other: Xinv 1 Xspu To achieve this, we apply random projection matrix to the dataset after sampling and rerun the same experiments. Based on the rotational invariance of logistic regression (Ng 2204), we expect the results to not change by much. As shown Figure this indeed the case, showing that our result is also applicable to linearly transformed features.
E MAIN EXPERIMENTS
E.1 DATASETS
All datasets are added to or implemented by the Domainbed (Gulrajani & Lopez- Paz 2020) framework for better comparison among the algorithms.
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E.1.1 SUBPOPULATION SHIFTS WITH KNOWN GROUPS
Waterbirds (Sagawa et al. 2019) is binary bird-type classification dataset adapted from CUB dataset (Wah et 2011 and Places dataset (Zhou et al. 2017). There are total of four groups: waterbirds on water (G1). waterbirds on land (G2), landbirds on water (Ga), and landbirds on land (G4). In the training set, there are 3498 (73%), 184 (4%), 56 (1%), and 1057 (22%) images for the 4 groups. Apparently, there is a strong spurious correlation between the bird type and the background in the training set. The majority groups validation and test sets, there are an equal number of landbirds or waterbirds allocated to water and land backgrounds. We follow the standard train, val, and test splits.
91 and G1. and the minority groups are G2, G3. For the
fiom CelebA (Sagawa et al. 2019) is a binary hair color blondness prediction dataset adapted from Liu et al. (2015). There are four groups: (Gi) non-blond woman (71629 samples, 44%); (92) non-blond man (66874 samples, 41 %)); (9x) blond woman (22880 samples, 14%); (G4) blond man (1387 samples, 1 1%). There exists a strong spurious correlation between gender man and non-blondness. The majority groups are G1. G2. G3. and the minority group is G1 We follow the standard train, val, and test splits.
CivilComments (Borkan al. 2019) is a binary text toxicity classification task. We used the version implemented in WILDS (Koh et al. 202T). There are 16 overlapping groups based on 8 identities including male, female, LGBT, black, white, Christian, Muslim, other religion. In this work, we use the default setting and create 4 groups only based on the black identity.
E.1.2 SUBPOPULATION SHIFTS WITHOUT GROUP INFORMATION
We create new subpopulation-shiff datasets adapted from real-world datasets including Waterbirds and CelebA:
Waterbirdst: Based on the groups in Waterbirds, we create two new environments e1; 62. Each environment has half the data from each majority group (Gi, G4). Then we allocate 1/3 of the data from each minority group (G2, G3) ej and the rest 2/3 to eg. This resembles the CMNIST setup so that e2 has twice the amount of the minority data as e1, such that background is not an invariant feature across the two environments because predicting with only the background features will result in different error rates across environments.
CelebA+: Similar to Waterbirds+, we create two new environments c1, c2. Each environment has half the data from each majority group (G1, G2, Ga). Then we allocate 1/3 of the data from the minority group (G4) to e and the rest 2/3 to e2. Under this allocation, gender is not an invariant feature across the two environments because predicting hair blondness with only the gender features will result in different eиror rates across environments.
CivilComments Although there 16 overlapping groups with 8 identities, we only create 4 groups based on the black identity. Thus, the setup can exactly follow Waterbirdst.
We also use the synthetic ColoredMNIST dataset:
CMNIST (Arjovsky et al., 2019) is one of the most commonly experimented synthetic domain generalization datasets. Based on MNIST, digits 0-4 and 5-9 are categorized into classes 0 and 1, respectively. Additional two-color information is added to the dataset by creating a separate channel, resulting in the input size being (28 28 x 2). There are three environments mark the correlations between the color and the digits. In particular, in 61, 10% of the green instances 1 with the rest 90% being 0, and 10% of the red instances are with the rest 90% being 1. Thus, using the color information to predict with green and predict O with red, gives 90%, 80%, and 10% accuracy for the three environments. By default, the labels are corrupted with 25% uniform label noise.
{0.1, 0.2, 0.9) that
1.3 DOMAIN SHIFTS
We choose four domain-shift datasets from the Domainbed framework (Gulrajani & Lopez- Paz 2020). All four datasets have input sizes (3, 224, 224).
PACS Li et al. 2017) is 7-class image classification dataset specifically designed for domain generalization, encompassing 4 distinct domains: Photo (1670 images), Art Painting (2048 images), Cartoon (2344 images), and Sketch (3929 images). There 9991 samples in total.
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VLCS (Fang et al. 2013) is 5-class image dataset with domains: Caltech101, LabelMe, SUN09, and VOC2007. Each domain is a standalone image classification dataset released by distinct research teams. There are 10729 samples in total.
Office-Home /Venkateswara et al. 2017) is a 65-class image classification dataset for objects that be found typically in office home environments. The dataset has 4 domains: Art, Clip Art, Product, and Real- Real-World. There are 15588 samples in total.
Terra Incognita (Beery al., 2018) is 10-class wild-animal classification dataset with 4 domains: L100, L38,, L43, L46. Each domain represents a distinct location where camera traps are placed for photography. There 24788 samples in total.
E.2 ALGORITHMS
We choose a few most influential and well-cited algorithms with theoretical guarantees for compar- isons.
1. Mixup Zhang et al. 2017) is simple yet effective approach that has been widely studied from both empirical and theoretical perspectives. interpolates the input and the loss of the output to create pseudo-training samples for better nut-of-distribution generalization.
2. IRM (Arjovsky et al., 2019): IRM is the first algorithm proposed to learn invariant/causal predictors from multiple environments with deep learning. It augments the ERM with additional objective of achieving optimal loss for all training environments simultaneously, preventing the model from learning spurious correlations that are only applicable certain environments.
3. V-REx (Krueger et al., 2021): V-REx extends IRM with a simplified regularizer that minimizes the variance of risks across environments. has the benefit of being robust to covariate shifts in the input distribution. It is also theoretically analyzed for homoskedastic environments (consistent noise level).
GroupDRO (Sagawa et al., 2019): GroupDRO prioritizes optimizing the worst-group risk to address subpopulation shifts. It is one of the first algorithms proposed to specifically address sub- population shifts by minimizing the worst-group risk directly. By utilizing the group information effectively, it has been considered the gold standard amongst many subsequent works that try to deal with subpopulation shifts without group labels.
F.3 EXPERIMENTAL. SETUP
We use Domainbed (Gulrajani & Lopez-Paz Paz 2020 experiments for all 9 datasets. We report the performance (average and standard error) of the models trained out of 3 trials and each trial searches across 20 sets of different hyperparameters according the default setting. We train all models for 5000 steps. for CMNIST, and 16 for the rest real-world datasets. We use simple Convolutional Neural Network (CNN) for CMNIST, and ResNet 50 pretrained on Imagenet for the rest datasets. For all datasets, we add 10%, 25% label noise to test the performance of different algorithms.
For Waterbirds, CelebA, and their derivative Waterbirds+, CelebA+ datasets, we perform standard data augmentation according to Sagawa et al. (2019) using Pytorch (This is slightly different from the default augmentation implemented by Domainbed). Since Waterbirds and CelebA have predefined train, val, test splits and group information, we report both the average and worst- soottg (WG) test accuracy of the model selected according to the best WG performance in the standard validation split with early stopping, which follows most of the algorithms evaluated on these datasets.
For CMNIST, we do not perform any data augmentation. The evaluation is only done on the 3rd environment where the minority groups from the training environments become the majority. Due to the simplicity of the dataset, we have tested more fine- iine-grained label noise 7) E (0,0.05, 0.1, 0.15, 0.2).
For the domain-shift datasets, we adopt Domainbed's 's data augmentation (Gulrajani & Lopez- Paz 2020). Since there are multiple different environments, we run a taking the environments one by the test set and the rest as the training set. Then, report the average accuracy of the models across all environments selected using 20% of the test data without early stopping.
cross-test by
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F ADDITIONAL MAIN RESULTS
F.1 COMILTTMENNTS (NLP DATASET)
CivilComments (Borkan al. 2019) is binary toxicity classification task. We used the version implemented in WILDS Koh et al. 2021). There are 16 overlapping groups based on 8 identities including male, female, LGBT, black, white, Christian, Muslim, other religion. In this work, similar to Waterbirds and CelebA, we create 4 groups only based on black identity. We also simulate the wwo-environment subpopulation shift setting as done for Waterbirds+ and CelebA+, and we call CivilCommntts..
We use the base uncased DistilIBERT model as the feature extractor. We feed the final-layer representation of the [CLS] token into linear classifier for toxicity detection. We report the worst- group error with the worst-group selection. Note that Mixup cannot be directly applied to the NLP dataset since the input is discrete words that cannot be interpolated.
As shown the Table below, GroupDRO has a clear advantage over other algorithms in the noise- setting. However, ERM still performs competitively compared to other invariance learning algorithms such as IRM and V-REx, even under label noise setting, which is consistent with the existing empirical observations.
Algorithm/y
0.1
0.25
Avg
ERM
89.8
67.9:bo.4
55.4+o.1
71.0
IRM
89.2
66.9
54.2:
70.1
GroupDRO
91.9..0.2 0.2
69.7
55.4
72.3
VREx
89.6
69.0=
56.0
71.6
Table Worst-group accuracy (%) for subpopulation shift on CivilComments Dataset.
Algorithm/n
0.1
0.25
Avg
ERM
69.2
19.8
52.5::1.0
60.5
IRM
60.3
54.8
50.3
55.1
GroupDRO
73.3
60.2:tu.9
52.2
61.9
VREx
64.5
58.7
54.9
59.4
Table 4: Worst-group accuracy (%) for subpopulation shift on CivilComments+ Dataset.
F.2 EVALUATING MORE ALGORITHMS
We added evaluations of four additional algorithms used for domain adaptation and domain general- ization, including:
1. CORAL Sun & Saenko 2016): A moment matching technique that matches the mean and variance the features between the training and test domains.
IB IRM (Ahuja et al. 202T): An IRM based algorithm that is combined with information bottleneck as additional regularization.
Fish (Shi et al. 202I): A gradient matching technique that aligns the gradients of different training domains by maximizing their inner products.
4. Fishr (Rame et al., 2022); A gradient matching technique based on regularizing the inter-domain variance the intra-domain gradient variance.
We show the results for three datasets from each category, including corrupted CMNIST (synthetic subpopulation shift), Waterbirds+ (real-wort subpopulation shift), and PACS (real-world domain shift). We observe that similar trend still holds compared other DG algorithms presented in the main paper. Thus, we hypothesize that these results can extended to other datasets that are similar to the ones evaluated.
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F.2.1
CMNIST
Algorithm/1
0.1
0.25
Avg
ERM
97.1
1..8
39.5
71.5
Mixup
97.3 97.3-t0.2
02.
24.7+11.9
68.1
GroupDRO
97.3=0.1
84.5
48.3#1.0
76.7
IRM
96.8
86.5.13.8
73.2.t9.11
85.5
VREx
96.7
84.1
57.8++2.0
79.5
CORAL
97.1
77.3
37.2
70.5
IB IRM
96.5
83.7
71.3
83.8
Fish
97.2+t0.2
78.8
37.7=1.3
71.2
Fishr
97.3 0.0
82.0
67.6
82.3
F.2.2
WATERBIRDS+
Algorithm/n
0.1
0.25
Avg
ERM
81.5
64.622..5
50.4
65.5
IRM
78.5
63.3
47.8#1.8
63.2
GroupDRO
82.6+t.4
65.6
51.2+t0.7
66.5
Mixup
79.8+
61.8
51.3+11.5
64.3
VREx
78.3
66.6
51.9:to.5
65.6
CORAL
80.8
64.311.
53.5
66.2
IB JJMM
74.5
60.8
48.6
61.3
Fish
82.3
66.4
49.8
66.2
Fishr
79.4
62.1
53.5#1..
65.0
F.2.3 PACS
Algorithm/n
0.1
0.25
Avg
ERM
82.0
74.5
78.3
IRM
80
71.0#1.9
75.7
GroupDRO
82.4
74.7:t0.4
78.5
Mixup
83.6
75.2:t0.9
79.4
VREx
81.4+tu.2
73.5
77.4
CORAL
82.6=bu.4
75.9
79.2
IB IRM
77.0
67.8
72.4
Fish
83.1
77.0
80.1
Fishr
81.7
74.5
78.1
G LIMITATIONS
G.1 FEATURE AVAILABILITY
In practice, when the feature extractors for image and text commonly non-linear, it is uncertain whether the invariant and spurious features can be perfectly extracted by the deep learning model. As our setup assumes the availability of invariant and spurious features, our theoretical results in Section 4.1 under linear settings may not be directly applicable in the real world.
We would like discuss aspects regarding the potential limitations due to the availability of invariant, spurious, and nuisance features:
1. Failure mode of ERM: Let's briefly consider two other scenarios where the ideal feature extractor does not exist:
(a) Spurious features not fully extracted: This can be viewed as having weaker spurious correlation and our results still apply. (b) Invariant features are not fully extracted: The model has to exploit more spurious correlations and nuisance features, it already fails generalize.
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As such, we believe that showing the failure mode of ERM under such an ideal situation actually encompasses some other failure modes without the ideal feature extractors. The assumption of having orthogonal and spurious features is not restrictive to demonstrate the failure mode of ERM.
2. Advantage of DG algorithms in practice: Typically, DG algorithms aim to learn the invariant representations and require the existence invariant features to function. If the invariant features cannot be fully extracted, there is also no guarantee for DG algorithms to have the advantage over ERM. The robustness to label noise does not necessarily imply that the invariant representations could be learned. Rather, it has the implication of less memorization and not converging to the suboptimal solution when label noise is present. Thus, the unavailability of the ideal sets of features could be responsible for DG algorithms not having clear advantage over ERM in real-world noisy datasets. However, without enough evidence, we do think that pretraining the model with better representations is more responsible for the reduced performance gaps between different DG algorithms vs. ERM compared to the lack of an ideal feature extractor.
G.2 LINEAR SEPARABILITY
Our theoretical analysis is also based the assumption that the invariant features linearly separable in the noise-free setting. However, in practice, the invariant features can achieve the Bayes optimal error rate 0%. making the invariant features non-linearly separable. Our theorem then is not directly applicable. Future work that explicitly allows non-separable invariant features will improve the analysis.
However, we think that our assumption is in fact more inclusive than it is restrictive. This is because the linear separability is only assumed for the noise-free data distribution. In the cases when the dataset is not linearly separable, we can view that as result of adding some form of label noise to the clean data. higher noise level corresponds to worse non-separability. However, we acknowledge that this analogy is by no means comprehensive of all cases.
G.3 GENERALIZATION OTHER DG METHOSS
Our analysis Section SacttionJaaddeee adddresses the noise robustness of two invariance learning algorithms: IRM and V-REx. However, DG algorithms are designed for various aspects of the training regime with diverse mechanisms (discussed Section 2). not all domain generalization methods can be theoretically proven possess such desirable property. For example, from the empirical observation in Appendix [.2, algorithms such as Fish which aligns the gradient of samples between environments may not be robust noise on the CMNIST dataset. More comprehensive analyses needed to incorporate all algorithms.
H NOISE MEMORIZATION ON WATERBIRDS DATA
Algorithm
Waterbirdst 0.1
Waterbirdst 0.25
Waterbirds 0.1
Waterbirds 0.25
ERM
28.3
35.1
33.4
58.4#s.6
IRM
27.3
38.7+1.8
30.4+
54.3+
GroupDRO
23.4+22
36.3
32.9+o..1
57.3
Mixup VREx
28.2+33.6
34.6
34.9
57.2+t..0
24.8
37.8
35.3
48.1
Table 5: Training noise memorization on Waterbirds+
As shown in Table 5 the memorization issue is much less severe. The likely reason is because carly-stopping is applied to Waterbirds, CelebA, and CivilComments, so that model checkpoints in the carly-training steps with less memorization allows the models to fit the downstream tasks faster. Thus, good performance can be obtained without prolonged training compared to training from scratch.
selected. Besides, pretraining ImageNet
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I IMPLEMENTATION DETAILS OF ERM
The classic environment-agneen agnostii ERM randomly samples training data from pool of data from all environments. For the wwwtroommet--alannee ERM algorithm implemented by Domainbed (Gulrajani & Lopez- Paz 2020). in each step, an equal amount of data is sampled from each environment according batch size. Thus, the actual "batch size" equals to num data batch batch_size, size. Datasets with more environments are trained effectively with more data points.
J RESULTS
All these results are obtained using 5000 training steps and datasct-dependent batch size.
J.1 CMNIST
J.1.1 AVERAGES
Algorithm/y
0.05
0.1
0.15
0.25
Avg
ERM
97.1+0.1
89.2+t..6
77.8
63.1 +3.8
39.5-h1.2
73.3
IRM
96.8+..3
89.5
86.5
76.6
93.2
84.5
GroupDRO
97.3
91.1
84.5
74.9
48.3
79.2
Mixup
97.3
90.8
82.3
66.4
24.7#11.9
72.3
VREx
96.7
90.9
84.1
82.3
57.8:t2.0
82.3
J.1.2 NOISE MEMORIZATION
Algorithm/n
0.05
0.1
0.15
0.25
Avg
ERM
26.6
31.7=6.4
53.7
74.5
46.6
IRM
16.6.33.6
16.6
32.0#12.2
26.8
23.0
GroupDRO
10.0
15.7+11.7
26.2
54.5t22.
26.6
Mixup
17.5#1.0
22.1=1.1
38.1
77.7#1.9
38.9
VREx
8.3
16.6-12.0
18.3
41.2.t2.2
21.1
J.2 WATERBIRDS
J.2.1
WORST-GROUP TEST ACCURACY
Algorithm
0.1
0.25
Avg
ERM
86.7+1.
61.4 61.4
48.8+
65.6
IRM
87.0+1.44
58.8
48.3
64.7
GroupDRO
84.9
61.910.77
47.7
64.8
Mixup
88.0
56.9
50.4
65.1
VREx
87.4
59.2
46.8
64.5
J.2.2 AVERAGE TEST ACCURACY
Algorithm
0.1
0.25
Avg
ERM
91.9
63.6
50.3
68.6
IRM
91.8+t0.4
63.5
50.6
68.6
GroupDRO
91.9
63.6#tu.
51.1
68.9
Mixup
92.6
62.2
51.2
68.7
VREx
92.6
63.7
51.0
69.1
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J.3 WATERBIRDS+
J.3.1
WORST-GROUP TEST ACCURACY
Algorithm
0.1
0.25
Avg
ERM
81.5
64.6
50.4
65.5
IRM
78.5
63.3
47.8+1..8
63.2
GroupDRO
82.6
65.61...
51.2
66.5
Mixup
79.8
61.812..0
51.3
64.3
VREx
78.3
66.6
51.9
65.6
J.3.2 AVERAGE TEST ACCURACY
Algorithm
0.1
0.25
Avg
ERM
90.3
73.
55.7+0.2
73.2
IRM
89.0
12.0
56.4
72.5
GroupDRO
90.4
70.6
54.3#1.3
71.8
Mixup
89.7
72.2
54.7=
72.2
VREx
89.6
71.6:1
55.2
72.1
CELEBA
J.4.1 WORST-GROUP TEST ACCURACY
Algorithm
CelebA
CelebA 0.1
CelebA 0.25
Avg
ERM
18.1
63.1
50.2+1..2
67.1
IRM
86.9
61.7+1.8
51.2
66.6
GroupDRO
88.0
59.3
49.0+11.4
65.4
Mixup VREx
87.6:
63.0
49.5
66.7
89.3
60.2
49.5
66.3
J.4.2 AVERAGE TEST ACCURACY
Algorithm
0.1
0.25
Avg
ERM
93.5
67.2
52.3
71.0
IRM
93.5
67.1
52.1
70.9
GroupDRO
93.8
67.0
52.1
70.9
Mixup
93.5
67.6:b..2
52.6
71.2
VREx
93.5
67.8
52.5
71.2
J.5 CELEBA+
J.5.1 WORST-GROUP TEST ACCURACY
Algorithm
0.1
0.25
Avg
ERM
71.7
65.4
55.0
64.0
IRM
75.4
62.4
53.0
63.6
GroupDRO
71.9
61.7
57.0
63.5
Mixup
71.1
64.6#
52.6
62.8
VREx
73.9
69.9
50.0
64.6
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J.5.2 AVERAGE TEST ACCURACY
Algorithm
CelebA
CelebA 0.1
CelebA 0.25
Avg
ERM
91.9
76.2
60.3
76.1
IRM
89.6
76.5
59.8.to.7
75.3
GroupDRO
89.9
76.9
60.5
75.8
Mixup
91.6+1.3
76.6
60.2
76.1
VREx
90.5+
74.8
60.3
75.2
Fishr
92.3+
75.8
60.7
76.3
J.6 PACS
J.6.1
PACS 1=0.1
Algorithm
C
P
Avg
ERM
81.1+0.4
76.4
95.1
75.3+1.
82.0
IRM
77.0 75.7
96.0
72.7+2.3
80.4
GroupDRO
81.7
78.0
94.0
75.8
82.4
Mixup
84.2
77.8
96.3
76.0
83.6
VREx
80.3
75.9:11.
95.0
74.4+14
81.4
J.6.2 PACS 1==0.25
Algorithm
A
C
P
S
Avg
ERM
73.6
67.9+
87.4+1.0
69.3 1.5
74.5
IRM
72.8 64.9
90.0
56.4
71.0
GroupDRO
74.4
67.7
89.9
66.9
74.7
Mixup
72.4
69.9
90.5
67.8
75.2
VREx
69.8
68.4
89.4
66.2
73.5
J.7 VLCS
J.7.1 VLCS 10.1
Algorithm
C
L
S
V
Avg
ERM
97. 97.2
64.4
68.8+1.0
69.8
75.0
IRM
96.7
64.0
67.8
69.9
74.6
GroupDRO
96.1
64.6+t..2
69.2
70.6
75.1
Mixup
97.1
65.5
69.2
70.1
75.5
VREx
95.1
65.9
68.210..4
70.6
75.0
J.7.2
VLCCS 00.25
Algorithm
Avg
ERM
93.6
62.7
64.6#bo.9
66.8
71.9
IRM
94.5
62.1
61.6
63.1
70.3
GroupDRO
93.6#111
62.4
63.8
65.1
71.2
Mixup
94.1#
62.5
64.0=
66.9
71.9
VREx
93.9
63.0
63.6
66.8
71.8
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J.8
OFFICEHOME
J.8.1
OFFICEHOME v=0.1
Algorithm
A
C
P
R
Avg
ERM
67.4+
47.4+t..4
71.1+0.3
72.7+0.3
62.2
IRM
55.6
46.3
70.7
72.1
61.2
GroupDRO
55.8
47.0
69.5
72.8+0.4
61.3
Mixup
57.0
51.8ttu.
73.2
73.8
63.9
VREx
54.7#112
47.6
68.9
71.1
60.6
J.8.2 OFFICEHOME 1
Algorithm
A
C
P
R
Avg
ERM
47.9 41.3114
64.3
66.0
54.9
IRM
46.9#1.9
39.1+224
63.2
66.5
53.9
GroupDRO
48.6#t..4
39.2
62.4
66.1
54.1
Mixup
53.1
42.0
66.3
68.4
57.5
VREx
47.1
39.1
61.0
64.6
53.0
J.9
TERRAINCOONITA
J.9.1
TERRAINCOGNITA 0
Algorithm
L100
L38
L43
L46
Avg
ERM
58.1#1.9
52.1+1.0
57.9
40.5
52.1
IRM
47.6#$3.3
51.9t112
53.4#1.0
40.4
48.3
GroupDRO
58.0#1..
49.1
59.3
40.8
51.8
Mixup
62.4
51.44:2.0
58.3
40.6
53.2
VREx
49.7
46.7+bo.9
56.0
40.8
48.3
J.9.2 T'RRAINCOGNITA -0.25
Algorithm
L100
L38
L43
L46
Avg
ERM
52.3
48.8
55.8
37.2
48.5
IRM
41.5
46.1
51.0
39.0
44.4
GroupDRO
56.9
48.7
55.4
41.55t.6
50.6
Mixup
62.2
47.5
55.3
40.2
51.3
VREx
51.0112.6
51.5::1.2
54.6:to.5
38.51t..9
48.9
30


Reviews:
[{"review_id": "xw9GiPIq3L", "reviewer": "Reviewer_TZVP", "summary": "As many previous studies have numerically demonstrated, no domain generalization method clearly outperforms the empirical risk minimization in general. This study investigates when and why DG methods better generalize than the empirical risk minimization and vice versa, through the lens of label-noise and subpopulation shifts. Particularly, the authors demonstrate the empirical risk minimization's tendency to learn spurious correlations (or domain-specific features) rather than invariant features for overparameterized models determined by both degrees of spurious correlation and label noise. Moreover, the authors also investigated that some domain generalization methods can learn invariant features over spurious correlations, resulting in better generalizability in the presence of noisy labels. Extensive numerical experiments were provided.", "questions": "- I think a simpler simulation scenario, such as the linear case given in Section 4.1, might be more appropriate to demonstrate the theory. As mentioned in the weaknesses, the computer vision scenarios presented in the experimental study, require an ideal feature extractor that can provide invariant, spurious, and nuisance features. However, there is no guarantee that such representations were learned. \n- The authors might need an ideal feature extractor that provides the disjoint sets of invariant, spurious, and nuisance predictors or a proxy of such extractor to demonstrate the hypothesis. \n- It seems like the overall problem setting is also relevant to the fairness problem. Is there any relevant study from algorithmic fairness literature that the authors know of?\n- Is it possible to extend the discussion in Section 4, which is focused on subpopulation shift, to analyze the domain shift? Particularly, for the cases where the ERM or DG might fail.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- Theoretical analysis on when and why the domain generalization methods perform better or worse than the empirical risk minimization has rarely been studied. This paper provides a concrete and interesting one.\n- Well-written and easy to follow. Assumptions for the analysis have been made clear.", "weaknesses": "- The analyses provided in this study are based on a linear setting, assuming that we have disjoint sets of invariant, spurious, and nuisance predictors. This seems to be a reasonable assumption for theoretical analysis. However, in real-world cases, we might not be able to have such disjoint sets of predictors. For example, in the computer vision tasks given in the experimental study, it is not straightforward that we have such predictors unless a neural network learns such appropriate representations, which I think is hardly possible.\n- Even for tabular data, we might need a proper transformation to have such ideal sets of predictors.\n- So, such difficulty in having an appropriate representation might be responsible for the failure of the noise robustness to translate to better generalizability in the experimental study.\n- In short, the theory is sound, however, the conclusion from the experimental study is not fully convincing, and seems to need further exploration.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "53BDCwlb3X", "reviewer": "Reviewer_jUs6", "summary": "This paper explores whether the Domain Generalization (DG) algorithm outperforms the classic Empirical Risk Minimization (ERM) algorithm in the presence of labeled noise, and why.", "questions": "Q1. I would like to know the relationship between Theorem 4.2 and \"Specifically, our finite-sample analysis reveals that label noise exacerbates the effect of spurious correlations for ERM, undermining generalization. Conversely, we illustrate that DG algorithms exhibit implicit label-noise robustness during finite-sample training even when spurious correlation is present.\" in the abstract? How does this theorem reflect label noise?", "limitations": "", "rating": 3, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "S1. The writing expression of this paper is relatively clear, but it is still not standardized enough, such as Eqn 1 should be written as Eqn (1).\n\nS2. The research motivation of this paper is to explore the effectiveness of DG compared to ERM under labeled noise settings. This is positive for the study of DG, after all, there is no clear empirical evidence that the existing DG algorithms perform the classic ERM across standard benchmarks.", "weaknesses": "W1. The failure of validation on real data is pessimistic, which seriously reduces the importance of the settings discussed in this paper, as real data does not fit well with simple noise settings.\n\nW2. The main theoretical results of this paper have poor readability. The conclusion described in Theorem 4.2 is not very intuitive. It is difficult to associate this principle with the main contributions described in the abstract of this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "y4SGZhMhtA", "reviewer": "Reviewer_4qg5", "summary": "This study offers a thorough exploration, both theoretically and empirically, to ascertain the situations where Domain Generalization (DG) algorithms outperform Empirical Risk Minimization (ERM) counterparts. The findings reveal that DG algorithms exhibit greater resilience in the presence of label noise during subpopulation shifts. On the other hand, ERM approaches tend to be susceptible to capitalizing on spurious correlations, particularly in overparameterized models. The study backs these observations with a blend of theoretical insights and empirical evidence.", "questions": "Q1 - Can your theoretical framework comprehensively explain all domain generalization methods that address subpopulation shift and spurious correlations, or are there any inherent limitations?\n\nQ2 - What motivated the selection of the methods used in the experimental section? Why were these particular methods chosen?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "S1 -The paper upholds a commendable level of clarity in its exposition, effectively conveying intricate ideas in an easily comprehensible fashion. Its tone is suitably professional, aligning with the subject matter, which contributes to a satisfying reading experience. The structural organization into subsections facilitates navigation and swift access to specific information. Furthermore, the notation employed is consistently precise, enabling readers to grasp the mathematical elements of the paper with ease. Notably, the theoretical statements are  thoughtfully elucidated through illustrative examples and substantiated by empirical findings.\n\nS2 - The claims presented in the paper are substantiated through a combination of theoretical and empirical evidence. \n\nS3 - The paper's commitment to reproducibility is highly commendable. The detailed and transparent presentation of the experimental setup, data sources, and code availability significantly enhances the reliability and trustworthiness of the research findings.\n\nS4 - This paper thoroughly explores different scenarios for empirical validation and uses a diverse set of datasets (i.e, classification tasks). The authors have selected datasets that are commonly featured in the existing literature on this research topic.", "weaknesses": "W1 - The section discussing related work appears somewhat limited. Domain generalization is a machine learning technique designed to train models for effective performance on unfamiliar data originating from multiple domains or distributions. As such, it covers a broad and diverse research landscape, accommodating various scenarios and types of data shift. To enhance reader comprehension and provide a more thorough context, it would be beneficial for the authors to initiate with a comprehensive overview of domain generalization before delving into the specific scenario they focus on, which involves challenges such as label noise, spurious correlation, and subpopulation shifts. This approach can mitigate potential misunderstandings and offer a more holistic understanding for the readers.\n\nW2 -  The paper primarily presents an empirical analysis, and it is commendable that the authors have incorporated a diverse set of datasets, which enhances the generalizability of their findings. However, it's worth noting that the diversity observed in the choice of datasets is not reflected in the selection of Domain Generalization (DG) methods considered for the analysis. In order to strengthen the general statements proposed in the paper, it would be valuable to expand the range of DG algorithms under examination. This broader inclusion of methods can further validate and reinforce the claims made in the paper, offering a more comprehensive view of the research landscape.\n\nW3 - The experiments of the paper are limited to tasks related to image classification, with no inclusion of other types of data, such as tabular data. This focus on image classification tasks allows the authors to delve deeply into this specific domain and gain insights relevant to this context. However, it's important to note that the findings and conclusions may not be directly applicable to other data types or domains, and this limitation should be kept in mind when interpreting the results.\n\nW4 - Section 5 only provides a description of IRM, but it would be valuable to have a more extensive discussion about other methods as well.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "PRe9mGcqSS", "reviewer": "Reviewer_qd9a", "summary": "This paper theoretically analyzes the benefits of the DG algorithm over ERM from the perspective of noise robustness. Label noise increases the model’s dependence on the spurious features of ERM. In contrast, DG algorithms have better label-noise robustness with the regularization which prevents capturing spurious correlation.", "questions": "1.\tIn all tables in experiments, the authors should add references to Mixeup, GroupDRO, IRM, VREx.\n2.\tThe author should provide another theorem to illustrate that the classifier with a smaller norm becomes more favored by the model.\n3.\tThe authors should provide further explanations on the regularization term in the practical surrogate. This should include an explanation of why it has this specific form and why it incorporates first-order derivatives.", "limitations": "", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The authors theoretically analyze the benefits of the DG algorithm over ERM from the perspective of noise robustness.\n2. Experimental results show that DG algorithms are more robust to label noise memorization.\n3. The authors provide discussions about why noise robustness couldn’t lead to better performance of the DG algorithm in practical scenarios.", "weaknesses": "1.\tMy primary concern relates to the assumption of linear separability, as it can be challenging to meet this condition in real-world settings, especially when the invariant features are sparse. Moreover, it is also hard to have orthogonal invariant and spurious features. I think a more general analysis should be given to support the main idea.\n\n2.\tIn practical scenarios, the minimum-norm classifier may not be the most suitable choice, which implies that Thm4.2 may not be applicable. \n\n3.\tThe theorem does not sufficiently prove the ideas presented in this paper. Thm4.2 only suggests that the ERM algorithm favors spurious features over invariant ones, which leads to poor generalization performance.  However, for the DG algorithm, the authors only incorporate a regularization term and analyze the gradient with varying lambda. No formal theorem is there to demonstrate the superior performance of the DG algorithm.\n\n4.\tThe crucial information is not expressed clearly. In page 4, how comes \"the classifier become either $w_spu$ or $w_inv$\"?  Is it an assumption or is it a mathematically grounded result? I think the authors should justify such an extreme claim.\n\n5.\tThe description of the Lemma C.1 is unclear. The Lemma measures the cost of memorizing a mislabeled or non-spuriously correlated sample. Based on the derivation process, it should be a conclusion that holds with high probability, but the lemma presents it deterministically.\n\n6.\tThe absence of mathematical symbol definitions in this paper reduces its readability. Some of the symbols are not described (just mentioned) in this paper. If possible, give their mathematical definitions. \n\n7.\tThe experiments do not effectively support the theorem. In the experiment section, I recommend that the authors discuss the values of the weights' norms, such as $w^(inv)_inv$, $w^(spu)_spu$, $w^(inv)$, and $w^(spu)$, to further support the proof of Thm4.2.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)