# domain_generalization_using_causal_matching__afa97534.pdf

Domain Generalization using Causal Matching

Divyat Mahajan 1 Shruti Tople 2 Amit Sharma 1

In the domain generalization literature, a common objective is to learn representations independent of the domain after conditioning on the class label. We show that this objective is not sufficient: there exist counter-examples where a model fails to generalize to unseen domains even after satisfying class-conditional domain invariance. We formalize this observation through a structural causal model and show the importance of modeling within-class variations for generalization. Specifically, classes contain objects that characterize specific causal features, and domains can be interpreted as interventions on these objects that change non-causal features. We highlight an alternative condition: inputs across domains should have the same representation if they are derived from the same object. Based on this objective, we propose matching-based algorithms when base objects are observed (e.g., through data augmentation) and approximate the objective when objects are not observed (Match DG). Our simple matching-based algorithms are competitive to prior work on out-of-domain accuracy for rotated MNIST, Fashion-MNIST, PACS, and Chest-Xray datasets. Our method Match DG also recovers ground-truth object matches: on MNIST and Fashion-MNIST, top-10 matches from Match DG have over 50% overlap with ground-truth matches.

1. Introduction

Domain generalization is the task of learning a machine learning model that can generalize to unseen data distributions, after training on more than one data distributions. For example, a model trained on hospitals in one region may be deployed to another, or an image classifier may be deployed on slightly rotated images. Typically, it is assumed that

1Microsoft Research, India 2Microsoft Research, UK.. Correspondence to: Divyat Mahajan <divyatmahajan@gmail.com>.

Proceedings of the 38 th International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s).

the different domains share some stable features whose relationship with the output is invariant across domains (Piratla et al., 2020) and the goal is to learn those features. A popular class of methods aim to learn representations that are independent of domain conditional on class (Li et al., 2018c;d; Ghifary et al., 2016; Hu et al., 2019), based on evidence of their superiority (Zhao et al., 2019) to methods that learn representations that are marginally independent of domain (Muandet et al., 2013; Ganin et al., 2016).

In this work, we show that the class-conditional domaininvariant objective for representations is insufficient. We provide counter-examples where a feature representation satisfies the objective but still fails to generalize to new domains, both theoretically and empirically. Specifically, when the distribution of the stable features to be learnt varies across domains, class-conditional objective is insufficient to learn the stable features (they are optimal only when the distribution of stable features is the same across domains). Differing distributions of stable features within the same class label is common in real-world datasets, e.g., in digit recognition, the stable feature shape may differ based on people s handwriting, or medical images may differ based on variation in body characteristics across people. Our investigation reveals the importance of considering withinclass variation in the stable features.

To derive a better objective for domain generalization, we represent the within-class variation in stable features using a structural causal model, building on prior work (Heinze Deml & Meinshausen, 2019) from single-domain generalization. Specifically, we construct a model for the data generation process that assumes each input is constructed from a mix of stable (causal) and domain-dependent (non-causal) features, and only the stable features cause the output. We consider domain as a special intervention that changes the non-causal features of an input, and posit that an ideal classifier should be based only on the causal features. Using d-separation, we show that the correct objective is to build a representation that is invariant conditional on each object, where an object is defined as a set of inputs that share the same causal features (e.g., photos of the same person from different viewpoints or augmentations of an image in different rotations, color or background). When the object variable is observed (e.g., in self-collected data or by dataset augmentation), we propose a perfect-match regularizer for

Domain Generalization using Causal Matching

domain generalization that minimizes the distance between representations of the same object across domains.

In practice, however, the underlying objects are not always known. We therefore propose an approximation that aims to learn which inputs share the same object, under the assumption that inputs from the same class have more similar causal features than those from different classes. Our algorithm, Match DG is an iterative algorithm that starts with randomly matched inputs from the same class and builds a representation using contrastive learning such that inputs sharing the same causal features are closer to one another. While past work has used contrastive loss to regularize the empirical risk minimization (ERM) objective (Dou et al., 2019), we demonstrate the importance of a two-phase method that first learns a representation independent of the ERM loss, so that classification loss does not interfere with the learning of stable features. In datasets with data augmentations, we extend Match DG to also use the perfect object matches obtained from pairs of original and augmented images (MDGHybrid).

We evaluate our matching-based methods on rotated MNIST and Fashion-MNIST, PACS and Chest X-ray datasets. On all datasets, the simple methods Match DG and MDGHybrid are competitive to state-of-the-art methods for out-ofdomain accuracy. On the rotated MNIST and Fashion MNIST datasets where the ground-truth objects are known, Match DG learns to makes the representation more similar to their ground-truth matches (about 50% overlap for top-10 matches), even though the method does not have access to them. Our results with simple matching methods show the importance of enforcing the correct invariance condition.

Contributions. To summarize, our contributions include: 1). An object-invariant condition for domain generalization that highlights a key limitation of previous approaches, 2). When object information is not available, a two-phase, iterative algorithm to approximate object-based matches. Also, the code repository can be accessed at: https:// github.com/microsoft/robustdg

2. Related Work

Learning common representation. To learn a generalizable classifier, several methods enforce the learnt representation Φ(x) to be independent of domain marginally or conditional on class label, using divergence measures such as maximum mean discrepancy (Muandet et al., 2013; Li et al., 2018b;c), adversarial training with a domain discriminator (Ganin et al., 2016; Li et al., 2018d; Albuquerque et al., 2020a), discriminant analysis (Ghifary et al., 2016; Hu et al., 2019), and other techniques (Ghifary et al., 2015).

Among them, several works (Zhao et al., 2019; Johansson et al., 2019; Akuzawa et al., 2019) show that the class-

conditional methods (Li et al., 2018c;d; Ghifary et al., 2016; Hu et al., 2019) are better than those that enforce marginal domain-invariance of features (Muandet et al., 2013; Ganin et al., 2016; Li et al., 2018b; Albuquerque et al., 2020a), whenever there is a varying distribution of class labels across domains. We show that the class-conditional invariant is also not sufficient for generalizing to unseen domains.

Causality and domain generalization. Past work has shown the connection between causality and generalizable predictors (Peters et al., 2016; Christiansen et al., 2020). There is work on use of causal reasoning for domain adaptation (Gong et al., 2016; Heinze-Deml & Meinshausen, 2019; Magliacane et al., 2018; Rojas-Carulla et al., 2018) that assumes Y X direction and other work (Arjovsky et al., 2019; Peters et al., 2016) on connecting causality that assumes X Y . Our SCM model unites these streams by introducing Ytrue and labelled Y and develop an invariance condition for domain generalization that is valid under both interpretations. Perhaps the closest to our work is by (Heinze-Deml & Meinshausen, 2019) who use the object concept in single-domain datasets for better generalization. We extend their SCM to the multi-domain setting and use it to show the inconsistency of prior methods. In addition, while (Heinze-Deml & Meinshausen, 2019) assume objects are always observed, we also provide an algorithm for the case when objects are unobserved.

Matching and Contrastive Loss. Regularizers based on matching have been proposed for domain generalization. (Motiian et al., 2017) proposed matching representations of inputs from the same class. (Dou et al., 2019) used a contrastive (triplet) loss to regularize the ERM objective. In contrast to regularizing based on contrastive loss, our algorithm Match DG proceeds in two phases and learns a representation independent of the ERM objective. Such an iterative 2-phase algorithm has empirical benefits, as we will show in Suppl. D.4. Additionally, we propose an ideal object-based matching algorithm when objects are observed.

Other work. Others approaches to domain generalization include meta-learning (Li et al., 2018a; Balaji et al., 2018), dataset augmentation (Volpi et al., 2018; Shankar et al., 2018), parameter decomposition (Piratla et al., 2020; Li et al., 2017), and enforcing domain-invariance of the optimal P(Y |Φ(x)) (Arjovsky et al., 2019; Ahuja et al., 2020). We empirically compare our algorithm to some of them.

3. Insufficiency of class-conditional invariance

Consider a classification task where the learning algorithm has access to i.i.d. data from m domains, {(di, xi, yi)}n i=1 (Dm, X, Y)n where di Dm and Dm D is a set of m domains. Each training input (d, x, y) is sampled from an unknown distribution

Domain Generalization using Causal Matching

(a) Simple Example

(b) Slab Dataset (Slab (y-axis) is the stable feature)

Figure 1. Two datasets showing the limitations of class-conditional domain-invariance objective. a) The CDM predictor is domain-invariant given the class label but does not generalize to the target domain; b) Colors denote the two ground-truth class labels. For class prediction, the linear feature exhibits varying level of noise across domains. The stable slab feature also has noise but it is invariant across domains.

Pm(D, X, Y ). The domain generalization task is to learn a single classifier that generalizes well to unseen domains d Dm and to new data from the same domains (Shankar et al., 2018). The optimum classifier can be written as: f = arg minf F E(d,x,y) P[l(y(d), f(x(d)))], where (d, x, y) P over (D, X, Y).

As mentioned above, a popular line of work enforces that the learnt representation Φ(x) be independent of domain conditional on the class (Li et al., 2018c;d; Ghifary et al., 2016; Hu et al., 2019), Φ(x) D|Y . Below we present two counter-examples showing that the class-conditional objective is not sufficient.

3.1. A simple counter-example

We construct an example where Φ(x) D|Y , but still the classifier does not generalize to new domains. Consider a two dimensional problem where x1 = xc + αd; x2 = αd where xc and αd are unobserved variables, and αd varies with domain (Figure 1(a)). The true function depends only on the stable feature xc, y = f(xc) = I(xc 0). Suppose there are two training domains with α1 = 1 for domain 1 and α2 = 2 for domain 2, and the test domain has α3 = 0 (see Figure 1(a)). Suppose further that the conditional distribution of XC given Y is a uniform distribution that changes across domains: for domain 1, Xc|Y = 1 U(1, 3); Xc|Y = 0 U( 2, 0); and for domain 2, Xc|Y = 1 U(0, 2); Xc|Y = 0 U( 3, 1). Note that the distributions are picked such that φ(x1, x2) = x1 satisfies the conditional distribution invariant, φ(x) D|Y . The optimal ERM classifier based on this representation, (I(x1 1.5) has 100% train accuracy on both domains. But for the test domain with αd = 0; Xc|Y = 1 U(0, 2); Xc|Y = 0 U( 2, 0), the classifier fails to generalize. It obtains 62.5% test accuracy (and 25% accuracy on the positive class), even though its representation satisfies class-conditional domain invariance. In comparison, the ideal representation is x1 x2 which attains 100% train accuracy and 100% test domain accuracy,

and does not satisfy the class-conditional invariant.

The above counter-example is due to the changing distribution of xc across domains. If P(Xc|Y ) stays the same across domains, then class-conditional methods would not incorrectly pick x1 as the representation. Following (Akuzawa et al., 2019), we claim the following (proof in Suppl. B.3). Proposition 1. Under the domain generalization setup as above, if P(Xc|Y ) remains the same across domains where xc is the stable feature, then the class-conditional domaininvariant objective for learning representations yields a generalizable classifier such that the learnt representation Φ(x) is independent of the domain given xc. Specifically, the entropy H(d|xc) = H(d|Φ, xc).

However, if P(XC|Y ) changes across domains, then we cannot guarantee the same: H(d|xc) and H(d|Φ, xc) may not be equal. For building generalizable classifiers in such cases, this example shows that we need an additional constraint on Φ, H(d|xc) = H(d|Φ, xc); i.e. domain and representation should be independent conditioned on xc.

3.2. An empirical study of class-conditional methods

As a more realistic example, consider the slab dataset introduced for detecting simplicity bias in neural networks (Shah et al., 2020) that contains a feature with spurious correlation. It comprises of two features and a binary label; (x1) has a linear relationship with the label and the other feature (x2) has a piece-wise linear relationship with the label which is a stable relationship. The relationship of the linear feature with the label changes with domains (A.1); we do so by adding noise with probability ϵ = 0 for domain 1 and ϵ = 0.1 for domain 2. On the third (test) domain, we add noise with probability 1 (see Figure 1(b)). We expect that methods that rely on the spurious feature x1 would not be able to perform well on the out-of-domain data.

The results in Table 1 (implementation details in Appendix A.1) show that ERM is unable to learn the slab feature, as evident by poor generalization to the target domain, de-

Domain Generalization using Causal Matching

spite very good performance on the source domains. We also show that methods based on learning invariant representations by unconditional (DANN, MMD, CORAL) and conditional distribution matching (CDANN, C-MMD, C-CORAL), and matching same-class inputs (Random Match) (Motiian et al., 2017) fail to learn the stable slab feature. Note that Proposition 1 suggested the failure of conditional distribution matching (CDM) algorithms when the distribution of stable feature (slab feature) is different across the source domains. However, the slab dataset has similar distribution of stable feature (slabs) across the source domains, yet the CDM algorithms fail to generalize to the target domain. It can be explained by considering the spurious linear feature, which can also satisfy the CDM constraint by shifting the y-conditional distributions along the linear feature. We conjecture that the model may first learn the linear feature due to its simplicity (Shah et al., 2020), and then retain the spurious linear feature upon further optimization since it satisfies the CDM constraint. This shows that the CDM methods can empirically fail even when there is an equal distribution of stable features across domains.

How can we ensure that a model learns the stable, generalizable feature x2? We turn to our example above, where the required invariant was that the representation Φ(x) should be independent of domain given the stable feature. We apply this intuition and construct a model that enforces that the learnt representation be independent of domain given x2. We do so by minimizing the ℓ2-norm of the representations for data points from different domains that share the same slab value (details of the Perfect Match method in Section 4.3). The results improve substantially: out-of-domain accuracy is now 78%.

In the next section, we formalize the intuition of conditioning on stable features xc using a causal graph, and introduce the concept of objects that act as proxies of stable features.

4. A Causal View of Domain Generalization

4.1. Data-generating process

Figure 2(a) shows a structural causal model (SCM) that describes the data-generating process for the domain generalization task. For intuition, consider a task of classifying the type of item or screening an image for a medical condition. Due to human variability or by design (using data augmentation), the data generation process yields variety of images for each class, sometimes multiple views for the same object. Here each view can be considered as a different domain D, the label for item type or medical condition as the class Y , and the image pixels as the features X. Photos of the same item or the same person correspond to a common object variable, denoted by O. To create an image, the data-generating process first samples an object and view

Table 1. Slab Dataset: Source domains with noisy linear component with probability 0.0 and 0.1, target domain with noise 1.0. Mean and standard deviation over 10 different seed values for each method. The results for DANN (Ganin et al., 2016), CDANN (Li et al., 2018d), MMD, C-MMD (Li et al., 2018b), CORAL, CCORAL (Sun & Saenko, 2016) were computed by using their implementations in Domain Bed (Gulrajani & Lopez-Paz, 2020).

Method Source 1 Source 2 Target

ERM 100.0 (0.0 ) 96.0 (0.25) 57.6 (6.58) DANN 99.9 (0.07) 94.8 (0.25) 53.0 (1.41) MMD 99.9 (0.01) 95.9 (0.27) 62.9 (5.01) CORAL 99.9 (0.01) 96.0 (0.27) 63.1 (5.86)

Rand Match 100.0 (0.0) 96.1 (0.22) 59.5 (3.50) CDANN 99.9 (0.01) 96.0 (0.27) 55.9 (2.47) C-MMD 99.9 (0.01) 96.0 (0.27) 58.9 (3.43) C-CORAL 99.9 (0.01) 96.0 (0.27) 64.7 (4.69)

Perf Match 99.9 (0.05) 97.8 (0.28) 77.8 (6.01)

Object Domain

(a) Image classification.

(b) General SCM.

Figure 2. Structural causal models for the data-generating process. Observed variables are shaded; dashed arrows denote correlated nodes. Object may not be observed.

(domain) that may be correlated to each other (shown with dashed arrows). The pixels in the photo are caused by both the object and the view, as shown by the two incoming arrows to X. The object also corresponds to high-level causal features XC that are common to any image of the same object, which in turn are used by humans to label the class Y . We call XC as causal features because they directly cause the class Y .

The above example is typical of a domain generalization problem; a general SCM is shown in Figure 2(b), similar to the graph in (Heinze-Deml & Meinshausen, 2019). In general, the underlying object for each input x(d) i may not be observed. Analogous to the object-dependent (causal) features XC, we introduce a node for domain-dependent high-level features of the object XA. Changing the domain can be seen as an intervention: for each observed x(d) i , there are a set of (possibly unobserved) counterfactual inputs x(d ) j where d = d , such that all correspond to the same object (and thus share the same XC). For completeness, we also show the true unobserved label of the object which led to its generation as Ytrue (additional motivation for the

Domain Generalization using Causal Matching

causal graph is in Suppl. B.1). Like the object O, Y may be correlated with the domain D. Extending the model in (Heinze-Deml & Meinshausen, 2019), we allow that objects can be correlated with the domain conditioned on Ytrue. As we shall see, considering the relationship of the object node becomes the key piece for developing the invariant condition. The SCM corresponds to the following nonparametric equations.

o := go(ytrue, ϵo, ϵod) xc = gxc(o) xa := gxa(d, o, ϵxa) x := gx(xc, xa, ϵx)y := h(xc, ϵy)

where go, gxc, gxa, gx and h are general non-parametric functions. The error ϵod is correlated with domain d whereas ϵo, ϵxa, ϵx and ϵy are mutually independent error terms that are independent of all other variables. Thus, noise in the class label is independent of domain. Since xc is common to all inputs of the same object, gxc is a deterministic function of o. In addition, the SCM provides conditionalindependence conditions that all data distributions P must satisfy, through the concept of d-separation (Suppl. B.2) and the perfect map assumption (Pearl, 2009).

4.2. Identifying the invariance condition

From Figure 2(b), XC is the node that causes Y . Further, by d-separation, the class label is independent of domain conditioned on XC, Y D|XC. Thus our goal is to learn y as h(xc) where h : C Y. The ideal loss-minimizing function f can be rewritten as (assuming xc is known):

arg min f E(d,x,y)l(y, f(x)) = arg min h E[l(y, h(xc))] (1)

Since XC is unobserved, this implies that we need to learn it through a representation function Φ : X C. Together, h(Φ(x)) leads to the desired classifer f : X Y.

Negative result on identification. Identification of causal features is a non-trivial problem (Magliacane et al., 2018). We first show that x C is unidentifiable given observed data P(X, Y, D, O) over multiple domains. Given the same probability distribution P(X, Y, D, O), multiple values of XC are possible. Substituting for o in the SCM equations, we obtain, y = h(gxc(o), ϵy); x = gx(gxc(o), gxa(d, o, ϵxa), ϵx). By choosing gx and h appropriately, different values of gxc (that determine xc from o) can lead to the same observed values for (y, d, o, x). The proof for the following proposition is in Supp. B.4. Proposition 2. Given observed data distribution P(Y, X, D, O) that may also include data obtained from interventions on domain D, multiple values of XC yield exactly the same observational and interventional distributions and hence Xc is unidentifiable.

4.3. A perfect-match invariant

In the absence of identifiability, we proceed to find an invariant that can characterize Xc. By the d-separation criterion,

we see that XC satisfies two conditions: 1) XC D|O, 2) XC O; where O refers to the object variable and D refers to a domain. The first is an invariance condition: XC does not change with different domains for the same object. To enforce this, we stipulate that the average pairwise distance between Φ(x) for inputs across domains for the same object is 0, P

Ω(j,k)=1;d =d dist(Φ(x(d) j ), Φ(x(d ) k )) = 0. Here Ω: X X {0, 1} is a matching function that is 1 for pairs of inputs across domains corresponding to the same object, and 0 otherwise.

However, just the above invariance will not work: we need the representation to be informative of the object O (otherwise even a constant Φ minimizes the above loss). Therefore, the second condition stipulates that XC should be informative of the object, and hence about Y . We add the standard classification loss, leading to constrained optimization,

fperfectmatch = arg minh,Φ Pm d=1 Ld(h(Φ(X)), Y )

s.t. P Ω(j,k)=1;d =d dist(Φ(x(d) j ), Φ(x(d ) k )) = 0 (2)

where Ld(h(Φ(X), Y )) = Pnd i=1 l(h(Φ(x(d) i ), y(d) i ). Here f represents the composition h Φ. E.g., a neural network with Φ(x) as its rth layer, and h being the rest of the layers.

Note that there can be multiple Φ(x) (e.g., linear transformations) that are equally good for the prediction task. Since xc is unidentifiable, we focus on the set of stable representations that are d-separated from D given O. Being independent of domain given the object, they cannot have any association with Xa, the high-level features that directly depend on domain (Figure 2b). The proof for the next theorem is in Suppl. B.5.

Theorem 1. For a finite number of domains m, as the number of examples in each domain nd , 1. The set of representations that satisfy the condition P Ω(j,k)=1;d =d dist(Φ(x(d) j ), Φ(x(d ) k )) = 0 contains the optimal Φ(x) = XC that minimizes the domain generalization loss in (1). 2. Assuming that P(Xa|O, D) < 1 for every high-level feature Xa that is directly caused by domain, and for P-admissible loss functions (Miller et al., 1993) whose minimization is conditional expectation (e.g., ℓ2 or crossentropy), a loss-minimizing classifier for the following loss is the true function f , for some value of λ.

fperfectmatch = arg minh,Φ Pm d=1 Ld(h(Φ(X)), Y )+

λ P Ω(j,k)=1;d =d dist(Φ(x(d) j ), Φ(x(d ) k )) (3)

4.4. Past work: Learning common representation

Using the SCM, we now compare the proposed invariance condition to domain-invariant and class-conditional domain-invariant objectives. d-separation results show that

Domain Generalization using Causal Matching

both these objectives are incorrect: in particular, the classconditional objective Φ(x) D|Y is not satisfied by XC, (XC D|Ytrue) due to a path through O. Even with infinite data across domains, they will not learn the true XC. The proof is in Suppl. B.6.

Proposition 3. The conditions enforced by domaininvariant (Φ(x) D) or class-conditional domaininvariant (Φ(x) D|Y ) methods are not satisfied by the causal representation XC. Thus, without additional assumptions, the set of representations that satisfy any of these conditions does not contain XC, even as n .

5. Match DG: Matching without objects

When object information is available, Eq. (3) provides a loss objective to build a classifer using causal features. However, object information is not always available, and in many datasets there may not be a perfect counterfactual match based on same object across domains. Therefore, we propose a two-phase, iterative contrastive learning method to approximate object matches.

The object-invariant condition from Section 4.2 can be interpreted as matching pairs of inputs from different domains that share the same XC. To approximate it, our goal is to learn a matching Ω: X X {0, 1} such that pairs having Ω(x, x ) = 1 have low difference in xc and x c. We make the following assumption.

Assumption 1. Let (x(d) i , y), (x(d ) j , y) be any two points

that belong to the same class, and let (x(d) k y ) be any other point that has a different class label. Then the distance in causal features between xi and xj is smaller than that between xi and xk or xj and xk: dist(x(d) c,i , x(d ) c,j )

dist(x(d) c,i , x(d ) c,k ) and dist(x(d) c,j , x(d ) c,i ) dist(x(d) c,j , x(d ) c,k ).

5.1. Two-phase method with iterative matches

To learn a matching function Ω, we use unsupervised contrastive learning from (Chen et al., 2020; He et al., 2019) and adapt it to construct an iterative Match DG algorithm that updates the both the representation and matches after each epoch. The algorithm relies on the property that two inputs from the same class have more similar causal features than inputs from different classes.

Contrastive Loss. To find matches, we optimize a contrastive representation learning loss that minimizes distance between same-class inputs from different domains in comparison to inputs from different classes across domains. Adapting the contrastive loss for a single domain (Chen et al., 2020), we consider positive matches as two inputs with the same class but different domains, and negative matches as pairs with different classes. For every positive match pair (xj, xk), we propose a loss where τ is

Algorithm 1 Match DG

In: Dataset (di, xi, yi)n i=1 from m domains, τ, t Out: Function f : X Y Create random match pairs ΩY . Build a p q data matrix M. Phase I while notconverged do

for batch M do

Minimize contrastive loss (4). end for if epoch % t == 0 then

Update match pairs using Φepoch. end if end while Phase II Compute matching based on Φ. Minimize the loss (3) with learnt match function Φ to obtain f.

a hyperparameter, B is the batch size, and sim(a, b) = Φ(xa)T Φ(xb)/ Φ(xa) Φ(xb) is the cosine similarity.

l(xj, xk) = log esim(j,k)/τ

esim(j,k)/τ + PB i=0,yi =yj esim(j,i)/τ (4)

Iterative matching. Our key insight is to update the positive matches during training. We start training with a random set of positive matches based on the classes, but after every t epochs, we update the positive matches based on the nearest same-class pairs in representation space and iterate until convergence. Hence for each anchor point, starting with an initial set of positive matches, in each epoch a representation is learnt using contrastive learning; after which the positive matches are themselves updated based on the closest same-class data points across domains in the representation. As a result, the method differentiates between data points of the same class instead of treating all of them as a single unit. With iterative updates to the positive matches, the aim is to account for intra-class variance across domains and match data points across domains that are more likely to share the same base object. In Suppl. D.6, we compare the gains due to the proposed iterative matching versus standard contrastive training.

Obtaining the final representation completes Phase I of the algorithm. In Phase II, we use this representation to compute a new match function based on closest same-class pairs and apply Eq. (3) to obtain a classifier regularized on those matches.

The importance of using two phases. We implement Match DG as a 2-phase method, unlike previous methods (Motiian et al., 2017; Dou et al., 2019) that employed class-based contrastive loss as a regularizer with ERM. This is to avoid the classification loss interfering with the goal of learning an invariant representation across domains (e.g., in datasets where one of the domains has many more samples than others). Therefore, we first learn the match function

Domain Generalization using Causal Matching

using only the contrastive loss. Our results in Suppl. D.4 show that the two-phase method provides better overlap with ground-truth perfect matches than optimizing classification and matching simultaneously.

To implement Match DG we build a p q data matrix containing q 1 positive matches for each input and then sample mini-batches from this matrix. The last layer of the contrastive loss network is considered as the learnt representation (see Algorithm 1; details are in Suppl. C.1).

5.2. MDG Hybrid

While Match DG assumes no information about objects, it can be easily augmented to incorporate information about known objects. For example, in computer vision, a standard practice is to augment data by performing rotations, horizontal flips, color jitter, etc. These self-augmentations provide us with access to known objects, which can included as perfect-matches in Match DG Phase-II by adding another regularizer to the loss from Eq 3. We name this method MDGHybrid and evaluate it alongside Match DG for datasets where we can perform self augmentations.

6. Evaluation

We evaluate out-of-domain accuracy of Match DG on two simulated benchmarks by Piratla et al. (2020), Rotated MNIST and Fashion-MNIST, on PACS dataset (Li et al., 2017), and on a novel Chest X-rays dataset. In addition, using the simulated datasets, we inspect the quality of matches learnt by Match DG by comparing them to groundtruth object-based matches. For PACS and Chest X-rays, we also implement MDGHybrid that uses augmentations commonly done while training neural networks. We compare to 1) ERM: Standard empirical risk minimization, 2) ERM-Rand Match that implements the loss from Eq. (3) but with randomly selected matches from the same class (Motiian et al., 2017), 3) other state-of-the-art methods for each dataset. For all matching-based methods, we use the cross-entropy loss for Ld and ℓ2 distance for dist in Eq.(3). Details of implementation and the datasets are in Suppl. C.1. All the numbers are averaged over 3 runs with standard deviation in brackets.

Rotated MNIST & Fashion-MNIST. The datasets contain rotations of grayscale MNIST handwritten digits and fashion article images from 0 to 90 with an interval of 15 (Ghifary et al., 2015), where each rotation angle represents a domain and the task is to predict the class label. Since different domains images are generated from the same base image (object), there exist perfect matches across domains. Following CSD, we report accuracy on 0 and 90 together as the test domain and the rest as the train domains; since these test angles, being extreme, are the hardest to generalize

to (standard setting results are in Suppl. D.1, D.2).

PACS. This dataset contains total 9991 images from four domains: Photos (P), Art painting (A), Cartoon (C) and Sketch (S). The task is to classify objects over 7 classes. Following (Dou et al., 2019), we train 4 models with each domain as the target using Resnet-18, Resnet-50 and Alexnet.

Chest X-rays. We introduce a harder real-world dataset based on Chest X-ray images from three different sources: NIH (Wang et al., 2017), Chex Pert (Irvin et al., 2019) and RSNA (rsn, 2018). The task is to detect whether the image corresponds to a patient with Pneumonia (1) or not (0). To create spurious correlation, all images of class 0 in the training domains are translated vertically downwards; while no such translation is done for the test domain.

Model Selection. While using a validation set from the test domain may improve classification accuracy, it goes against the problem motivation of generalization to unseen domains. Hence, we use only data from source domains to construct a validation set (except when explicitly mentioned in Table 4, to compare to past methods that use test domain validation).

6.1. Rotated MNIST and Fashion MNIST

Table 2 shows classification accuracy on rot MNIST and rot Fashion MNIST for test domains 0 & 90 using Resnet-18 model. On both datasets, Match DG outperforms all baselines. The last column shows the accuracy for an oracle method, ERM-Perf Match that has access to ground-truth perfect matches across domains. Match DG s accuracy lies between ERM-Rand Match and ERM-Perf Match, indicating the benefit of learning a matching function. As the number of training domains decrease, the gap between Match DG and baselines is highlighted: with 3 source domains for rot Fashion MNIST, Match DG achieves accuracy of 43.8% whereas the next best method ERM-Rand Match achieves 38.4%.

We also evaluate on a simpler 2-layer Le Net (Motiian et al., 2017), and the model from (Gulrajani & Lopez-Paz, 2020) to compare Match DG to prior works (Ilse et al., 2020; Ganin et al., 2016; Shankar et al., 2018; Goodfellow et al., 2014); the results are in Suppl. D.1, D.2.

Why Match DG works? We compare the matches returned by Match DG Phase I (on Resnet-18 network) to the groundtruth perfect matches and find that it has significantly higher overlap than matching based on ERM loss (Table 3). We report three metrics on the representation learnt: percentage of Match DG matches that are perfect matches, %-age of inputs for which the perfect match is within the top-10 ranked Match DG matches, and mean rank of perfect matches measured by distance over the Match DG representation.

On all three metrics, Match DG finds a representation whose

Domain Generalization using Causal Matching

Table 2. Accuracy for Rotated MNIST & Fashion-MNIST datasets on target domains of 0 and 90 . Accuracy for CSD (Piratla et al.,

2020), MASF (Dou et al., 2019), IRM (Arjovsky et al., 2019) are reproduced from their code. Results for the other versions of Rotated MNIST with all test angles (Let Net (Motiian et al., 2017), Domain Bed (Gulrajani & Lopez-Paz, 2020)) are in Suppl. D.1, D.2.

Dataset Source ERM MASF CSD IRM Rand Match Match DG Perf Match (Oracle)

Rotated MNIST

15, 30, 45, 60, 75 93.0 (0.11) 93.2 (0.2) 94.5 (0.35) 92.8 (0.53) 93.4 (0.26) 95.1 (0.25) 96.0 (0.41)

30, 45, 60 76.2 (1.27) 69.4 (1.32) 77.7 (1.88) 75.7 (1.11) 78.3 (0.55) 83.6 (1.44) 89.7 (1.68)

30, 45 59.7 (1.75) 60.8 (1.53) 62.0 (1.31) 59.5 (2.61) 63.8 (3.92) 69.7 (1.30) 80.4 (1.79)

Rotated Fashion MNIST

15, 30, 45, 60, 75 77.9 (0.13) 72.4 (2.9) 78.7 (0.38) 77.8 (0.02) 77.0 (0.42) 80.9 (0.26) 81.6 (0.46)

30, 45, 60 36.1 (1.91) 29.7 (1.73) 36.3 (2.65) 37.8 (1.85) 38.4 (2.73) 43.8 (1.33) 54.0 (2.79)

30, 45 26.1 (1.10) 22.8 (1.26) 24.2 (1.69) 26.6 (1.06) 26.9 (0.34) 33.0 (0.72) 41.8 (1.78)

Table 3. Overlap with perfect matches. top-10 overlap and the mean rank for perfect matches for Match DG and ERM over all training domains. Lower is better for mean rank.

Dataset Method Overlap (%) Top 10 Overlap (%) Mean Rank

ERM 15.8 (0.42) 48.8 (0.78) 27.4 (0.89) Match DG (Default) 28.9 (1.24) 64.2 (2.42) 18.6 (1.59)

Match DG (Perf Match) 47.4 (2.25) 83.8 (1.46) 6.19 (0.61)

Fashion MNIST

ERM 2.1 (0.12) 11.1 (0.63) 224.3 (8.73) Match DG (Default) 17.9 (0.62) 43.1 (0.83) 89.0 (3.15)

Match DG (Perf Match) 56.2 (1.79) 87.2 (1.48) 7.3 (1.18)

matches are more consistent with ground-truth perfect matches. For both rot MNIST and rot Fashion MNIST datasets, about 50% of the inputs have their perfect match within top-10 ranked matches based on the representation learnt by Match DG Phase I. About 25% of all matches learnt by Match DG are perfect matches. For comparison, we also show metrics for an (oracle) Match DG method that is initialized with perfect matches: it achieves better overall and Top-10 values. Similar results for Match DG Phase 2 are in Suppl. D.4. Mean rank for rot Fashion MNIST may be higher because of the larger sample size 10, 000 per domain; metrics for training with 2000 samples are in Suppl. D.5. To see how the overlap with perfect matches affects accuracy, we simulate random matches with 25%, 50% and 75% overlap with perfect matches (Suppl. Tbl. D.3). Accuracy increases with the fraction of perfect matches, indicating the importance of capturing good matches.

Match DG vs. IRM on zero training error. Since neural networks often achieve zero training error, we also evaluate the effectiveness of the Match DG regularization under this regime. Fig. 3 shows the matching loss term as training proceeds for rot MNIST and rot Fashion MNIST. Even

(a) Match DG Penalty during training

(b) IRM Penalty during training

Figure 3. Match DG regularization penalty is not trivially minimized even as the training error goes to zero.

after the model achieves zero training error, we see that plain ERM objective is unable to minimize the matching loss (and thus Match DG penalty is needed). This is because Match DG regularization depends on comparing the (last layer) representations, and zero training error does not mean that the representations within each class are the same. In contrast, regularizations that are based on comparing loss between training domains such as the IRM penalty can be satisfied by plain ERM as the training error goes to zero (Fig. 3(b)); similar to Fig. (5) from (Krueger et al., 2020) where ERM can minimize IRM penalty on Colored MNIST.

6.2. PACS dataset

Res Net-18. On the PACS dataset with Res Net-18 architecture (Table 4), our methods are competitive to state-of-the-

Domain Generalization using Causal Matching

Table 4. Accuracy on PACS with Res Net 18 (default), and Resnet 18 with test domain validation. The results for Ji Gen (Carlucci et al., 2019), DDAIG (Zhou et al., 2020), Sag Net (Nam et al., 2019), DDEC (Asadi et al., 2019), were taken from the Domain Bed (Gulrajani & Lopez-Paz, 2020) paper. For G2DM (Albuquerque et al., 2020a), CSD (Piratla et al., 2020), RSC (Huang et al., 2020) it was taken from the respective paper. Extensive comparison with other works and std. dev. in results is in Supp E.1.

P A C S Average.

ERM 95.38 77.68 78.98 74.75 81.70 Ji Gen 96.0 79.42 75.25 71.35 80.41 G2DM 93.75 77.78 75.54 77.58 81.16 CSD 94.1 78.9 75.8 76.7 81.4 DDAIG 95.30 84.20 78.10 74.70 83.10 Sag Net 95.47 83.58 77.66 76.30 83.25 DDEC 96.93 83.01 79.39 78.62 84.46 RSC 95.99 83.43 80.31 80.85 85.15 Rand Match 95.37 78.16 78.83 75.13 81.87 Match DG 95.93 79.77 80.03 77.11 83.21 MDGHybrid 96.15 81.71 80.75 78.79 84.35

G2DM (Test) 94.63 81.44 79.35 79.52 83.34 Rand Match (Test) 95.57 79.09 79.37 77.60 82.91 Match DG (Test) 96.53 81.32 80.70 79.72 84.56 MDGHybrid (Test) 96.67 82.80 81.61 81.05 85.53

Table 5. Accuracy on PACS with architecture Res Net 50. The results for IRM (Arjovsky et al., 2019), CORAL (Sun & Saenko, 2016), were taken from the Domain Bed (Gulrajani & Lopez-Paz, 2020) paper. The result for RSC (Huang et al., 2020) was taken from their paper. Comparison with other works in Supp E.1.

P A C S Average.

Domain Bed (Res Net50) 97.8 88.1 77.9 79.1 85.7 IRM (Res Net50) 96.7 85.0 77.6 78.5 84.4 CORAL (Res Net50) 97.6 87.7 79.2 79.4 86.0 RSC (Res Net50) 97.92 87.89 82.16 83.35 87.83 Rand Match (Res Net50) 97.89 82.16 81.68 80.45 85.54 Match DG (Res Net50) 97.94 85.61 82.12 78.76 86.11 MDGHybrid (Res Net50) 98.36 86.74 82.32 82.66 87.52

art results averaged over all domains. The MDGHybrid has the highest average accuracy across domains, except compared to DDEC and RSC. These works do not disclose their model selection strategy (whether the results are using source or test domain validation). Therefore, we also report results of Match DG and MDGHybrid using test domain validation, where MDGHybrid obtains comparable results to the best-performing method. In addition, with DDEC (Asadi et al., 2019), it is not a fair comparison since they use additional style transfer data from Behance BAM! dataset during training.

Res Net-50. We implement Match DG on Resnet50 model (Table 5) used by the ERM in Domain Bed. Adding Match DG loss regularization improves the accuracy of Domain Bed, from 85.7 to 87.5 with MDGHybrid. Also, MDGHybrid performs better than the prior approaches us-

Table 6. Chest X-Rays data. As an upper bound, training ERM on the target domain itself yields 73.8%, 66.5%, and 59.9% accuracy for RSNA, Chex Pert, and NIH respectively.

RSNA Chex Pert NIH

ERM 55.1 (2.93) 60.9 (0.51) 53.4 (1.36) IRM 57.0 (0.75) 63.3 (0.25) 54.6 (0.88) CSD 58.6 (1.63) 64.4 (0.88) 54.7 (0.13) Rand Match 56.3 (3.38) 55.3 (2.25) 53.1 (0.13) Match DG 58.2 (1.25) 59.0 (0.25) 53.2 (0.65) MDGHybrid 64.3 (0.75) 60.6 (0.25) 57.6 (0.13)

ing Resnet50 architecture, except RSC (Huang et al., 2020), whose results (87.83) are close to ours (87.52). Note that we chose a subset of the best-performing baselines for Table 4, 5; an extensive comparison with other works is in Suppl. E.1. Suppl. E.2 gives the results using Alex Net network, and a t-SNE plot (Figure 5) to show the quality of representation learnt by Match DG.

6.3. Chest X-rays dataset

Table 6 provides results for the Chest X-rays dataset, where the spurious correlation of vertical translation with the class label in source domains may lead the models to learn an unstable relationship. With RSNA as the target domain, ERM obtains 79.8%, 81.8% accuracy on the source domains while its accuracy drops to 55.1% for the target domain. In contrast, MDGHybrid obtain the highest classification accuracy (8 % above ERM), followed by CSD and Match DG; while methods like ERM and IRM are more susceptible to spurious correlation. However, on Chex Pert as the target domain, CSD and IRM do better than ERM while matchingbased methods are not effective. We conjecture these varying trends might be due to the inherent variability in images in the source domains, indicating the challenges of building domain generalization methods for real-world datasets.

7. Conclusion

We presented a causal view of domain generalization that provides an object-conditional objective. Simple matchingbased methods perform competitively to state-of-the-art methods on PACS, indicating the importance of choosing the right invariance. The proposed Match DG uses certain assumptions when objects are unknown. More work needs to be done to develop better matching methods, as indicated by the mixed results on the Chest-Xrays dataset.

Acknowledgements. We would like to thank Adith Swaminathan, Aditya Nori, Emre Kiciman, Praneeth Netrapalli, Tobias Schnabel, Vineeth Balasubramanian and the reviewers who provided us valuable feedback on this work. We also thank Vihari Piratla who helped us with reproducing the CSD method and other baselines.

Domain Generalization using Causal Matching

Kaggle: Rsna pneumonia detection challenge, 2018. URL https://www.kaggle.com/c/ rsna-pneumonia-detection-challenge.

Ahuja, K., Shanmugam, K., Varshney, K., and Dhurandhar, A. Invariant risk minimization games. ar Xiv preprint ar Xiv:2002.04692, 2020.

Akuzawa, K., Iwasawa, Y., and Matsuo, Y. Adversarial invariant feature learning with accuracy constraint for domain generalization. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 315 331. Springer, 2019.

Albuquerque, I., Monteiro, J., Darvishi, M., Falk, T. H., and Mitliagkas, I. Generalizing to unseen domains via distribution matching, 2020a.

Albuquerque, I., Naik, N., Li, J., Keskar, N., and Socher, R. Improving out-of-distribution generalization via multi-task self-supervised pretraining. ar Xiv preprint ar Xiv:2003.13525, 2020b.

Arjovsky, M., Bottou, L., Gulrajani, I., and Lopez Paz, D. Invariant risk minimization. ar Xiv preprint ar Xiv:1907.02893, 2019.

Asadi, N., Sarfi, A. M., Hosseinzadeh, M., Karimpour, Z., and Eftekhari, M. Towards shape biased unsupervised representation learning for domain generalization. ar Xiv preprint ar Xiv:1909.08245, 2019.

Balaji, Y., Sankaranarayanan, S., and Chellappa, R. Metareg: Towards domain generalization using meta-regularization. In Advances in Neural Information Processing Systems, pp. 998 1008, 2018.

Carlucci, F. M., D Innocente, A., Bucci, S., Caputo, B., and Tommasi, T. Domain generalization by solving jigsaw puzzles. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2229 2238, 2019.

Chen, T., Kornblith, S., Norouzi, M., and Hinton, G. A simple framework for contrastive learning of visual representations. ar Xiv preprint ar Xiv:2002.05709, 2020.

Christiansen, R., Pfister, N., Jakobsen, M. E., Gnecco, N., and Peters, J. A causal framework for distribution generalization. ar Xiv e-prints, pp. ar Xiv 2006, 2020.

Cohen, J. P., Hashir, M., Brooks, R., and Bertrand, H. On the limits of cross-domain generalization in automated x-ray prediction. ar Xiv preprint ar Xiv:2002.02497, 2020.

Dou, Q., de Castro, D. C., Kamnitsas, K., and Glocker, B. Domain generalization via model-agnostic learning of semantic features. In Advances in Neural Information Processing Systems, pp. 6447 6458, 2019.

D Innocente, A. and Caputo, B. Domain generalization with domain-specific aggregation modules. In German Conference on Pattern Recognition, pp. 187 198. Springer, 2018.

Ganin, Y., Ustinova, E., Ajakan, H., Germain, P., Larochelle, H., Laviolette, F., Marchand, M., and Lempitsky, V. Domain-adversarial training of neural networks. The Journal of Machine Learning Research, 17(1):2096 2030, 2016.

Ghifary, M., Bastiaan Kleijn, W., Zhang, M., and Balduzzi, D. Domain generalization for object recognition with multi-task autoencoders. In Proceedings of the IEEE international conference on computer vision, pp. 2551 2559, 2015.

Ghifary, M., Balduzzi, D., Kleijn, W. B., and Zhang, M. Scatter component analysis: A unified framework for domain adaptation and domain generalization. IEEE transactions on pattern analysis and machine intelligence, 39(7):1414 1430, 2016.

Gong, M., Zhang, K., Liu, T., Tao, D., Glymour, C., and Sch olkopf, B. Domain adaptation with conditional transferable components. In International conference on machine learning, pp. 2839 2848, 2016.

Goodfellow, I. J., Shlens, J., and Szegedy, C. Explaining and harnessing adversarial examples. ar Xiv preprint ar Xiv:1412.6572, 2014.

Gulrajani, I. and Lopez-Paz, D. In search of lost domain generalization. ar Xiv preprint ar Xiv:2007.01434, 2020.

He, K., Fan, H., Wu, Y., Xie, S., and Girshick, R. Momentum contrast for unsupervised visual representation learning. ar Xiv preprint ar Xiv:1911.05722, 2019.

Heinze-Deml, C. and Meinshausen, N. Conditional variance penalties and domain shift robustness. ar Xiv preprint ar Xiv:1710.11469, 2019.

Hu, S., Zhang, K., Chen, Z., and Chan, L. Domain generalization via multidomain discriminant analysis. In Uncertainty in artificial intelligence: proceedings of the... conference. Conference on Uncertainty in Artificial Intelligence, volume 35. NIH Public Access, 2019.

Huang, Z., Wang, H., Xing, E. P., and Huang, D. Selfchallenging improves cross-domain generalization. ar Xiv preprint ar Xiv:2007.02454, 2, 2020.

Domain Generalization using Causal Matching

Ilse, M., Tomczak, J. M., Louizos, C., and Welling, M. Diva: Domain invariant variational autoencoders. In Medical Imaging with Deep Learning, pp. 322 348. PMLR, 2020.

Irvin, J., Rajpurkar, P., Ko, M., Yu, Y., Ciurea-Ilcus, S., Chute, C., Marklund, H., Haghgoo, B., Ball, R., Shpanskaya, K., et al. Chexpert: A large chest radiograph dataset with uncertainty labels and expert comparison. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 590 597, 2019.

Johansson, F. D., Sontag, D., and Ranganath, R. Support and invertibility in domain-invariant representations. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 527 536, 2019.

Krueger, D., Caballero, E., Jacobsen, J.-H., Zhang, A., Binas, J., Priol, R. L., and Courville, A. Out-of-distribution generalization via risk extrapolation (rex). ar Xiv preprint ar Xiv:2003.00688, 2020.

Li, D., Yang, Y., Song, Y.-Z., and Hospedales, T. M. Deeper, broader and artier domain generalization. In Proceedings of the IEEE international conference on computer vision, pp. 5542 5550, 2017.

Li, D., Yang, Y., Song, Y.-Z., and Hospedales, T. M. Learning to generalize: Meta-learning for domain generalization. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018a.

Li, D., Zhang, J., Yang, Y., Liu, C., Song, Y.-Z., and Hospedales, T. M. Episodic training for domain generalization. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1446 1455, 2019a.

Li, D., Yang, Y., Song, Y.-Z., and Hospedales, T. Sequential learning for domain generalization. ar Xiv preprint ar Xiv:2004.01377, 2020.

Li, H., Jialin Pan, S., Wang, S., and Kot, A. C. Domain generalization with adversarial feature learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5400 5409, 2018b.

Li, Y., Gong, M., Tian, X., Liu, T., and Tao, D. Domain generalization via conditional invariant representations. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018c.

Li, Y., Tian, X., Gong, M., Liu, Y., Liu, T., Zhang, K., and Tao, D. Deep domain generalization via conditional invariant adversarial networks. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 624 639, 2018d.

Li, Y., Yang, Y., Zhou, W., and Hospedales, T. M. Featurecritic networks for heterogeneous domain generalization. ar Xiv preprint ar Xiv:1901.11448, 2019b.

Maaten, L. v. d. and Hinton, G. Visualizing data using t-sne. Journal of machine learning research, 9(Nov): 2579 2605, 2008.

Magliacane, S., van Ommen, T., Claassen, T., Bongers, S., Versteeg, P., and Mooij, J. M. Domain adaptation by using causal inference to predict invariant conditional distributions. In Advances in Neural Information Processing Systems, pp. 10846 10856, 2018.

Matsuura, T. and Harada, T. Domain generalization using a mixture of multiple latent domains. In AAAI, pp. 11749 11756, 2020.

Miller, J. W., Goodman, R., and Smyth, P. On loss functions which minimize to conditional expected values and posterior probabilities. IEEE Transactions on Information Theory, 39(4):1404 1408, 1993.

Motiian, S., Piccirilli, M., Adjeroh, D. A., and Doretto, G. Unified deep supervised domain adaptation and generalization. In Proceedings of the IEEE International Conference on Computer Vision, pp. 5715 5725, 2017.

Muandet, K., Balduzzi, D., and Sch olkopf, B. Domain generalization via invariant feature representation. In International Conference on Machine Learning, pp. 10 18, 2013.

Nam, H., Lee, H., Park, J., Yoon, W., and Yoo, D. Reducing domain gap via style-agnostic networks. ar Xiv preprint ar Xiv:1910.11645, 2019.

Pearl, J. Causality. Cambridge university press, 2009.

Peters, J., B uhlmann, P., and Meinshausen, N. Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(5):947 1012, 2016.

Piratla, V., Netrapalli, P., and Sarawagi, S. Efficient domain generalization via common-specific low-rank decomposition. Proceedings of the International Conference of Machine Learning (ICML) 2020, 2020.

Rahman, M. M., Fookes, C., Baktashmotlagh, M., and Sridharan, S. Correlation-aware adversarial domain adaptation and generalization. Pattern Recognition, 100:107124, 2020.

Rojas-Carulla, M., Sch olkopf, B., Turner, R., and Peters, J. Invariant models for causal transfer learning. The Journal of Machine Learning Research, 19(1):1309 1342, 2018.

Sagawa, S., Koh, P. W., Hashimoto, T. B., and Liang, P. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. ar Xiv preprint ar Xiv:1911.08731, 2019.

Domain Generalization using Causal Matching

Shah, H., Tamuly, K., Raghunathan, A., Jain, P., and Netrapalli, P. The pitfalls of simplicity bias in neural networks. ar Xiv preprint ar Xiv:2006.07710, 2020.

Shankar, S., Piratla, V., Chakrabarti, S., Chaudhuri, S., Jyothi, P., and Sarawagi, S. Generalizing across domains via cross-gradient training. In International Conference on Learning Representations, 2018.

Sun, B. and Saenko, K. Deep coral: Correlation alignment for deep domain adaptation. In European conference on computer vision, pp. 443 450. Springer, 2016.

Volpi, R., Namkoong, H., Sener, O., Duchi, J. C., Murino, V., and Savarese, S. Generalizing to unseen domains via adversarial data augmentation. In Advances in Neural Information Processing Systems, pp. 5334 5344, 2018.

Wang, H., He, Z., Lipton, Z. C., and Xing, E. P. Learning robust representations by projecting superficial statistics out. ar Xiv preprint ar Xiv:1903.06256, 2019.

Wang, X., Peng, Y., Lu, L., Lu, Z., Bagheri, M., and Summers, R. M. Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2097 2106, 2017.

Wang, Y., Li, H., and Kot, A. C. Heterogeneous domain generalization via domain mixup. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3622 3626. IEEE, 2020.

Xu, M., Zhang, J., Ni, B., Li, T., Wang, C., Tian, Q., and Zhang, W. Adversarial domain adaptation with domain mixup. ar Xiv preprint ar Xiv:1912.01805, 2019.

Yan, S., Song, H., Li, N., Zou, L., and Ren, L. Improve unsupervised domain adaptation with mixup training. ar Xiv preprint ar Xiv:2001.00677, 2020.

Zhao, H., Combes, R. T. d., Zhang, K., and Gordon, G. J. On learning invariant representation for domain adaptation. ar Xiv preprint ar Xiv:1901.09453, 2019.

Zhao, S., Gong, M., Liu, T., Fu, H., and Tao, D. Domain generalization via entropy regularization. Advances in Neural Information Processing Systems, 33, 2020.

Zhou, K., Yang, Y., Hospedales, T. M., and Xiang, T. Deep domain-adversarial image generation for domain generalisation. In AAAI, pp. 13025 13032, 2020.