Learning Causal Semantic Representation for Out-of-Distribution Prediction

1 Introduction . Deep learning has initiated a new era of artificial intelligence where the potential of machine learning models is greatly unleashed . Despite the great success , these methods heavily rely on the assumption that data from training and test domains follow the same distribution ( i.e. , the IID assumption ) , while in practice the test domain is often out-of-distribution ( OOD ) , meaning that the test data distribute differently from the training data . Popular models for predicting the output ( or label , response , outcome ) y from the input ( or covariate ) x have been found erroneous when confronted with a distribution change , even from an essentially irrelevant perturbation like a position shift or background change for images [ 91 , 6 , 102 , 41 , 2 , 27 ] . These phenomena pose serious concerns on the robustness and trustworthiness of machine learning methods and severely impede them from risk-sensitive scenarios . Looking into the problem , although deep learning models allow extracting abstract representation for prediction with their powerful approximation capacity , the representation may unconsciously mix up semantic factors s ( e.g. , shape of an object ) and variation factors v ( e.g. , background , object position ) due to a correlation between them ( e.g. , desks often appear in a workspace background and beds in bedrooms ) , so the model also relies on the variation factors v for prediction via this correlation . However , this correlation tends to be superficial and spurious ( e.g. , a desk can also appear in a bedroom , but this does not make it a bed ) , and may change drastically in a new domain , making the effect from v misleading . So it is desired to learn a representation that identifies s against v. Formally , the essence of this goal is to leverage causal relations for prediction , since the fundamental distinction between s and v is that only s is the cause of y. Causal relations better reflect basic ∗Correspondence to : Chang Liu < changliu @ microsoft.com > . †Work done during an internship at Microsoft Research Asia . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . mechanisms of nature . They bring the merit to machine learning that they tend to be universal and invariant across domains [ 97 , 87 , 93 , 77 , 16 , 96 , 98 ] , thus provide the most transferable and reliable information to unseen domains . This causal invariance has been shown to lead to proper domain adaptation [ 97 , 123 ] , lower adaptation cost and lighter catastrophic forgetting [ 87 , 9 , 56 ] . In this work , we propose a Causal Semantic Generative model ( CSG ) following a causal consideration to separately model the semantic ( cause of prediction ) and variation latent factors , and develop OOD prediction methods with theoretical guarantees on identifiability and the boundedness of OOD prediction error . Addressing the complaint that OOD prediction and causality methods often require multi-domain or intervention data , we focus on the most common and also challenging tasks where only one single training domain is available , including OOD generalization and domain adaptation , where in the latter , unsupervised test-domain data are additionally available for training . The methods and theory are based on the causal invariance principle , which suggests to share generative mechanisms across domains , while the latent factor distribution ( i.e. , the prior p ( s , v ) ) changes . We argue that this causal invariance is more reliable than inference invariance in the other direction adopted by many existing methods [ 33 , 101 , 2 , 66 , 79 ] . For our method , we design novel and delicate reformulations of the ELBO objective so that we avoid the cost to build and learn two inference models . Theoretically , we prove that under certain conditions , CSG can identify the semantic factor on the single training domain , even in presence of an s-v correlation . We further prove the merits from this identification : prediction error is bounded for OOD generalization , and for domain adaptation , the test-domain prior is identifiable which leads to an accurate prediction . To sum up our contributions , • Up to our knowledge , we are the first to show a theoretical guarantee ( under appropriate conditions ) to identify the latent cause of prediction ( i.e. , the semantic factor ) on a single training domain , and also the first to show the theoretical benefits of this identification for OOD prediction . The results also contribute to generative representation learning for revealing what is learned . • We develop effective methods for OOD generalization and domain adaptation , and achieve mostly better performance than prevailing methods on real-world image classification tasks . 2 Related Work . OOD generalization with causality . There are trials that ameliorate discriminative models towards a causal behavior . Bahadori et al . [ 4 ] introduce a regularizer that reweights input dimensions based on their approximated causal effects to the output , and Shen et al . [ 102 ] reweight training samples by amortizing causal effects among input samples . Their linear input-output assumption is then extended [ 4 , 41 ] by learning a representation . Some recent works require identity data ( finer than label ) and enforce inference invariance via variance minimization [ 42 ] , or leverage a strong domain knowledge to augment images as an independent intervention on variation factors [ 79 ] . These methods introduce no additional generative modeling efforts , at the cost of limited capacity for invariant causal mechanisms . Domain adaptation/generalization with causality . There are methods developed under various causal assumptions [ 97 , 123 ] or using learned causal relations [ 93 , 77 ] . Zhang et al . [ 123 ] , Gong et al . [ 35 , 36 ] also consider certain ways of mechanism change . The considered causality is among directly observed variables , which may not well suit general data like image pixels where causality rather lies in the conceptual latent level [ 75 , 10 , 59 ] . To consider latent factors , there are domain adaptation [ 83 , 5 , 33 , 73 , 74 ] and generalization methods [ 80 , 101 , 113 ] that learn a representation with a domain-invariant marginal distribution . Remarkable results have been achieved . Nevertheless , it is found that this invariance is neither sufficient nor necessary to identify the true semantics or lower the adaptation error ( [ 54 , 125 ] ; see also Appx . E ) . Moreover , these methods are based on inference invariance , which may not be as reliable as causal invariance ( see Sec . 3.2 ) . There are also generative methods for domain adaptation/generalization that model latent factors . Cai et al . [ 18 ] and Ilse et al . [ 49 ] introduce a semantic factor and a domain-feature factor . They assume the two latent factors are independent in both generative and inference models , which is unrealistic . Correlated factors are then considered [ 3 ] . But all these works do not adapt the prior for domain change thus resort to inference invariance . Zhang et al . [ 121 ] consider a partially observed manipulation variable , while still assuming its independence from the output in both the joint and posterior , and the adaptation is inconsistent with causal invariance . The above methods also do not show guarantees to identify their latent factors . Teshima et al . [ 108 ] leverage causal invariance and adapt the prior , yet also assume latent independence and do not separate the semantic factor . They require some supervised test-domain data , and their deterministic and invertible mechanism also indicates inference invariance . In addition , most domain generalization methods require multiple training domains , with exceptions [ 89 ] that still seek to augment domains . In contrast , CSG leverages causal invariance , and has guarantee to identify the semantic factor from a single training domain , even with a correlation to the variation factor . Disentangled latent representations is also of interest in unsupervised learning . Despite empirical success [ 22 , 43 , 21 ] , Locatello et al . [ 70 ] conclude that it is impossible to guarantee the disentanglement in unsupervised settings . Subsequent works then introduce ways of supervision like a few latent variable observations [ 71 ] or sample similarity [ 20 , 72 , 104 ] . Identifiable VAE [ 57 ] and extensions [ 58 , 117 ] leverage the data of a cause variable of the latent variables and have established theoretical guarantees under a diversity condition . But the works do not depict domain change thus not suitable for OOD prediction . Instead of disentangling latent factors , we focus on identifying the semantic factor s ( Sec . 5.1 ) and its benefit for OOD prediction . Appx . D shows more related work . 3 The Causal Semantic Generative Model . To develop the model soberly based on causality , we require its formal definition : two variables have a causal relation , denoted as “ cause→ effect ” , if intervening the cause ( by changing external variables out of the considered system ) may change the effect , but not vice versa [ 85 , 88 ] . We follow this definition to build our model ( Fig . 1a ) by analyzing the example that an photographer takes a photo in a scene as x and labels it as y. Appx . C provides more explanations under other perspectives . ( 1 ) It is likely that neither y → x ( e.g. , intervening the label with noise by distracting the photographer does not change the image ) nor x→ y holds ( e.g. , intervening an image by breaking a camera sensor unit does not change how the photographer labels it ) , as also argued in [ 88 , Sec . 1.4 ; 59 ] . So we introduce a latent variable z to capture factors with causal relations . Also for this reason , we need a generative model ( vs. discriminative model that only learns x→ y ) . ( 2 ) The latent variable z as underlying generating factors ( e.g. , object shape and texture , background and illumination during imaging ) is plausible to cause both x ( e.g. , changing object shape or background makes a different image , but breaking the camera does not change the shape or background ) and y ( e.g. , the photographer would give a different label if the object shape had been different , but noise-corrupting the label does not change the shape ) . So we orient the edges in the generative direction z → ( x , y ) , as also adopted in [ 78 , 88 , 108 ] . This is in contrast to prior works [ 18 , 49 , 48 , 19 ] that treat y as the cause of a semantic factor , which , when y is also a noisy observation , makes unreasonable implications ( e.g. , adding noise to the labels in a dataset automatically changes object features and consequently the images , and changing the object features does not change the label ) . This difference is also discussed in [ 88 , Sec . 1.4 ; 59 ] . ( 3 ) We attribute all x-y relation to the existence of some latent factor [ 68 , “ purely common cause ” ; 51 ] and exclude x-y edges . This can be achieved as long as z holds sufficient information of data ( e.g. , with shape , background etc . fixed , breaking the camera does not change the label , and noisecorrupting the label does not change the image ) . Promoting this structure reduces arbitrariness in explaining x-y relation thus helps identify ( part of ) z . This is in contrast to prior works [ 63 , 121 , 19 ] that treat y as a cause of x as no latent variable is introduced between . ( 4 ) Not all latent factors are the causes of y ( e.g. , changing the shape may alter the label , while changing the background does not ) . We thus split the latent variable as z = ( s , v ) and remove the v → y edge , where s represents the semantic factor that causes y , and v describes the variation or diversity in generating x . This formalizes the intuition on the concepts in Introduction ( Sec . 1 ) . ( 5 ) The two factors s and v often have a relation ( e.g. , a desk/bed shape tends to appear with a workspace/bedroom background ) , but it is usually a spurious correlation ( e.g. , putting a desk in a bedroom does not automatically change the room as a workspace , nor does it turn the desk into a bed ) . So we keep the undirected s-v edge . This is in contrast to prior works [ 18 , 49 , 121 , 108 , 79 ] which assume independent latent variables . Although v is not a cause of y , modeling it explicitly is worth the effort since otherwise it would still be implicitly mixed into s anyway through the s-v correlation . We summarize these conclusions in the following definition . Definition 1 ( CSG ) . A Causal Semantic Generative Model ( CSG ) , p : = 〈p ( s , v ) , p ( x|s , v ) , p ( y|s ) 〉 , is a generative model on data variables x ∈ X ⊆ RdX and y ∈ Y with semantic s ∈ S ⊆ RdS and variation v ∈ V ⊆ RdV latent variables , following the graphical structure shown in Fig . 1a .

Summary: The paper addresses the problem of out-of-distribution prediction by proposing a causal model which captures the underlying data generation process. The causal model consists of a latent variable which generates both the input images and the associated output labels. The novelty of the paper stems from the fact that the authors split the latent variable into two parts. The first part (s) captures the semantics of the generation process and the second part (v) captures the spurious domain specific variations. Since maximizing the likelihood of the data under this model is intractable, the authors propose an approximate and efficient training and inference mechanism. The authors show theoretically that their model will learn semantic features which are responsible for the final prediction. The claims are backed by a collection of experiments on simple datasets.