Discovering Invariant Rationales for Graph Neural Networks

1 INTRODUCTION . The eye-catching success in graph neural networks ( GNNs ) ( Hamilton et al. , 2017 ; Kipf & Welling , 2017 ; Dwivedi et al. , 2020 ) provokes the rationalization task , answering “ What knowledge drives the model to make certain predictions ? ” . The goal of selective rationalization ( aka . feature attribution ) ( Chang et al. , 2020 ; Ying et al. , 2019 ; Luo et al. , 2020 ; Wang et al. , 2021c ) is to find a small subset of the input ’ s graph features — rationale — which best guides or explains the model prediction . Discovering the rationale in a model helps audit its inner workings and justify its predictions . Moreover , it has tremendous impacts on real-world applications , such as finding functional groups to shed light on protein structure prediction ( Senior et al. , 2020 ) . Two research lines of rationalization have recently emerged in GNNs . Post-hoc explainability ( Ying et al. , 2019 ; Luo et al. , 2020 ; Yuan et al. , 2021 ; Wang et al. , 2021c ) attributes a model ’ s prediction to the input graph with a separate explanation method , while intrinsic interpretability ( Veličković et al. , 2018 ; Gao & Ji , 2019 ) incorporates a rationalization module into the model to make transparent predictions . Here we focus on intrinsically interpretable GNNs . Among them , graph attention ( Veličković et al. , 2018 ) and pooling ( Lee et al. , 2019 ; Knyazev et al. , 2019 ; Gao & Ji , 2019 ; Ranjan et al. , 2020 ) operators prevail , which work as a computational block of a GNN to generate soft or hard masks on the input graph . They cast the learning paradigm of GNN as minimizing the empirical risk with the masked subgraphs , which are regarded as rationales to guide the model predictions . Despite the appealing nature , recent studies ( Chang et al. , 2020 ; Knyazev et al. , 2019 ) show that the current rationalization methods are prone to ∗Corresponding author . exploit data biases as shortcuts to make predictions and compose rationales . Typically , shortcuts result from confounding factors , sampling biases , and artifacts in the training data . Considering Figure 1 , when the most bases of House-motif graphs are Tree , a GNN does not need to learn the correct function to reach high accuracy for the motif type . Instead , it is much easier to learn from the statistical shortcuts linking the bases Tree with the most occurring motifs House . Unfortunately , when facing with out-of-distribution ( OOD ) data , such methods generalize poorly since the shortcuts are changed . Hence , such shortcut-involved rationales hardly reveal the truly critical subgraphs for the predicted labels , being at odds with the true reasoning process that underlies the task of interest ( Teney et al. , 2020 ) and human cognition ( Alvarez-Melis & Jaakkola , 2017 ) . Here we ascribe the failure on OOD data to the inability to identify causal patterns , which are stable to distribution shift . Motivated by recent studies on invariant learning ( IL ) ( Arjovsky et al. , 2019 ; Krueger et al. , 2021 ; Chang et al. , 2020 ; Bühlmann , 2018 ) , we premise different distributions elicit different environments of data generating process . We argue that the causal patterns to the labels remain stable across environments , while the relations between the shortcut patterns and the labels vary . Such environment-invariant patterns are more plausible and qualified as rationales . Aiming to identify rationales that capture the environment-invariant causal patterns , we formalize a learning strategy , Discovering Invariant Rationales ( DIR ) , for intrinsically interpretable GNNs . One major problem is how to get multiple environments from a standard training set . Differing from the heterogeneous setting ( Bühlmann , 2018 ) of existing IL methods , where environments are observable and attainable , DIR does not assume prophets about environments . It instead generates distribution perturbations by causal intervention — interventional distributions ( Tian et al. , 2006 ; Pearl et al. , 2016 ) — to instantiate environments and further distinguish the causal and non-causal parts . Guided by this idea , our DIR strategy consists of four modules : a rationale generator , a distribution intervener , a feature encoder , two classifiers . Specifically , the rationale generator learns to split the input graph into causal and non-causal subgraphs , which are respectively encoded by the encoder into representations . Then , the distribution intervener conducts the causal interventions on the non-causal representations to create perturbed distributions , with which we can infer the invariant causal parts . Then , the two classifiers are respectively built upon the causal and non-causal parts to generate the joint prediction , whose invariant risk is minimized across different distributions . On one synthetic and three real datasets , extensive experiments demonstrate the generalization ability of DIR to surpass current state-of-the-art IL methods ( Arjovsky et al. , 2019 ; Krueger et al. , 2021 ; Sagawa et al. , 2019 ) , and the interpretability of DIR to outperform the attention- and pooling-based rationalization methods ( Veličković et al. , 2018 ; Gao & Ji , 2019 ) . Our main contributions are : • We propose a novel invariant learning algorithm , DIR , for inherent interpretable models , improving the generalization ability and is suitable for any deep models . • We offer causality theoretic analysis to guarantee the preeminence of DIR . • We provide the implementation of DIR for graph classification tasks , which consistently achieves excellent performance on three datasets with various generalization types . 2 INVARIANT RATIONALE DISCOVERY . With a causal look at the data-generating process , we formalize the principle of discovering invariant rationales , which guides our discovery strategy . Throughout the paper , upper-cased letters like G denote random variables , while lower-case letters like g denote deterministic value of variables . 2.1 CAUSAL VIEW OF DATA-GENERATING PROCESS . Generating rationales for transparent predictions requires understanding the actual mechanisms of the task of interest . Without loss of generality , we focus on the graph classification task and present a causal view of the data-generating process behind this task . Here we formalize the causal view as a Structure Causal Model ( SCM ) ( Pearl et al. , 2016 ; Pearl , 2000 ) by inspecting on the causalities among four variables : input graph G , ground-truth label Y , causal part C , non-causal part S. Figure 2a illustrates the SCM , where each link denotes a causal relationship between two variables . • C → G ← S. The input graph G consists of two disjoint parts : the causal part C and the non-causal part S , such as the House motif and the Tree base in Figure 1 . Published as a conference paper at ICLR 2022 𝑺 𝒀 𝑮 𝑪 𝑮 𝑺 𝑪 𝒀 House Cycle Crane House Cycle Crane … ( b ) Illustration of Constructing Interventional Distributions House Cycle Crane 𝒅𝒐 ( 𝑺 = ∅ ) ∅ 𝒅𝒐 ( 𝑺 = ) 𝒅𝒐 ( 𝑺 = ) … … … 𝑺 𝒀 𝑮 𝑪 Original Distribution Multiple 𝒔-Interventional Distributions ( a ) SCM 𝑺 𝒀 𝑮 𝑪 𝑮 𝑺 𝑪 𝒀 House Cycle Crane House Cycle Crane … ( b ) Illustration of Constructing Interventional Distributions House Cycle Crane 𝒅𝒐 ( 𝑺 = ∅ ) ∅ 𝒅𝒐 ( 𝑺 = ) 𝒅𝒐 ( 𝑺 = ) … … … 𝑺 𝒀 𝑮 𝑪 Original Distribution Multiple 𝒔-Interventional Distributions ( b ) Interventional Distributions . Figure 2 : ( a ) Causal view of data-generating process ; ( b ) Illustration of interventional distributions . • C → Y . By “ causal part ” , we mean C is the only endogenous parent to determine the groundtruth label Y . Taking the motif-base example in Figure 1 again , C is the oracle rationale , which perfectly explains why the graph is labeled as Y . • C L9999K S. This dashed arrow indicates additional probabilistic dependencies ( Pearl , 2000 ; Pearl et al. , 2016 ) between C and S. We consider three typical relationships here : ( 1 ) C is independent of S , i.e. , C |= S ; ( 2 ) C is the direct cause of S , i.e. , C → S ; and ( 3 ) There exists a common cause E , i.e. , C ← E → S. See Appendix B for the corresponding examples . C L9999K S can create spurious correlations between the non-causal part S and the ground-truth label Y . Assuming C → S , C is a confounder between S and Y , which opens a backdoor path S ← C → Y , thus making S and Y spuriously correlated ( Pearl et al. , 2016 ) . We systematize such spurious correlations as Y 6⊥⊥ S. Wherein , we make feature induction assumption on S to avoid the confusion of the induced subset of S between C. See Appendix C for the formal assumption . Furthermore , data collected from different environments exhibit various spurious correlations ( Teney et al. , 2020 ; Arjovsky et al. , 2019 ) , e.g. , one mostly picks House motifs with Tree bases as the training data , while another selects House motifs with Wheel bases as the testing data . Hence , such spurious correlations are unstable and variant across different distributions . 2.2 TASK FORMALIZATION OF INVARIANT RATIONALIZATION . Oracle Rationale . With the causal theory ( Pearl et al. , 2016 ; Pearl , 2000 ) , for each variable X in a SCM , there exists a directed link from each of its parent variables PA ( X ) to X , if and only if the causal mechanism X = fX ( PA ( X ) , X ) persists , where X |= PA ( X ) is the exogenous noise of X . For simplicity , we omit the exogenous noise and simplify it as X = fX ( PA ( X ) ) . Hence , there exist a function fY : C → Y in our SCM , where the “ oracle rationale ” C satisfies : Y = fY ( C ) , Y |= S | C , ( 1 ) where Y |= S | C indicates that C shields Y from the influence of S , making the causal relationship C → Y invariant across different S. Rationalization . In general , only the pairs of input G and label Y are observed during training , while neither oracle rationale C nor oracle structural equation model fY is available . The absence of oracles calls for the study on intrinsic interpretability . We systematize an intrinsically-interpretable GNN as a combination of two modules , i.e. , h = hŶ ◦ hC̃ , where hC̃ : G→ C̃ discovers rationale C̃ from the observed G , and hŶ : C̃ → Ŷ outputs the prediction Ŷ to approach Y . Distinct from C and Y which are the variables in the causal mechanisms , C̃ and Ŷ represent the variables in the modeling process to approximateC and Y . To optimize these modules , most of current intrinsicallyinterpretable GNNs ( Veličković et al. , 2018 ; Lee et al. , 2019 ; Knyazev et al. , 2019 ; Gao & Ji , 2019 ; Ranjan et al. , 2020 ) adopt the learning strategy of minimizing the empirical risk : min hC̃ , hŶ R ( hŶ ◦ hC̃ ( G ) , Y ) , ( 2 ) where R ( · , · ) is the risk function , which can be the cross-entropy loss . Nevertheless , this learning strategy relies heavily on the statistical associations between the input features and labels , and can potentially exhibit non-causal rationales . Invariant Rationalization . We ascribe the limitation to ignoring Y |= S | C in Equation 1 , which is crucial to refine the causal relationship C → Y that is invariant across different S. By introducing this independence , we formalize the task of invariant rationalization as : min hC̃ , hŶ R ( hŶ ◦ hC̃ ( G ) , Y ) , s.t . Y |= S̃ | C̃ , ( 3 ) where S̃ = G \ C̃ is the complement of C̃ . This formulation encourages the rationale C̃ seeking the patterns that are stable across different distributions , while discarding the unstable patterns .

Summary: This paper proposes a novel algorithm DIR that allows the learned GNN model to make predictions based on causal patterns. The whole framework consists of four components where (1) the rationale generator splits the input into causal and spurious parts; (2) the distribution intervener replaces the spurious part; (3) the graph encoder embeds the graph based on the spurious and causal parts, respectively; (4) the shortcut and causal classifiers are applied to the spurious and causal embeddings, respectively. This algorithm is highlighted by its objective which encourages the predictions accurate while invariant to the spurious part. On both synthetic and real-world datasets, DIR outperforms the related baselines consistently. As a novel end2end rationalization method, As a novel end2end rationalization method, the underlying DIR principle is proved to be sufficient and necessary for satisfying the oracle function.