Deconfounding to Explanation Evaluation in Graph Neural Networks

Explainability of graph neural networks ( GNNs ) aims to answer “ Why the GNN made a certain prediction ? ” , which is crucial to interpret the model prediction . The feature attribution framework distributes a GNN ’ s prediction to its input features ( e.g. , edges ) , identifying an influential subgraph as the explanation . When evaluating the explanation ( i.e. , subgraph importance ) , a standard way is to audit the model prediction based on the subgraph solely . However , we argue that a distribution shift exists between the full graph and the subgraph , causing the out-ofdistribution problem . Furthermore , with an in-depth causal analysis , we find the OOD effect acts as the confounder , which brings spurious associations between the subgraph importance and model prediction , making the evaluation less reliable . In this work , we propose Deconfounded Subgraph Evaluation ( DSE ) which assesses the causal effect of an explanatory subgraph on the model prediction . While the distribution shift is generally intractable , we employ the front-door adjustment and introduce a surrogate variable of the subgraphs . Specifically , we devise a generative model to generate the plausible surrogates that conform to the data distribution , thus approaching the unbiased estimation of subgraph importance . Empirical results demonstrate the effectiveness of DSE in terms of explanation fidelity . 1 INTRODUCTION . Explainability of graph neural networks ( GNNs ) ( Hamilton et al. , 2017 ; Dwivedi et al. , 2020 ) is crucial to model understanding and reliability in real-world applications , especially when about fairness and privacy ( Ying et al. , 2019 ; Luo et al. , 2020 ) . It aims to provide insight into how predictor models work , answering “ Why the target GNN made a certain prediction ? ” . Towards this end , a variety of explainer models are proposed for feature attribution ( Selvaraju et al. , 2017 ; Ying et al. , 2019 ; Luo et al. , 2020 ; Vu & Thai , 2020 ) , which decomposes the predictor ’ s prediction as contributions ( i.e. , importance ) of its input features ( e.g. , edges , nodes ) . While feature attribution assigns the features with importance scores , it redistributes the graph features and creates a new distribution different from that of the original full graphs , from which a subgraph is sampled as the explanation . Such sampling process is referred to as feature removal ( Covert et al. , 2020 ) . Then , to assess the explanatory subgraph , the current evaluation frameworks use the feature removal principle — ( 1 ) only feed the subgraph into the target predictor , discarding the other features ; ( 2 ) measure the importance of the subgraph based on its information amount to recover the model ’ s prediction . Such subgraph-prediction correlations uncovered by the removal-based evaluator should offer a faithful inspection of the predictor ’ s decision-making process and assess the fidelity of the explainers reliably . However , feature removal brings the out-of-distribution ( OOD ) problem ( Frye et al. , 2020 ; Chang et al. , 2019 ; Lukas Faber , 2021 ) : the distribution shift from full graphs to subgraphs likely violates underlying properties , including node degree distribution ( Leskovec et al. , 2005 ) and domain-specific constraints ( Liu et al. , 2018 ) of the full graphs . For example , graph properties of chemical molecules , such as the valency rules , impose some constraints on syntactically valid molecules ( Liu et al. , 2018 ) ; hence , simply removing some bonds ( edges ) or atoms ( nodes ) creates invalid molecular subgraphs that never appear in the training dataset . Such OOD subgraphs could manipulate the predictor ’ s Under review as a conference paper at ICLR 2022 𝑮 : Full Graph 𝑮𝒔 : Subgraph 𝒀 : Predicate Logits 𝐬 𝓖 House Cycle Crane 0.21 0.70 Target Predictor Crane 𝓖Cycle House𝓖𝒔𝟏𝑫 : Distribution Shift 𝓖𝒔𝟐 𝒔∗ 𝐬 Front-door Adjustment𝓖𝒔𝟐𝓖𝒔𝟏 ( a ) Feature Removal to Evaluate Explanatory Subgraph Gs 𝑮 : Full Graph 𝑮𝒔 : Subgraph 𝒀 : Predicate Logits 𝐬 𝓖 House Cycle Crane 0.21 0.70 Target Predictor Crane 𝓖Cycle House𝓖𝒔𝟏𝑫 : Distribution Shift 𝓖𝒔𝟐 𝒔∗ 𝐬 Front-door Adjustment𝓖𝒔𝟐𝓖𝒔𝟏 ( b ) SCM I Figure 1 : ( a ) A real example in TR3 . The GNN predictor classifies the full graph as ‘ House ” . On subgraphs Gs1 and Gs2 , the prediction probabilities of being “ House ” are respectively 0.21 and 0.70 . ( b ) The structural causal model represents the causalities among variables : G as the input graph , D as the unobserved distribution shift , Gs as the explanatory subgraph , and Y as the model prediction . outcome arbitrarily ( Dai et al. , 2018 ; Zügner et al. , 2018 ) , generates erroneous predictions , and limits the reliability of the evaluation process . Here we demonstrate the OOD effect by a real example in Figure 1a , where the trained ASAP ( Ranjan et al. , 2020 ) predictor has classified the input graph as “ House ” for its attached motif ( see Section 4 for more details ) . On the ground-truth explanation Gs1 , the output probability of the “ House ” class is surprisingly low ( 0.21 ) . While for Gs2 with less discriminative information , the outputs probability of the “ House ” class ( 0.70 ) is higher . Clearly , the removal-based evaluator assigns the OOD subgraphs with unreliable importance scores , which are unfaithful to the predictor ’ s decision . The OOD effect has not been explored in evaluating GNN explanations , to the best of our knowledge . We rigorously investigate it from a causal view ( Pearl et al. , 2016 ; Pearl , 2000 ; Pearl & Mackenzie , 2018 ) . Figure 1b represents our causal assumption via a structural causal model ( SCM ) ( Pearl et al. , 2016 ; Pearl , 2000 ) , where we target the causal effect of Gs on Y . Nonetheless , as a confounder between Gs and Y , distribution shift D opens the spurious path Gs ← D → Y . By “ spurious ” , we mean that the path lies outside the direct causal path from Gs to Y , making Gs and Y spuriously correlated and yielding an erroneous effect . And one can hardly distinguish between the spurious correlation and causative relations ( Pearl et al. , 2016 ) . Hence , auditing Y on Gs suffers from the OOD effect and wrongly evaluates the importance of Gs . Motivated by our causal insight , we propose a novel evaluation paradigm , Deconfounded Subgraph Evaluator ( DSE ) , to faithfully measure the causal effect of explanatory subgraphs on the prediction . 𝑮 : Full Graph 𝑮𝒔 : Subgraph 𝒀 : Predicate Logits 𝑮𝐬𝑮 𝒀 𝑫 𝑮𝒔∗ 𝑮𝐬𝑮 𝒀 𝑫 Front-door Adjustment reliably and further guide explainers to generate faithful explanations . In a nutshell , our contributions are : • From a causal perspective , we argue that the OOD effect is the confounder that causes spurious correlations between subgraph importance and model prediction . • We propose a deconfounding paradigm , DSE , which exploits the front-door adjustment to mitigate the out-of-distribution effect and evaluate the explanatory subgraphs unbiasedly . • We validate the effectiveness of our framework over various explainers , target GNN models , and datasets . Significant boosts are achieved over the conventional feature removal techniques . Code and datasets are available at : https : //anonymous.4open.science/r/DSE-24BC/ . 2 A CAUSAL VIEW OF EXPLANATION EVALUATION . Here we begin with the causality-based view of feature removal in Section 2.1 and present our causal assumption to inspect the OOD effect in Section 2.2 . 2.1 PROBLEM FORMULATION . Without loss of generality , we focus on the graph classification task : a well-trained GNN predictor f takes the graph variable G as input and predicts the class Y ∈ { 1 , · · · , K } , i.e. , Y = f ( G ) . Generation of Explanatory Subgraphs . Post-hoc explainability typically considers the question “ Why the GNN predictor f made certain prediction ? ” . A prevalent solution is building an explainer model to conduct feature attribution ( Ying et al. , 2019 ; Luo et al. , 2020 ; Pope et al. , 2019 ) . It decomposes the prediction into the contributions of the input features , which redistributes the probability of features according to their importance and sample the salient features as an explanatory subgraph Gs . Specifically , Gs can be a structure-wise ( Ying et al. , 2019 ; Luo et al. , 2020 ) or featurewise ( Ying et al. , 2019 ) subgraph of G. In this paper , we focus on the structural features . That is , for graph G = ( N , E ) with the edge set E and the node set N , the explanatory subgraph Gs = ( Ns , Es ) consists of a subset of edges Es ⊂ E and their endpoints Ns = { u , v| ( u , v ) ∈ Es } . Evaluation of Explanatory Subgraphs . Insertion-based evaluation by feature removal ( Covert et al. , 2020 ; Dabkowski & Gal , 2017 ) aims to check whether the subgraph is the supporting substructure 1 that alone allows a confident classification . We systematize this paradigm as three steps : ( 1 ) divide the full graph G into two parts , the subgraph Gs and the complement Gs ; ( 2 ) feed Gs into the target GNN f , while discarding Gs ; and ( 3 ) obtain the model prediction on Gs , to assess its discriminative information to recover the prediction on G. Briefly , at the core of the evaluator is the subgraphprediction correlation . However , as discussed in Section 1 , the OOD effect is inherent in the removal-based evaluator , hindering the subgraph-prediction correlation from accurately estimating the subgraph importance . 2.2 STRUCTURAL CAUSAL MODEL . To inspect the OOD effect rigorously , we take a causal look at the evaluation process with a Structural Causal Model ( SCM I ) in Figure 1b . We denote the abstract data variables by the nodes , where the directed links represent the causality . The SCM indicates how the variables interact with each other through the graphical definition of causation : • G→ Gs ← D. We introduce an abstract distribution shift variable D to sample a subgraph Gs from the edge distributions of the full graph G. • Gs → Y ← D. We denote Y as the prediction variable ( e.g. , logits output ) , which is determined by ( 1 ) the direct effect from Gs , and ( 2 ) the confounding effect caused by D. In particular , the former causation that led to the result is the focus of this work . We suggest readers to refer to Appendix A where we offer an elaboration of D. With our SCM assumption , directly measuring the importance of explanatory subgraphs is distracted by the backdoor path ( Pearl , 2000 ) , Gs ← D → Y . This path introduces the confounding associations between Gs and Y , which makes Gs and Y spuriously correlated , i.e. , biases the subgraph-prediction correlations , thus making the evaluator invalid . How to mitigate the OOD effect and quantify Gs ’ s genuine causal effect on Y remains largely unexplored in the literature and is the focus of our work .

Summary: This paper has done an excellent work of finding the out-of-distribution between the subgraph and graph as the confounder. Further, this paper proposes a conditional variational graph auto-encoder in assessing the causal effects of subgraph on the prediction. They also introduce a surrogate variable to denote this out-of-distribution effect. Through adversarial training, the effects of the proposed model is correctly verified.