Learning Neural Causal Models with Active Interventions

Discovering causal structures from data is a challenging inference problem of fundamental importance in all areas of science . The appealing scaling properties of neural networks have recently led to a surge of interest in differentiable neural network-based methods for learning causal structures from data . So far , differentiable causal discovery has focused on static datasets of observational or interventional origin . In this work , we introduce an active intervention-targeting mechanism which enables quick identification of the underlying causal structure of the data-generating process . Our method significantly reduces the required number of interactions compared with random intervention targeting and is applicable for both discrete and continuous optimization formulations of learning the underlying directed acyclic graph ( DAG ) from data . We examine the proposed method across multiple frameworks in a wide range of settings and demonstrate superior performance on multiple benchmarks from simulated to real-world data . 1 INTRODUCTION . Learning causal structure from data is a challenging but important task that lies at the heart of scientific reasoning and accompanying progress in many disciplines ( Sachs et al. , 2005 ; Hill et al. , 2016 ; Lauritzen & Spiegelhalter , 1988 ; Korb & Nicholson , 2010 ) . While there exists a plethora of methods for the task , computationally and statistically more efficient algorithms are highly desirable ( Heinze-Deml et al. , 2018 ) . As a result , there has been a surge in interest in differentiable structure learning and the combination of deep learning and causal inference ( Schölkopf et al. , 2021 ) . Such methods define a structural causal model with smoothly differentiable parameters that are adjusted to fit observational data ( Zheng et al. , 2018 ; Yu et al. , 2019 ; Zheng et al. , 2020 ; Bengio et al. , 2019 ; Lorch et al. , 2021 ; Annadani et al. , 2021 ) , although some methods can accept interventional data , thereby significantly improving the identification of the underlying data-generating process ( Ke et al. , 2019 ; Brouillard et al. , 2020 ; Lippe et al. , 2021 ) . However , the improvement critically depends on the experiments and interventions available to the learner . Despite advances in high-throughput methods for interventional data in specific fields ( Dixit et al. , 2016 ) , the acquisition of interventional samples in general settings tends to be costly , technically impossible or even unethical for specific interventions . There is , therefore , a need for efficient usage of the available interventional samples and efficient experimental design to keep the number of interventions to a minimum . A significant amount of prior work exists in causal structure learning that leverages active learning and experimental design to improve identifiability in a sequential manner . These approaches are either graph theoretical ( He & Geng , 2008 ; Eberhardt , 2012 ; Hyttinen et al. , 2013 ; Hauser & Bühlmann , 2014 ; Shanmugam et al. , 2015 ; Kocaoglu et al. , 2017b ; a ; Lindgren et al. , 2018 ; Ghassami et al. , 2018 ; 2019 ; Greenewald et al. , 2019 ; Squires et al. , 2020 ) , Bayesian ( Murphy , 2001 ; Tong & Koller , 2001 ; Masegosa & Moral , 2013 ; Cho et al. , 2016 ; Ness et al. , 2017 ; Agrawal et al. , 2019 ; Zemplenyi & Miller , 2021 ) or rely on Invariant Causal Prediction ( Gamella & Heinze-Deml , 2020 ) . These methods are typically computationally very expensive and do not scale well with respect to the number of variables or dataset size ( Heinze-Deml et al. , 2018 ) . A promising alternative is the use of active learning in a continuous optimization framework for causal structure learning from joint data . However , since the applicability of existing scores / heuristics for selecting inter- vention targets is limited for existing frameworks ( see §A.1 ) , current approaches rely on random and independent interventions and do not leverage the acquired evidence from processed experiments . We thus propose a novel method of active selection of intervention targets that can easily be incorporated into many differentiable causal discovery algorithms . Since most of these algorithms treat the adjacency matrix of the causal graph as a learned soft-adjacency , it is readily available for parametrized sampling of different hypothesis graphs . Our method looks for an intervention target that gives maximum disagreement between post-interventional sample distributions under these hypothesis graphs . We conjecture that interventions on such nodes will contain more information about the causal structure and hence enable more efficient learning . To the best of our knowledge , our paper is the first approach to combine both a continuous optimization framework and active causal structure learning from observational and interventional data . We summarize our contributions as follows : • We propose a novel approach for selecting interventions ( single and multi-target ) which identify the underlying graph efficiently and can be used for any differentiable causal discovery method . • We introduce a novel , scalable two-phase DAG sampling procedure which efficiently generates hypothesis DAGs based on a soft-adjacency matrix . • We examine the proposed intervention-targeting method across multiple differentiable causal discovery frameworks in a wide range of settings and demonstrate superior performance against established competitive baselines on multiple benchmarks from simulated to real-world data . • We provide empirical insights on the distribution of selected intervention targets and its connection to the ( causal ) topological order of the variables in the underlying system . 2 PRELIMINARIES . Structural Causal Model . An SCM ( Peters et al. , 2017 ) is defined over a set of random variables X1 , . . . , XM or justX for short and a directed acyclic graph ( DAG ) G = ( V , E ) over variable nodes V = { 1 , . . .M } . The random variables are connected by edges in E via functions fi and jointly independent noise variables Ui through Xi = fi ( Xpa ( i ) , Ui ) where Xpa ( i ) are Xi ’ s parents in G , and directed edges in the graph represent direct causation . The conditionals P ( Xi|Xpa ( i ) ) define the conditional distribution of Xi given its parents . Interventions . Interventions on Xi change the conditional distribution of P ( Xi|Xpa ( i ) ) to a different distribution , hence affecting the outcome of Xi . Interventions can be perfect ( hard ) or imperfect ( soft ) . Hard interventions entirely remove the dependencies of a variable Xi on its parents Xpa ( i ) , hence defining the conditional probability distribution of Xi by some P̃ ( Xi ) rather than P ( Xi|Xpa ( i ) ) . A more general form of intervention is the soft intervention , where the intervention changes the effect of the parents of Xi on itself by modifying the conditional distribution from Pi ( Xi|Xpa ( i ) ) to an alternative P̃i ( Xi|Xpa ( i ) ) . Dependency Structure Discovery from Interventions ( DSDI ) . We evaluate the proposed method under multiple continuous-optimization causal learning frameworks from fused ( observational and interventional ) data ( Bareinboim & Pearl , 2016 ) , one of them being DSDI ( Ke et al. , 2019 ) . The work of DSDI reformulates the problem of causal discovery from discrete data as a continuous optimization problem using neural networks . The framework proposes to learn the causal graph adjacency matrix as a matrix parameter γ of a neural network , and is trained using a 3-stage iterative procedure . The first stage involves sampling graphs under the model ’ s current belief in the graph structure and then training functional parameters P specifying the conditionals of the sampled graphs using observational data . Note how sampling a graph from γ can specify element-wise multiplications by 0 or 1 in the first layer of neural networks for the conditionals , to remove unallowed edges , with other network parameters P applicable to any sampled graph . The next stage is to evaluate the sampled graphs under interventional data and score these graphs accordingly . The final step is to update the learned adjacency matrix with the scores from stage 2 . This method performs competitively compared to many other methods . However , all intervention targets in stage 2 of DSDI are random and independent , a strategy that scales poorly to larger graphs . A better approach would have been active intervention targeting , elaborated in the next section . Differentiable Causal Discovery from Interventional Data ( DCDI ) . We also consider the work of DCDI ( Brouillard et al. , 2020 ) , which addresses causal discovery from continuous data as a continuous-constrained optimization problem using neural networks to model parameters of Gaus- lo g 1 0 D k sian distributions or normalizing flows ( Rezende & Mohamed , 2015 ) which represent conditional distributions . Unlike DSDI ’ s iterative training of the structural and functional parameters , DCDI optimizes the causal adjacency matrix and functional parameters jointly over the fused data space . But like DSDI , DCDI uses random and independent interventions . 3 ACTIVE INTERVENTION TARGETING . We present a score-based intervention design strategy , called Active Intervention Targeting ( AIT ) , which is applicable to many discrete and continuous optimization formulations of causal structure learning algorithms . Furthermore , we show how our proposed method can be integrated into recent differentiable causal discovery frameworks for guided exploration using interventional data . Assumptions . The proposed method assumes access to a belief state γ in the graph structure ( e.g. , in the form of a distribution over graphs , probabilistic adjacency matrix or a set of hypothetical graphs ) and functional parameters P characterizing the conditional relationships between variables ( constrained by the graphs sampled from the belief specified by γ ) . The proposed model does not have to assume causal sufficiency per se . However , it inherits the assumptions of the selected base framework , and this may include causal sufficiency depending on the base algorithm of choice . In case the underlying framework can handle unobserved variables and offers a generative method for interventional samples , then our method is also applicable 3.1 A SCORE FOR INTERVENTION TARGETING . Algorithm 1 Active Intervention Targeting ( AIT ) Input : Functional Parameters P , Graph Belief State γ , Interventional Target Space I Output : Intervention Target Ik∗ 1 : G ← Sample a set of hypothesis graphs from γ 2 : for each intervention target Ik in I do 3 : Pk ← Perform intervention Ik on P 4 : for each graph Gi in G do 5 : Sk , i← Draw samples from Pk on Gi 6 : Sk , i← Set variables in Ik to 0 7 : end for 8 : Compute : Dk ← ∑ i ( µ k i −µ k ) 2∑ i ∑ j ( S k , i j −µ k i ) 2 9 : end for . 10 : Target Intervention Ik∗ ← argmaxk ( Dk ) Given a graph belief state γ with its corresponding functional parameters P , and a possible set of intervention targets I ( single-node and multi-node intervention targets ) , we wish to select the most informative intervention target ( s ) Ik∗ ∈ I with respect to identifiability of the underlying structure . Such target ( s ) presumably yield relatively high discrepancies between samples drawn under different hypothesis graphs , indicating larger uncertainty about the target ’ s relation to its parents and/or children . We thus construct an F-test-inspired score to determine the Ik∗ exhibiting the highest discrepancies between post-interventional sample distributions generated by likely graph structures under fixed functional parameters P . In order to compare sample distributions over different graphs , we distinguish between two sources of variation : Variance between graphs ( VBG ) and variance within graphs ( VWG ) . While VBG characterizes the variance of sample means over multiple graphs , VWG accounts for the sample variance when a specific graph is fixed . As in DSDI and DCDI , we mask the contribution of the intervened variables Ik to VBG and VWG , and construct our discrepancy score D as a ratio D = VBGVWG . This discrepancy score attains high values for intervention targets of particular interest ( see Figure 1b for a landscape visualization ) . While VBG itself indicates for which intervention targets the model is unsettled about , an extension to the proposed variance ratio enables more control over the region of interest . Given a fixed set of graphs G and a fixed interventional sample size across all graphs , let us assume a scenario where multiple intervention targets attain high VBG . Assessing VWG allows us to distinguish between two extreme cases : ( a ) targets with sample populations that exhibit large VWG ( b ) targets with sample populations that exhibit low VWG . While high VBG in ( a ) might be induced by an insufficient sample size due to high variance in the interventional distribution itself , ( b ) clearly indicates high discrepancy between graphs and should be preferentially studied . Computational Details . We begin by sampling a set of graphs G = { Gi } , i = 1 , 2 , 3 , . . . from our graph structure belief state γ , however parametrized . This G will remain fixed for all considered interventions for the current experimental round . Then , we fix an intervention target Ik and apply the corresponding intervention to P , resulting in partially altered functional parameters Pk where some conditionals have been changed . Next , we draw interventional samples Sk , i from Pk on every graph Gi ∈ G and set variables in Ik to zero to mask of their contribution to the variance . Having collected all samples over the considered graphs for the specific intervention target Ik , we compute VBGk and VWGk as : VBGk = ∑ i ( µki − µk ) 2 and VWGk = ∑ i ∑ j ( Sk , ij − µ k i ) 2 where µk is a vector of the same dimension as any sample in S and denotes the overall sample-mean of the interventional setting , µki the corresponding mean for a specific graph Gi and S k , i j is the j-th sample of the i-th graph configuration . Finally , we construct the discrepancy score Dk of Ik as : Dk ← VBGk VWGk . In contrast to the original definition of the F-Score , we can ignore the normalization constants due to equal group size and degree-of-freedoms . Although the dependence between the variables is apparent from the connected causal structure , we approximate the variance of the multidimensional samples as the trace over the covariance matrix by assuming that the variables are independent . An outline of the method is provided in Algorithm 1 .

Summary: Structure learning from observational data is a long-standing challenge that could benefit from creative uses of interventional data. The authors explore the use of active learning in a continuous optimization framework to better traverse and narrow down the space of potential graphs that describe the data. The contribution is a heuristic to choose the "best" intervention target based on a comparison between samples from an intervened SCM with the current best guess of an underlying causal discovery algorithm (in which the proposal is embedded).