Learning Neural Causal Models from Unknown Interventions

1 INTRODUCTION . A major challenge of contemporary deep learning is to generalize well outside the assumptions of independent and identically distributed data , when we care about generalization or fast adaptation to distributions other than the main training distribution . For this purpose , we propose an approach that starts by distinguishing between : ( a ) an underlying causal model , ( b ) observational distributions derived from it , an ( c ) interventional distributions arising from interventions upon its variables , whether they be known or unknown . Fortunately , these distinctions can be addressed by the paradigm of Structural Causal Models ( SCMs ) ( Pearl , 1995 ; Peters et al. , 2017 ) and a wide body of associated literature . Unlike other frameworks using SCMs we also account for interventions performed by agents other than an experimenter . By treating the interventions of other agents as unknown interventions that lead to changes in the underlying data distribution , the present work is a contribution towards the use of a meta-learning for causal model induction . Estimating the underlying causal structure from data is an open and challenging problem ( Pearl , 2009 ; Imbens & Rubin , 2015 ) . A lot of prior work has examined learning causal structure based on observational data ( Chickering , 2002 ; Tsamardinos et al. , 2006 ; Goudet et al. , 2017 ; Hauser & Bühlmann , 2012 ; Spirtes et al. , 2000 ; Sun et al. , 2007 ; Zhang et al. , 2012 ; Shimizu et al. , 2006 ; Hoyer et al. , 2009 ; Daniusis et al. , 2012 ; Budhathoki & Vreeken , 2017 ; Kalainathan et al. , 2018 ) . However , many real-world datasets have an inherent distributional heterogeneity due to different interventions to the variables composing the model . In these situations interventional approaches are needed ( e.g . Heckerman et al. , 1995 ; Cooper & Yoo , 1999 ; Hauser & Bühlmann , 2012 ; Peters et al. , 2016 ; Rothenhäusler et al. , 2015 ; Ghassami et al. , 2017 ) . Established approaches for causal inference are often either relying on restrictive assumptions or on conditional independence testing , which is hard ( Shah & Peters , 2018 ) . Furthermore , most of these approaches either assume full knowledge of the intervention or make strong assumptions about its form ( Heinze-Deml et al. , 2018 ) . However , in the real world , interventions are also not always performed by an experimenter . They can be performed by other agents , or by environmental changes in ways that are unknown , or by a naive learner ( like a robot ) which does not know precisely yet how its low-level actions change high-level causal variables . In this paper , we look at the setting where interventions are unknown , and our goal is to discover causal graphs given unknown-intervention samples . The challenging aspect of this setting is to not only learn the causal graph structure , but also predict the intervention accurately . In this setting , we need to make sure to : ( 1 ) avoid an exponential search over all possible DAGs , ( 2 ) handle unknown interventions , ( 3 ) model the effect of interventions , and ( 4 ) model the underlying causal structure . One possibility for learning a causal structure ( through SCM modelling ) is to perform many experiments in which one executes interventions . Thus , such interventions modify the effect of the intervened upon variable from its parents in the corresponding DAG , which the model has to quickly adapt to . We can make parallel connections to meta-learning , where the inner loop can be considered as fast adaptation to the distribution change , and outer loop can be considered as learning the stationary meta-parameters of the model . For causal induction , one can consider each distribution which arises as a result of an intervention as a meta-example , and use a meta-learning objective for fast adaptation in response to an intervention . One can think of model parameters as being composed of slow- and fast-changing parameters . The slow parameters are analogous to the meta-parameters in meta-learning and are used for ( 1 ) intervention prediction in order to handle the unknown intervention and for ( 2 ) modeling the underlying causal structure . On the other hand , the fast parameters are used to model the effect of interventions . An explicit search over the exponentially-growing space of all possible DAGs is avoided by modeling the conditionals of the structural causal model using function approximators , with one neural network per variable . The belief over whether one node i is a direct causal parent of another node j corresponds to a dropout probability for the i-th input of network j ( which predicts variable j given its parents ) . This cheaply represents all 2M 2 possible model graphs , with the graph search implicitly achieved by learning these dropout probabilities . We thus propose a new method for fast adaptation and learning of neural causal models by framing the problem in a meta-learning setting , similar to ( Dasgupta et al. , 2019 ; Bengio et al. , 2019 ) . Our contributions Our key contributions can be summarized as follows : • Handle causal induction to the case where the variable on which a soft intervention took place is not known by the learner , and show that better results can be obtained when the learner attempts to infer that information and uses it to appropriately change parameters and meta-parameters . • We bypass the issue of having to optimize over and represent an exponentially large set of discrete causal graphs by learning an efficiently parametrized ensemble of SCMs , • Show that our algorithm correctly identifies the causal graph and use the learned graph for generalization to an unseen environment . 2 PRELIMINARIES . A Structural Causal Model ( SCM ) ( Peters et al. , 2017 ) over a finite number M of random variables Xi is a set of structural assignments Xi : = fi ( Xpa ( i , C ) , Ni ) , ∀i ∈ { 0 , . . . , M − 1 } ( 1 ) where Ni is jointly-independent noise and pa ( i , C ) is the set of parents ( direct causes ) of variable i under configuration C of the SCM directed acyclic graph , i.e. , C ∈ { 0 , 1 } M×M , with cij = 1 if node i has node j as a parent ( equivalently , Xj ∈ Xpa ( i , C ) ; i.e . Xj is a direct cause of Xi ) . The n-th power of an adjacency matrix , Cn , counts the number of length-n walks from node i to node j of the graph in element cij . The trace of the n-th power of an adjacency matrix , Tr ( Cn ) , counts the number of length-n cycles in the graph . Causal structure learning is the recovery of the ground-truth C from observational and interventional studies . Functional and structural meta-parameters Let us consider the simplest SCMs , those with M = 2 random variables ( lets say random variables A and B ) . Only three DAGs exist relating them ; they are A → B , B → A or A⊥B . These can be represented as the following . If A causes B , then the SCM between A and B can be represented as B = fθB ( cBA ·A , B ) , where cBA = 1 , B ∼ NB is a sample of an independent noise factor . If B causes A , then the SCM can be presented as A = fθA ( cAB ·B , A ) , where cAB = 1 , A ∼ NA is a sample of an independent noise factor . If A and B are independent , then we have the same equations but we set cAB = 0 and cBA = 0 . We may now think of learning the structural causal model as learning the two probabilities of cAB or cBA being 1 ( versus 0 ) , representing our belief in the causal relationship between A and B . We can parametrize these probabilities differentiably using P ( cAB = 1 ) = σ ( γAB ) and P ( cBA = 1 ) = σ ( γBA ) , with γ real numbers and σ ( x ) = 11+e−x . We can also parametrize the structural equations fθA and fθB in a differentiable manner , using conditional probability tables ( CPTs ) or neural networks . Hence , the problem becomes simultaneously learning the structural meta-parameters γ and the functional meta-parameters θ. Functional meta-parameters θ may be easily learned by maximum likelihood or a proxy , and using backpropagation . However , the γ are more difficult to infer . Bengio et al . ( 2019 ) propose to learn them by using observations of how the distribution changes sparsely , with the transfer generalization ( the adaptation rate after the sparse change ) being the training objective for γ . Simultaneously inferring both θ and γ is still more difficult . An M -variable SCM over random variables Xi , i ∈ { 0 , . . . , M − 1 } can induce a super-exponential number of adjacency matrices C. The super-exponential explosion in the number of potential graph connectivity patterns and the super-exponentially growing storage requirements of their defining conditional probability tables make CPT-based parametrizations of the structural assignments fi increasingly unwieldy as M scales . As shown below , neural networks with cij-masked inputs can provide a more manageable parametrization . For more background about different kinds of intervention we ask the reader to refer Appendix A.3 . 3 PROPOSED FRAMEWORK : META LEARNING FOR CAUSAL INDUCTION . Our framework disentangles the slow-changing meta-parameters , which reflect the stationary properties discovered by the learner , and the fast-changing parameters , which adapt in response to interventional changes in distribution . We consider two kinds of meta-parameters : the causal graph structure γ and the model ’ s slow weights θslow , along with the meta-learning objective for both of them . We also consider one kind of parameter : the model ’ s fast weights , θfast . We will call θ = θslow + θfast the sum of the slow , stationary meta-parameters and the fast , adaptational parameters . 3.1 TASK DESCRIPTION . Our task setup deviates from most common deep learning modeling setups . For the purposes of this work , we restrict ourselves to inference of randomly-generated or manually-provided ground-truth SCMs of M categorical random variables causally related via a DAG . The model is permitted to see ( 1 ) data from the original ground-truth model , and ( 2 ) data from a modified ground-truth model with a random intervention applied . In our experiments , at most one intervention is concurrently performed . When an intervention is performed , a single node is randomly and uniformly chosen among all M nodes , and its ground-truth distribution soft-intervened upon . The learner model is aware of the samples having come from an intervention distribution , but is not aware of the identity of the intervention node , and so must predict it . Each run of sampling steps under a given intervention is referred to as an episode . The learner , over a large number of episodes , will experience all nodes being intervened upon , and should be able to infer the SCM from these interventions .

Summary: This paper proposes a MAML objective to learn causal graphs from data. The data in question is randomized but the algorithm does not have access to the identity of the intervention variable. So there is an added layer of complexity of deciphering which variable was intervened on. The MAML objective, in this case, links the causal structure to the slow-moving parameter theta_slow.