Matching a Desired Causal State via Shift Interventions

1 Introduction . Consider an experimental biologist attempting to turn cells from one type into another , e.g. , from fibroblasts to neurons ( Vierbuchen et al. , 2010 ) , by altering gene expression . This is known as cellular reprogramming and has shown great promise in recent years for regenerative medicine ( Rackham et al. , 2016 ) . A common approach is to model gene expression of a cell , which is governed by an underlying gene regulatory network , using a structural causal model ( Friedman et al. , 2000 ; Badsha et al. , 2019 ) . Through a set of interventions , such as gene knockouts or overexpression of transcription factors ( Dominguez et al. , 2016 ) , a biologist can infer the structure of the underlying regulatory network . After inferring enough about this structure , a biologist can identify the intervention needed to successfully reprogram a cell . More generally , transforming a causal system from an initial state to a desired state through interventions is an important task pervading multiple applications . Other examples include closed-loop control ( Touchette and Lloyd , 2004 ) and pathway design of microstructures ( Wodo et al. , 2015 ) . With little prior knowledge of the underlying causal model , this task is intrinsically difficult . Previous works have addressed the problem of intervention design to achieve full identifiability of the causal model ( Hauser and Bühlmann , 2014 ; Greenewald et al. , 2019 ; Squires et al. , 2020a ) . However , since interventional experiments tend to be expensive in practice , one wishes to minimize the number of trials and learn just enough information about the causal model to be able to identify the intervention that will transform it into the desired state . Furthermore , in many realistic cases , the set 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . of interventions which can be performed is constrained . For instance , in CRISPR experiments , only a limited number of genes can be knocked out to keep the cell alive ; or in robotics , a robot can only manipulate a certain number of arms at once . Contributions . We take the first step towards the task of causal matching ( formalized in Section 2 ) , where an experimenter can perform a series of interventions in order to identify a matching intervention which transforms the system to a desired state . In particular , we consider the case where the goal is to match the mean of a distribution . We focus on a subclass of interventions called shift interventions , which can for example be used to model gene over-expression experiments ( Triantafillou et al. , 2017 ) . These interventions directly increase or decrease the values of their perturbation targets , with their effect being propagated to variables which are downstream ( in the underlying causal graph ) of these targets . We show that there always exists a unique shift intervention ( which may have multiple perturbation targets ) that exactly transforms the mean of the variables into the desired mean ( Lemma 1 ) . We call this shift intervention the matching intervention . To find the matching intervention , in Section 3 we characterize the Markov equivalence class of a causal graph induced by shift interventions , i.e. , the edges in the causal graph that are identifiable from shift interventions ; in particular , we show that the resulting Markov equivalence classes can be more refined than previous notions of interventional Markov equivalence classes . We then propose two active learning strategies in Section 4 based on this characterization , which are guaranteed to identify the matching intervention . These active strategies proceed in an adaptive manner , where each intervention is chosen based on all the information gathered so far . In Section 5 , we derive a worst-case lower bound on the number of interventions required to identify the matching intervention and show that the proposed strategies are optimal up to a logarithmic factor . Notably , the proposed strategies may use exponentially fewer interventions than previous active strategies for structure learning . Finally , in Section 6 , we demonstrate also empirically that our proposed strategies outperform previous methods as well as other baselines in various settings . 1.1 Related Works . Experimental Design . Previous work on experimental design in causality has considered two closely related goals : learning the most structural information about the underlying DAG given a fixed budget of interventions ( Ghassami et al. , 2018 ) , and fully identifying the underlying DAG while minimizing the total number or cost ( Shanmugam et al. , 2015 ; Kocaoglu et al. , 2017 ) of interventions . These works can also be classified according to whether they consider a passive setting , i.e. , the interventions are picked at a single point in time ( Hyttinen et al. , 2013 ; Shanmugam et al. , 2015 ; Kocaoglu et al. , 2017 ) , or an active setting , i.e. , interventions are decided based on the results of previous interventions ( He and Geng , 2008 ; Agrawal et al. , 2019 ; Greenewald et al. , 2019 ; Squires et al. , 2020a ) . The setting addressed in the current work is closest to the active , full-identification setting . The primary difference is that in order to match a desired mean , one does not require full identification ; in fact , as we show in this work , we may require significantly less interventions . Causal Bandits . Another related setting is the bandit problem in sequential decision making , where an agent aims to maximize the cumulative reward by selecting an arm at each time step . Previous works considered the setting where there are causal relations between regrets and arms ( Lattimore et al. , 2016 ; Lee and Bareinboim , 2018 ; Yabe et al. , 2018 ) . Using a known causal structure , these works were able to improve the dependence on the total number of arms compared to previous regret lower-bounds ( Bubeck and Cesa-Bianchi , 2012 ; Lattimore et al. , 2016 ) . These results were further extended to the case when the causal structure is unknown a priori ( de Kroon et al. , 2020 ) . In all these works the variables are discrete , with arms given by do-interventions ( i.e. , setting variables to a given value ) , so that there are only a finite number of arms . In our work , we are concerned with the continuous setting and shift interventions , which corresponds to an infinite ( continuous ) set of arms . Correlation-based Approaches . There are also various correlation-based approaches for this task that do not make use of any causal information . For example , previous works have proposed scorebased ( Cahan et al. , 2014 ) , entropy-based ( D ’ Alessio et al. , 2015 ) and distance-based approaches ( Rackham et al. , 2016 ) for cellular reprogramming . However , as shown in bandit settings ( Lattimore et al. , 2016 ) , when the system follows a causal structure , this structure can be exploited to learn the optimal intervention more efficiently . Therefore , we here focus on developing a causal approach . 2 Problem Setup . We now formally introduce the causal matching problem of identifying an intervention to match the desired state in a causal system under a given metric . Following Koller and Friedman ( 2009 ) , a causal structural model is given by a directed acyclic graph ( DAG ) G with nodes [ p ] = { 1 , . . . , p } , and a set of random variables X = { X1 , ... , Xp } whose joint distribution P factorizes according to G. Denote by paG ( i ) = { j 2 [ p ] | j ! i } the parents of node i in G. An intervention I ⇢ [ p ] with multiple perturbation targets i 2 I either removes all incoming edges to Xi ( hard intervention ) or modifies the conditional probability P ( Xi|XpaG ( i ) ) ( soft intervention ) for all i 2 I . This results in an interventional distribution PI . Given a desired joint distribution Q over X , the goal of causal matching is to find an optimal matching intervention I such that PI best matches Q under some metric . In this paper , we address a special case of the causal matching problem , which we call causal mean matching , where the distance metric between PI and Q depends only on their expectations . We focus on causal mean matching for a class of soft interventions , called shift interventions ( Rothenhäusler et al. , 2015 ) . Formally , a shift intervention with perturbation targets I ⇢ [ p ] and shift values { ai } i2I modifies the conditional distribution as PI ( Xi = x+ ai|XpaG ( i ) ) = P ( Xi = x|XpaG ( i ) ) . Here , the shift values { ai } i2I are assumed to be deterministic . We aim to find I ⇢ [ p ] and { ai } i2I 2 R|I| such that the mean of PI is closest to that of Q , i.e. , minimizes d ( EPI ( X ) , EQ ( X ) ) for some metric d. In fact , as we show in the following lemma , there always exists a unique shift intervention , which we call the matching intervention , that achieves exact mean matching.1 Lemma 1 . For any causal structural model and desired mean EQ ( X ) , there exists a unique shift intervention I⇤ such that EPI⇤ ( X ) = EQ ( X ) . We assume throughout that the underlying causal DAG G is unknown . But we assume causal sufficiency ( Spirtes et al. , 2000 ) , which excludes the existence of latent confounders , as well as access to enough observational data to determine the joint distribution P and thus the Markov equivalence class of G ( Andersson et al. , 1997 ) . It is well-known that with enough interventions , the causal DAG G becomes fully identifiable ( Yang et al. , 2018 ) . Thus one strategy for causal mean matching is to first use interventions to fully identify the structure of G , and then solve for the matching intervention given full knowledge of the graph . However , in general this strategy requires more interventions than needed . In fact , the number of interventions required by such a strategy can be exponentially larger than the number of interventions required by a strategy that directly attempts to identify the matching intervention , as illustrated in Figure 1 and proven in Theorem 2 . In this work , we consider active intervention designs , where a series of interventions are chosen adaptively to learn the matching intervention . This means that the information obtained after performing each intervention is taken into account for future choices of interventions . We here focus on the noiseless setting , where for each intervention enough data is obtained to decide the effect of each intervention . Direct implications for the noisy setting are discussed in Appendix G. To incorporate realistic cases in which the system can not withstand an intervention with too many target 1To lighten notation , we use I to denote both the perturbation targets and the shift values of this intervention . variables , as is the case in CRISPR experiments , where knocking out too many genes at once often kills the cell , we consider the setting where there is a sparsity constraint S on the maximum number of perturbation targets in each intervention , i.e. , we only allow I where |I| S .

Summary: This paper considers the problem of identifying a shift intervention to match a given joint distribution of a causal model to a desired joint distribution at the expectation level. It defines the causal mean matching criteria, shows the existence of shift interventions for matching the desired means, and characterizes the shift-interventional Markov equivalence class (shift-I-MEC), which is a more refined subclass of interventional MEC (I-MEC). This refinement is due to the interventions being a certain type of soft interventions, that is the shift interventions.  Following a chain graph decomposition of the interventional essential graph, clique tree and supermodular optimization approaches are proposed for active learning of the matching intervention for causal mean matching. A lower bound for the number of shift-interventions and upper bound for the proposed algorithms are provided. Importantly, the upper bound is a logarithmic factor of the lower bound. Proposed approach is compared to random search and structure learning based baselines in simulations with synthetic data.