Near-Optimal Multi-Perturbation Experimental Design for Causal Structure Learning

1 Introduction . The problem of finding the causal relationships between a set of variables is ubiquitous throughout the sciences . For example , scientists are interested in reconstructing gene regulatory networks ( GRNs ) of biological cells [ 11 ] . Directed Acyclic Graphs ( DAGs ) are a natural way to represent causal structures , with a directed edge from variable X to Y representing X being a direct cause of Y [ 33 ] . Learning the causal structure of a set of variables is fundamentally difficult . With only observational data , in general we can only identify the true DAG up to a set of DAGs called its Markov Equivalence Class ( MEC ) [ 35 ] . Empirically , for sparse DAGs the size of the MEC grows exponentially in the number of nodes [ 17 ] . Identifiability can be improved by intervening on variables , meaning one perturbs a subset of the variables and then observes more samples from the system [ 8 , 16 , 40 ] . There exist various inference algorithms for learning causal structures from a combination of observational and interventional data [ 16 , 36 , 40 , 28 , 34 ] . Here we focus on the identification of DAGs that have no unobserved confounding variables . Performing experiments is often expensive , however . Thus , we are interested in learning as much about the causal structure as possible given some constraints on the interventions . In this work , we focus on the batched setting , where several interventions are performed in parallel . This is a natural setting in scientific domains like reconstructing GRNs . Existing works propose meaningful objective 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . functions for this batched causal structure learning problem and then give algorithms that have provable guarantees [ 3 , 13 ] . However , these works focus on the setting where only a single random variable is perturbed per intervention . It is an open question as to whether there exist efficient algorithms for the multiple-perturbation setting , where more than one variable is perturbed in each intervention . For the task of reconstructing GRNs , it is now possible for experimenters to perturb multiple genes in a single cell [ 2 , 7 ] . Figures 1 b ) and c ) illustrate a specific example where a two-node intervention completely identifies a DAG in half as many interventions as single-node interventions . In general , it is possible for a set of q-node interventions to orient up to q-times more edges in a DAG than single-node interventions ( see the supplementary material for a more general example ) . While multiperturbation interventions can be more informative , designing them is algorithmically challenging because it leads to an exponentially larger search space : any algorithm must now select a set of sets . Our main contribution is to provide efficient algorithms for different objective functions with accompanying performance bounds . We demonstrate empirically on both synthetic and GRN graphs that our algorithms result in greater identifiability than existing approaches that do not make use of multiple perturbations [ 13 , 40 ] , as well as a random strategy . We begin by introducing the notation and the objective functions considered in this work in Section 2 , before reviewing related work in Section 3 . In Section 4 we present our algorithms along with proofs of their performance guarantees . Finally , in Section 5 we demonstrate the superior empirical performance of our method over existing baselines on both synthetic networks and on data generated from models of real GRNs . 2 Background and Problem Statement . Causal DAGs Consider a causal DAG G = ( [ p ] , E ) where [ p ] : = { 1 , ... , p } is a set of nodes and E is a set of directed edges . Let ( i , j ) ∈ E iff there is an edge from node i to node j . Each node i is associated with a random variable Xi . In the GRN example , Xi would be the measurement of the gene expression level for gene i . An edge from i → j would represent gene i having a causal effect on the expression of gene j . The functional dependence of a random variable on its parents can be described by a structural equation model ( SEM ) . The probability distribution over X = ( X1 , . . . , Xp ) is related to G by the Markov property , meaning each variable Xi is conditionally independent of its non-descendants given its parents [ 33 ] . From conditional independence tests one can determine the MEC of G , a set of DAGs with the same conditional independancies between variables . All members of the MEC share the same undirected skeleton and colliders [ 35 ] . A collider is a pair of edges ( i , k ) , ( j , k ) ∈ E such that ( i , j ) , ( j , i ) /∈ E. The essential graph of G , Ess ( G ) , is a partially directed graph , with directed edges where all members of the MEC share the same edge direction , and with undirected edges otherwise [ 5 ] . Ess ( G ) uniquely represents the MEC of G. These MECs can be large , so we seek to perform interventions on the nodes to reduce the MEC to a smaller set of possible DAGs . Interventions We use the term intervention to refer to a set I ⊂ [ p ] of perturbation targets ( variables ) . We assume all interventions are hard interventions , meaning intervening on a set I removes the incoming edges to the random variables XI : = ( Xi ) i∈I and sets their joint distribution to some interventional distribution PI [ 9 ] . In the GRN reconstruction example this corresponds to , for example , running an experiment where we knockout all genes in set I . Some of our results extend easily to the alternative model of soft interventions [ 25 ] , as we discuss in the supplementary material . We use I = 2 [ p ] to refer to the set of all possible interventions . Our goal will be to select a batch ξ of interventions , where ξ is a multiset of some I ∈ I . For practical reasons , we typically have constraints on the number of interventions , i.e. , |ξ| ≤ m and on the number of variables involved in each intervention |I| ≤ q , ∀I ∈ ξ. Namely , there are at most m interventions per batch and each intervention contains at most q nodes . A constraint on the number of perturbations per intervention is natural in reconstructing GRNs , since perturbing too many genes in one cell will leave it unlikely to survive . We refer to the set of ξs satisfying these constraints as Cm , q . The observational distribution ( no intervention ) is given by ξ = ∅ . For any set of interventions ξ and DAG G , there is a set of ξ-Markov equivalent DAGs . These are the set of DAGs that have the same set of conditional independencies under all I ∈ ξ and under the observational distribution . This set of DAGs is no larger than the MEC of G and can similarly be characterized by an essential graph Essξ ( G ) [ 16 ] . We will always assume that there exist no unobserved common causes of any pair of nodes in G. We also assume that the distribution of the random variables satisfies faithfulness with respect to G [ 33 ] . Choosing optimal interventions We seek to maximize an objective function F that quantifies our certainty in the true DAG . In general our goal is to determine argmax ξ∈Cm , q F ( ξ ) . A natural choice for F is given by Agrawal et al . [ 3 ] . They assume that there exist parameters θ that determine the functional relationships between random variables . For example , this could be the coefficients in a linear model . Given existing data D , we try to choose ξ that maximizes Objective 1 ( Mutual Information ( MI ) ) . FMI ( ξ ) = EG|DEy|G , θ̂ , ξ [ ŨM.I ( y , ξ ; D ) ] , ( 1 ) where y is the set of samples from the interventions , θ̂ is the current estimate of the parameters given D and G , and ŨM.I ( y , ξ ; D ) is the mutual information between the posterior over G and the samples y . Each intervention produces one sample in y . The use of mutual information means the objective aims to , in expectation over all observed samples and true DAGs , minimize the entropy of the posterior distribution over DAGs . There already exist a number of algorithms for determining the posterior over DAGs from observational or experimental data [ 40 , 36 , 16 ] . Infinite sample objectives Finding algorithms that optimize the MI objective is difficult because we have to account for noisy observations and limited samples . To remove this complexity , we study the limiting case of infinitely many samples per unique intervention . The constraints given by Cm , q still stipulate that there can be only m unique interventions , but each intervention can be performed with an infinite number of samples . We also assume that an essential graph is already known ( i.e. , we have infinite observational samples and infinite samples for any experiments performed so far ) . For objectives with infinite samples per intervention , we treat ξ as a set of interventions , not a multiset , since there is no change in objective value for choosing an intervention twice . Consider ξ′ to be the set of interventions contained in our dataset before our current batch . In this setting , maximizing Objective 1 reduces to maximizing Objective 2 ( Mutual info . inf . samples ( MI-∞ ) ) . F∞ ( ξ ) = − 1 |G| ∑ G∈G log2|Essξ∪ξ ′ ( G ) | , ( 2 ) where Essξ∪ξ ′ ( G ) refers to the updated essential graph after performing interventions in ξ . The objective aims to , on average across possible true DAGs , minimize the log of the essential graph size after performing the interventions . The derivation of this objective is given in the supplementary material . Ghassami et al . [ 13 ] study a different objective in the infinite-sample setting . The objective seeks to , on average across possible G given the current essential graph , orient as many edges as possible . Let R ( ξ , G , ξ′ ) be the set of edges oriented by ξ if the true DAG is G , and the essential graph is given by the ξ′-MEC . Objective 3 ( Edge-orientation ( EO ) ) . FEO ( ξ ) = 1 |G| ∑ G∈G |R ( ξ , G , ξ′ ) | . ( 3 ) The function R is computed as follows . Firstly , ∀I ∈ ξ , orient undirected edge i− j in Essξ ′ ( G ) if i ∈ I but j /∈ I or vice-versa . Secondly , execute the Meek Rules [ 26 ] , which allow inferring additional edge orientations ( discussed in the supplementary material ) on the resulting partially directed graph . Finally , output the set of all edges oriented . Agrawal et al . [ 3 ] show that this objective is not consistent ; however , this is because they fix G to be the MEC instead of using the most up-to-date essential graph for each batch . In the supplementary material , we show that the version of the objective we work with is indeed consistent . We will drop the dependence of R on ξ′ for readability . Below , we provide algorithms with near-optimality guarantees for Objectives 2 and 3 , while motivating a practical algorithm for Objective 1 .

Summary: This paper tackles the problem of causal structure learning from interventional data. Specifically, they seek a series of interventional experiments which can rapidly identify the correct causal graph, or reduce some measure of uncertainty about the true causal structure. In contrast to previous work, this paper considers the more general setting in which interventions can be made on more than one variable simultaneously (within one experiment). To select an optimal sequence of subsets on which to intervene, the authors begin by discussing 3 objectives functions that could be used to select “good” sets of interventions: two based on MI, one based on orienting edges. All objectives are grounded in existing literature. The authors propose two different algorithms to actually solve the experimental design problem. First, they propose Double Greedy Continuous. As the name implies, this method is greedy both with respect to adding interventions to the experiment sequence and with respect to adding variables to intervene on in a single interventional experiment. The authors provide strong theoretical guarantees for this method with respect to their edge-orienting objective. For the MI objective, a suitable sub-modularity property does not hold for double greedy. Instead, the authors propose a new algorithm, Separating Systems Greedy. This is greedy with respect to adding interventional experiments, but to choose sets of variables to intervene on, this algorithm uses an exhaustive search over a separating system. The authors prove guarantees relating this algorithm to their infinite data MI objective.