SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Kirankumar Shiragur* Eric and Wendy Schmidt Center, Broad Institute of MIT and Harvard
Jiaqi Zhang*  
LIDS, Massachusetts Institute of Technology  
Broad Institute of MIT and Harvard
Caroline Uhler  
LIDS, Massachusetts Institute of Technology  
Broad Institute of MIT and Harvard
Learning causal structures from interventional data is a fundamental problem with broad applications across various fields. While many previous works have focused on recovering the entire causal graph, in practice, there are scenarios where learning only part of the causal graph suffices. This is called targeted causal discovery. In our work, we focus on two such well-motivated problems: subset search and causal matching. We aim to minimize the number of interventions in both cases.
Towards this, we introduce the Meek separator, which is a subset of vertices that, when intervened, decomposes the remaining unoriented edges into smaller connected components. We then present an efficient algorithm to find Meek separators that are of small sizes. Such a procedure is helpful in designing various divide-and-conquer-based approaches. In particular, we propose two randomized algorithms that achieve logarithmic approximation for subset search and causal matching, respectively. Our results provide the first known average-case provable guarantees for both problems. We believe that this opens up possibilities to design near-optimal methods for many other targeted causal structure learning problems arising from various applications.
Discovering the causal structure among a set of variables is an important problem permeating multiple fields including biology, epidemiology, economics, and social science [FLNP00, RHB00, SGSH00, Pea03]. A common way to represent the causal structure is through a directed acyclic graph (DAG), where an arc between two variables encodes a direct causal effect [SGSH00]. The goal of causal discovery is thus to recover this DAG from data. With observational data, a DAG is generally only identifiable up to its Markov equivalence class (MEC) [VP90, AMP97]. Identifiability can be improved by performing interventions on the variables. In particular, a more refined MEC can be identified with both hard and soft interventions [HB12, YKU18], where a hard intervention eliminates the dependency between its target variables and their parents in the DAG and a soft intervention modifies this dependency without removing it [ES07].
As intervention experiments tend to be expensive in practice, a critical problem is to design algorithms to select interventions that minimize the number of trials needed to learn about the structure. Previous works have considered both fully identifying the DAG while minimizing the total number/cost of interventions [HB14, SKDV15, KDV17, GKS+19, SMG+20] and learning the most about the
*Equal contributions. Alphabetical order.
underlying DAG given a fixed budget [GSKB18]. While recovering the entire causal graph yields a holistic view of the relationships between variables, it is sometimes sufficient to learn only part of the causal graph for a particular downstream task. This is sometimes termed targeted causal discovery [SU22] and it has arisen in various different applications recently [ASY+19, GHD20, ZSU21, CS23]. The benefit of targeted causal discovery is that the number of required interventions can be significantly less than that needed to fully identify the DAG. In our work, we consider two such problems, subset search and causal matching, described in the following.
Subset Search. Proposed by [CS23], the problem of subset search aims to recover a subset of the causal relationships between variables. Formally, let  $\mathcal{G} = (V,E)$  denote the underlying DAG. Given a subset of target edges  $T\subseteq E$ , the goal is to recover the orientation of edges in  $T$  with the minimum number of interventions. Subset search problems arise in many applications, including local graph discovery for feature selection [ATS03], out-of-distribution generalization in machine learning [HDPM18], and learning gene regulatory networks [DL05]. As a concrete example, consider the study of melanoma  $[\mathrm{FMT^{+}}21]$ , a type of skin cancer. To understand its development for potential treatments, it may be of interest to identify the causal relationships between genes that are known to be relevant to melanoma. In this case, the problem can be formulated as recovering the subset of edges between melanoma-related genes.
Causal Matching. Motivated by many sequential design problems, [ZSU21] considered causal matching, where the goal is to identify an intervention which transforms a causal system to a desired state. Formally, let  $V$  be the system variables and  $P$  be its joint distribution which factorizes according to the underlying DAG  $\mathcal{G}$ . Given a desired joint distribution  $Q$  over  $V$ , the goal is to identify an intervention  $I$  such that the interventional distribution  $P^I$  best matches  $Q$  under some metric. Similar to [ZSU21], we will focus on a special form of this problem, causal mean matching, where the metric between  $P^I$  and  $Q$  is characterized by the mean discrepancy  $\| \mathbb{E}_{P^I}[V] - \mathbb{E}_Q[V] \|_2$ . Such problems arise in, for example, cellular reprogramming in genomics, which is of great interest for regenerative medicine [CD12]. The aim of this field is to reprogram easily accessible cell types into some desired cell types using genetic interventions. Since genes regulate each other via an underlying network, a targeted strategy can infer just enough about the structure in order to match the desired state.
Contributions. One of our primary contributions is an efficient algorithm for finding an intervention set  $\mathcal{I}$  of small size that, when intervened, decomposes the remaining unoriented edges into connected components of smaller sizes. This procedure is powerful in its flexibility to be used to design various divide-and-conquer based algorithms. In particular, we demonstrate this on the two targeted causal discovery problems of subset search and causal matching. $^2$
- For the subset search problem, we obtain an efficient randomized algorithm that in expectation achieves a logarithmic approximation to the minimum number of required interventions (verification number; defined in Section 2).
- For the causal mean matching problem, we derive a randomized algorithm that on average achieves a logarithmic competitive ratio against the optimal number of required interventions.
For both problems, we obtain exponential improvements – our analysis gives the first non-trivial competitive ratio in expectation; in contrast, prior works [CS23, ZSU21] show that no deterministic algorithm can achieve a better than linear approximation against the optimal solution for all instances.
Philosophically, the results in our work are a step towards the study of targeted causal discovery problems arising from various applications, where recovering the entire causal graph is unnecessary.
Organization. In Section 2, we provide formal definitions and useful results. In Section 3, we state all our main results. We provide our algorithm for Meek separator and its analysis in Section 4. We use the Meek separator subroutine to solve for subset search in Section 5 and causal matching in Section 6. In Section 7, we demonstrate empirically our proposed algorithms on synthetic data.
2.1 Basic Graph Definitions
Let  $\mathcal{G} = (V, E)$  be a graph on  $|V| = n$  vertices. We use  $V(\mathcal{G})$ ,  $E(\mathcal{G})$  and  $A(\mathcal{G}) \subseteq E(\mathcal{G})$  to denote its vertices, edges, and oriented arcs respectively. When the referred graph is clear from the
2In this work, we assume causal sufficiency [SGSH00] and noiseless setting with given observational data.
context, we use  $V$  (or  $E$ ) instead of  $V(\mathcal{G})$  (or  $E(\mathcal{G})$ ) for simplicity. The graph  $\mathcal{G}$  is fully oriented if  $A(\mathcal{G}) = E(\mathcal{G})$ , and partially oriented otherwise. For any two vertices  $u, v \in V$ , we write  $u \sim v$  if they are adjacent and  $u \not\sim v$  otherwise. To specify the arc orientations, we use  $u \to v$  or  $u \gets v$ . For any subset  $V' \subseteq V$  and  $E' \subseteq E$ , let  $\mathcal{G}[V']$  and  $\mathcal{G}[E']$  denote the vertex-induced and edge-induced subgraphs respectively. Consider a vertex  $v \in V$  in a directed graph, let  $\mathrm{Pa}(v)$ ,  $\mathrm{Anc}(v)$  and  $\mathrm{Des}(v)$  denote the parents, ancestors and descendants of  $v$  respectively. Let  $\mathrm{Des}[v] = \mathrm{Des}(v) \cup \{v\}$  and  $\mathrm{Anc}[v] = \mathrm{Anc}(v) \cup \{v\}$ . The skeleton  $\mathrm{skel}(\mathcal{G})$  of a graph  $\mathcal{G}$  refers to the graph where all edges are made undirected. A  $\nu$ -structure refers to three distinct vertices  $u, v, w$  such that  $u \to v \gets w$  and  $u \not\sim w$ . A simple cycle is a sequence of  $k \geq 3$  vertices where  $v_1 \sim v_2 \sim \ldots \sim v_k \sim v_1$ . The cycle is directed if at least one of the edges is directed and all oriented arcs are in the same direction along the cycle. A partially oriented graph is a chain graph if it contains no directed cycle. In the undirected graph  $\mathcal{G}[E \setminus A]$  obtained by removing all arcs from a chain graph  $\mathcal{G}$ , each connected component in  $\mathcal{G}[E \setminus A]$  is called a chain component. We use  $CC(\mathcal{G})$  to denote the set of all such chain components, where each chain component  $\mathcal{H} \in CC(\mathcal{G})$  is a subgraph of  $\mathcal{G}$  and  $V = \dot{\cup}_{\mathcal{H} \in CC(\mathcal{G})} V(\mathcal{H})$ . For any partially oriented graph, an acyclic completion or consistent extension refers to an assignment of orientations to undirected edges such that the resulting fully oriented graph has no directed cycles.
2.2 Graphical Concepts in Causal Models
Directed acyclic graphs (DAGs) are fully oriented chain graphs, where vertices represent random variables and their joint distribution  $P$  factorizes according to the DAG:  $P(v_{1},\ldots ,v_{n}) = \prod_{i = 1}^{n}P(v_{i}\mid \mathsf{Pa}(v))$ . We can associate a valid permutation or topological ordering  $\pi :V\to [n]$  to any (partially oriented) DAG such that oriented arcs  $u\rightarrow v$  satisfy  $\pi (u) < \pi (v)$  (and assigning undirected edges  $u\sim v$  as  $u\rightarrow v$  when  $\pi (u) < \pi (v)$  is an acyclic completion). Note that such valid permutation is not necessarily unique. Two DAGs  $\mathcal{G}_1,\mathcal{G}_2$  are in the same Markov equivalence class (MEC) if any positive distribution  $P$  which factorizes according to  $\mathcal{G}_1$  also factorizes according  $\mathcal{G}_2$ . For any DAG  $\mathcal{G}$ , we denote its MEC by  $[\mathcal{G}]$ . It is known that DAGs in the same MEC share the same skeleton and v-structures [VP90, AMP97]. A moral DAG is a DAG without v-structures. Figure 1 illustrates this definition. The essential graph  $\mathcal{E}(\mathcal{G})$  of  $[\mathcal{G}]$  is a partially oriented graph such that an arc  $u\rightarrow v$  is oriented if  $u\rightarrow v$
in every DAG in MEC  $[\mathcal{G}]$ , and an edge  $u\sim v$  is undirected if there exists two DAGs  $\mathcal{G}_1,\mathcal{G}_2\in [\mathcal{G}]$  such that  $u\rightarrow v$  in  $\mathcal{G}_1$  and  $u\gets v$  in  $\mathcal{G}_2$ . An edge  $u\sim v$  is a covered edge [Chi95, Definition 2] if  $\operatorname {Pa}(u)\setminus \{v\} = \operatorname {Pa}(v)\setminus \{u\}$ . We now give some useful definition and result for graph separators:
Definition 1 ( $\alpha$ -separator and  $\alpha$ -clique separator [CSB22]). Let  $A, B, C$  be a partition of the vertices  $V$  of a graph  $\mathcal{G}$ . We say that  $C$  is an  $\alpha$ -separator if no edge joins a vertex in  $A$  with a vertex in  $B$  and  $|A|, |B| \leq \alpha \cdot |V|$ . We call  $C$  is an  $\alpha$ -clique separator if it is an  $\alpha$ -separator and a clique.
Lemma 2 (Theorem 1,3 in [GRE84], instantiated for unweighted graphs). Let  $\mathcal{G} = (V,E)$  be a chordal graph with  $|V|\geq 2$ . Denote  $\omega (\mathcal{G})$  as the size of its largest clique. There exists a  $1 / 2$ -clique separator  $C$  involving at most  $\omega (\mathcal{G}) - 1$  vertices. The clique  $C$  can be computed in  $\mathcal{O}(|E|)$  time.
2.3 Interventions and Verifying Sets
An intervention  $I \subseteq V$ , associated with random variables  $V$  with joint distribution  $P$  that factorizes according to  $\mathcal{G}$ , is an experiment where the conditional distributions  $P(v \mid \mathrm{Pa}(v))$  for  $v \in I$  are changed. Hard interventions refer to changes that eliminate the dependency between  $v$  and  $\mathrm{Pa}(v)$ , while soft interventions modify this dependency without removing it [ES07]. Let  $P^I$  denote the interventional distribution. Observational data is a special case where  $I = \varnothing$ . An intervention is atomic if  $|I| = 1$  and bounded if  $|I| \leq k$ . We call a set of interventions  $\mathcal{I} \subseteq 2^V$  an intervention set.
<sup>3</sup>We denote  $A \dot{\cup} B$  as the union of two disjoint sets  $A$  and  $B$ .
4 Edge  $u\sim v$  joins vertex  $u$  with vertex  $v$
A clique is a fully connected graph, i.e.,  $u \sim v$  for every pair of distinct vertices in the graph.
${}^{6}$  A chordal graph is a graph where every cycle of length at least 4 has a chord (c.f. [BP93]).
(a) A moral DAG.
(b) A DAG with v-structure  $2\to 4\gets 3$
Figure 1: An example of moral DAGs. (a) a moral DAG without v-structures. (b) an immoral DAG.
Figure 2: An example illustrating essential graphs and connected chain components. Suppose the ground-truth DAG is given in Fig. 1a. (a) the essential graph of  $\mathcal{G}$ . (b) the  $\mathcal{I}$ -essential graph of  $\mathcal{G}$  after two atomic interventions  $\mathcal{I} = \{\{3\}, \{4\}\}$ , where edges oriented by  $\mathcal{I}$  are indicated in red. (c) three connected components of  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$  after removing the oriented edges.
With observational data, a DAG  $\mathcal{G}$  is generally only identifiable up to its MEC $^7$ , i.e.,  $\mathcal{E}(\mathcal{G})$  [AMP97]. Identifiability can be improved with interventional data and it is known that intervening on  $I$  allows us to infer the edge orientation of any edge cut by  $I$  and  $V \setminus I$  and possibly additional edges given by the Meek rules (Appendix A), for both hard [HB12] and soft [YKU18] interventions. $^8$  For intervention set  $\mathcal{I}$ , let the  $\mathcal{I}$ -essential graph  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$  of  $\mathcal{G}$  be the essential graph representing all DAGs in  $[\mathcal{G}]$  whose orientations of arcs cut by  $I$  and  $V \setminus I$  are the same as  $\mathcal{G}$  for all  $I \in \mathcal{I}$ . Figure 2 illustrates these concepts. The aforementioned results state that  $\mathcal{G}$  can be identified up to  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$  with observational and interventional data from  $\mathcal{I}$ . We state some useful properties about  $\mathcal{I}$ -essential graphs from [HB14]. First, every  $\mathcal{I}$ -essential graph is a chain graph with chordal chain components. This includes the case of  $\mathcal{I} = \emptyset$ . Second, orientations in one chain component do not affect orientations in other components. In other words, to fully orient any essential graph  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$ , it is necessary and sufficient to orient every chain component in  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$ .
A verifying set  $\mathcal{I}$  for a DAG  $\mathcal{G}$  [CSB22] is an intervention set that fully orients  $\mathcal{G}$  from  $\mathcal{E}(\mathcal{G})$ , possibly with repeated applications of Meek rules (see Appendix A). In other words, for any graph  $\mathcal{G} = (V,E)$  and any verifying set  $\mathcal{I}$  of  $\mathcal{G}$ , we have  $\mathcal{E}_{\mathcal{I}}(\mathcal{G}) = \mathcal{G}$ . A subset verifying set  $\mathcal{I}$  [CS23] for a subset of target edges  $T \subseteq E$  in a DAG  $\mathcal{G}$  is an intervention set that fully orients all arcs in  $T$  given  $\mathcal{E}(\mathcal{G})$ , possibly with repeated applications of Meek rules. Note that the subset verifying set depends on the target edges and the underlying ground truth DAG — the subset verifying set for the same  $T \subseteq E$  may differ across two different DAGs  $\mathcal{G}_1, \mathcal{G}_2$  in the same Markov equivalence class.
For bounded interventions of size at most  $k$ , the minimum verification number  $\nu_{k}(\mathcal{G},T)$  denotes the size of the minimum size subset verifying set for any DAG  $\mathcal{G} = (V,E)$  and subset of target edges  $T\subseteq E$ . We write  $\nu_{1}(\mathcal{G},T)$  when we restrict to atomic interventions. When  $k = 1$  and  $T = E$  (i.e., full graph identification), [CSB22] showed that it is necessary and sufficient to intervene on a minimum vertex cover of the covered edges in  $\mathcal{G}$ . For any intervention set  $\mathcal{I}\subseteq 2^{V}$ , we denote  $R(\mathcal{G},\mathcal{I}) = A(\mathcal{E}_{\mathcal{I}}(\mathcal{G}))\subseteq E$  as the set of oriented arcs in the  $\mathcal{I}$ -essential graph of a DAG  $\mathcal{G}$ . For cleaner notation, we write  $R(\mathcal{G},I)$  for one intervention  $\mathcal{I} = \{I\}$  for some  $I\subseteq V$ , and  $R(\mathcal{G},v)$  for one atomic intervention  $\mathcal{I} = \{\{v\}\}$  for some  $v\in V$ .
Definition 3. For any intervention set  $\mathcal{I} \subseteq 2^V$ , define  $\mathcal{G}^{\mathcal{I}} = \mathcal{G}[E \setminus R(G, \mathcal{I})]$  as the fully oriented subgraph of  $\mathcal{G}$  induced by the unoriented edges in  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$ . In addition, for  $u \in V$ , let  $\mathrm{Pa}_{\mathcal{G}, \mathcal{I}}(u) = \{x \in V : x \to u \in R(\mathcal{G}, \mathcal{I})\}$  as the recovered parents of  $u$  by  $\mathcal{I}$ .
Here we state all the main results of the paper. One of the primary contributions of our work is a randomized algorithm that outputs an intervention set  $\mathcal{I}$  of small size such that all connected components in the resulting  $\mathcal{I}$ -essential graph have small sizes. We now formally define such intervention sets as Meek separators. An example of  $1/2$ -Meek separator is shown in Figure 3.
Definition 4 ( $\alpha$ -Meek separator). We call an intervention set  $\mathcal{I}$  an  $\alpha$ -Meek separator of  $\mathcal{G}$  if each connected component  $\mathcal{H} \in CC(\mathcal{E}_{\mathcal{I}}(\mathcal{G}))$  satisfies:  $|V(\mathcal{H})| \leq \alpha |V(\mathcal{G})|$ .
Note that Meek separators differ from the traditional graph separators in Definition 1, where in the latter we have bounds on the sizes of connected components in  $\mathcal{G} \backslash \mathcal{I}$  instead of  $CC(\mathcal{E}_{\mathcal{I}}(\mathcal{G}))$ . Graph separators of small size may not exist. For example, consider a fully connected DAG  $\mathcal{G} = (V, E)$
<sup>7</sup>With additional parametric assumptions (e.g., [PB14]), additional identifiability can be achieved.
For both cases, a "faithfulness" assumption needs to be assumed for  $P$ ,  $P^I$  (c.f. [YKU18]).
(a) DAG  $\mathcal{G}$
(b)  $1 / 2$  -Meek separator  $\mathcal{I} = \{\{2\}\}$
(c)  $CC(\mathcal{E}_{\mathcal{I}}(\mathcal{G}))$
Figure 3: An example comparing a  $1/2$ -Meek separator with a  $1/2$ -graph separator. (a) the underlying true DAG  $\mathcal{G}$ . (b) the essential graph  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$  after intervening on the  $1/2$ -Meek separator  $\mathcal{I} = \{\{2\}\}$ , where edges oriented are indicated in red. (c) connected components of size  $\leq |V(\mathcal{G})|/2$  in  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$ . (d) the undirected version of  $\mathcal{G}$  with the  $1/2$ -graph separator  $\{2,3\}$  highlighted.
(d)  $1 / 2$  -graph separator  $\{2,3\}$
that forms a clique. Any  $\alpha$ -graph separator in  $\mathcal{G}$  must contain at least  $(1 - \alpha)|V|$  vertices, since every pair of vertices in the clique is connected. In contrast, we show that small-sized Meek separators always exist for any DAG. Moreover, we can efficiently find such separators by performing very few interventions, as given by the following theorem which we prove in Section 4.
Theorem 5 (Meek separator). Given an essential graph  $\mathcal{E}(\mathcal{G})$  of an unknown DAG  $\mathcal{G}$ , there exists a randomized procedure MeekSeparator (given in Algorithm 1) that runs in polynomial time and adaptively intervenes on a set of atomic interventions  $\mathcal{I}$  such that we can find a  $1/2$ -Meek separator of size at most 2 and  $\mathbb{E}[|\mathcal{I}|] \leq \mathcal{O}(\log \omega(\mathcal{G}))$ , where  $\omega(\mathcal{G})$  denotes the size of the largest clique in  $\mathcal{G}$ .
Although in the above result, we intervene on  $\mathcal{O}(\log \omega (\mathcal{G}))$  nodes, our proofs show that there always exists a Meek separator of size at most 2. However, to find such a Meek separator without knowing  $\mathcal{G}$  a priori, our algorithm needs to perform  $\mathcal{O}(\log \omega (\mathcal{G}))$  many interventions. The above result is significant because it can be used to design divide-and-conquer based approaches for various problems. Specifically, we use the Meek separator algorithm as a subroutine to develop approximation algorithms for the subset search and causal matching problems.
Subset Search. Our first application of the Meek separator result is that it can be used as a subroutine for approximately solving the subset search problem, as demonstrated in Algorithm 2. The analysis of our algorithm for subset search consists of two parts: (1) a lower bound on the number of interventions required for any algorithm (even with knowledge of  $\mathcal{G}$ ) to solve subset search, and (2) an upper bound on the number of interventions needed for our algorithm to solve subset search. By combining the two parts, we can bound the competitive ratio of our algorithm.
For the first part, we provide a lower bound for the subset verification number described below. Recall that the subset verification number is the minimum number of interventions needed to orient edges in  $T$  by any algorithm with full knowledge of  $\mathcal{G}$ . Therefore it is a natural lower bound on the number of interventions required for any algorithm to solve subset search.
Lemma 6 (Lower bound). Let  $\mathcal{G} = (V,E)$  be a DAG and  $T\subseteq E$  be a subset of target edges, then,  $\nu_{1}(\mathcal{G},T)\geq \max_{I\subseteq V}\sum_{\mathcal{H}\in CC(\mathcal{E}_{I}(\mathcal{G}))}\mathbb{1}(E(\mathcal{H})\cap T\neq \emptyset)$
For the upper bound, we show that our randomized algorithm, designed using the Meek separator subroutine, is competitive with respect to the aforementioned lower bound. Thus, it achieves a logarithmic approximation to the optimal number of interventions required to solve subset search.
Theorem 7 (Upper bound). Let  $\mathcal{G} = (V,E)$  be a DAG with  $|V| = n$  and  $T\subseteq E$  be a subset of target edges. Algorithm 2 that takes essential graph  $\mathcal{E}(\mathcal{G})$  and  $T$  as input, runs in polynomial time and adaptively intervenes on a set of atomic interventions  $\mathcal{I}\subseteq V$  that satisfies,  $\mathbb{E}[|\mathcal{I}|]\leq \mathcal{O}\bigl (\log n\cdot \log \omega (\mathcal{G})\bigr)\cdot \nu_1(\mathcal{G},T)$  and  $T\subseteq R(\mathcal{G},\mathcal{I})$ . Furthermore, in the special case of  $T = E$ , the solution returned by it satisfies,  $\mathbb{E}[|\mathcal{I}|]\leq \mathcal{O}(\log n)\cdot \nu_{1}(\mathcal{G})$  and  $E\subseteq R(\mathcal{G},\mathcal{I})$ .