Original: E FFICIENT N EURAL C AUSAL D ISCOVERY WITHOUT A CYCLICITY C ONSTRAINTS
Revised: E FFICIENT N EURAL C AUSAL D ISCOVERY WITHOUT A CYCLICITY C ONSTRAINTS

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Learning the structure of a causal graphical model using both observational and interventional data is a fundamental problem in many scientiﬁc ﬁelds.
Revised: Learning the structure of a causal graphical model using both observational and interventional data is a fundamental problem in many scientiﬁc ﬁelds.

Original: A promising direction is continuous optimization for score-based methods, which efﬁciently learn the causal graph in a data-driven manner. However, to date, those methods require constrained optimization to enforce acyclicity or lack convergence guarantees.
Revised: A promising direction is continuous optimization for score-based methods, which, however, require constrained optimization to enforce acyclicity or lack convergence guarantees.

Original: In this paper, we present ENCO, an efﬁcient structure learning method for directed, acyclic causal graphs leveraging observational and interventional data.
Revised: In this paper we present ENCO, an efﬁcient structure learning method for directed, acyclic causal graphs leveraging observational and interventional data.

Original: ENCO formulates the graph search as an optimization of independent edge likelihoods, with the edge orientation being modeled as a separate parameter.
Revised: ENCO formulates the graph search as an optimization of independent edge likelihoods, with the edge orientation being modeled as a separate parameter.

Original: Consequently, we provide for ENCO convergence guarantees when interventions on all variables are available, without having to constrain the score function with respect to acyclicity.
Revised: Consequently, we provide for ENCO convergence guarantees when interventions on all variables are available, without having to constrain the score function with respect to acyclicity.

Original: In experiments, we show that ENCO can efﬁciently recover graphs with hundreds of nodes, an order of magnitude larger than what was previously possible, while handling deterministic variables and discovering latent confounders.
Revised: In experiments we show that ENCO can efﬁciently recover graphs with hundreds of nodes, an order of magnitude larger than what was previously possible, while handling deterministic variables and discovering latent confounders

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Uncovering and understanding causal mechanisms is an important problem not only in machine learning (Pearl, 2009; Schölkopf et al., 2021) but also in various scientiﬁc disciplines such as computational biology (Friedman et al., 2000; Sachs et al., 2005), epidemiology (Robins et al., 2000;
Revised: Uncovering and understanding causal mechanisms is an important problem not only in machine learning (Schölkopf et al., 2021; Pearl, 2009) but also in various scientiﬁc disciplines such as computational biology (Friedman et al., 2000; Sachs et al., 2005), epidemiology (Robins et al., 2000;

Original: Vandenbroucke et al., 2016), and economics (Hicks et al., 1980; Pearl, 2009).
Revised: Vandenbroucke et al., 2016), and economics (Pearl, 2009; Hicks et al., 1980).

Original: A common task of interest is causal structure learning (Pearl, 2009; Peters et al., 2017), which aims at learning a directed acyclic graph (DAG) in which edges represent causal relations between variables.
Revised: A common task of interest is causal structure learning (Pearl, 2009; Peters et al., 2017), which aims at learning a directed acyclic graph (DAG) in which edges represent causal relations between variables.

Original: While observational data alone is in general not sufﬁcient to identify the DAG (Hauser & Bühlmann, 2012; Yang et al., 2018), interventional data can improve identiﬁability up to ﬁnding the exact graph (Eberhardt, 2008; Eberhardt et al., 2005).
Revised: While observational data alone is in general not sufﬁcient to identify the DAG (Yang et al., 2018; Hauser & Bühlmann, 2012), interventional data can improve identiﬁability up to ﬁnding the exact graph (Eberhardt et al., 2005; Eberhardt, 2008).

Original: Unfortunately, the solution space of DAGs grows super-exponentially with the variable count, requiring efﬁcient methods for large graphs.
Revised: Unfortunately, the solution space of DAGs grows super-exponentially with the variable count, requiring efﬁcient methods for large graphs.

Original: Current methods are typically applied to a few dozens of variables and cannot scale so well, which is imperative for modern applications like learning causal relations with gene editing interventions (Dixit et al., 2016; Macosko et al., 2015).
Revised: Current methods are typically applied to a few dozens of variables and cannot scale so well, which is imperative for modern applications like learning causal relations with gene editing interventions (Dixit et al., 2016; Macosko et al., 2015).

Original: A promising new direction for scaling up DAG discovery methods are continuous-optimization methods (Brouillard et al.,  2020; Ke et al., 2019; Yu et al., 2019; Zheng et al.,2018; 2020; Zhu et al., 2020).
Revised: A promising new direction for scaling up DAG discovery methods are continuous-optimization methods (Zheng et al., 2018; 2020; Zhu et al., 2020; Ke et al., 2019; Brouillard et al., 2020; Yu et al., 2019).

Original: In contrast to score-based and constrained-based (Guo et al., 2020; Peters et al., 2017) methods, continuous-optimization methods reinterpret the search over discrete graph topologies as a continuous problem with neural networks as function approximators, for which efﬁcient solvers are amenable.
Revised: In contrast to score-based and constrained-based (Peters et al., 2017; Guo et al., 2020) methods, continuous-optimization methods reinterpret the search over discrete graph topologies as a continuous problem with neural networks as function approximators, for which efﬁcient solvers are amenable.

Original: To restrict the search space to acyclic graphs, Zheng et al.
Revised: To restrict the search space to acyclic graphs, Zheng et al.

Original: ﬁrst proposed to view the search as a constrained optimization problem using an augmented Lagrangian procedure to solve it.
Revised: ﬁrst proposed to view the search as a constrained optimization problem using an augmented Lagrangian procedure to solve it.

Original: While several improvements have been explored (Brouillard et al., 2020; Lachapelle et al., 2020; Yu et al., 2019; Zheng et al., 2020), constrained optimization methods remain slow and hard to train.
Revised: While several improvements have been explored (Zheng et al., 2020; Brouillard et al., 2020; Yu et al., 2019; Lachapelle et al., 2020), constrained optimization methods remain slow and hard to train.

Original: Alternatively, it is possible to apply a regularizer (Ke et al., 2019; Zhu et al., 2020) to penalize cyclic graphs.
Revised: Alternatively, it is possible to apply a regularizer (Zhu et al., 2020; Ke et al., 2019) to penalize cyclic graphs.

Original: While simpler to optimize, methods relying on acyclicity regularizers commonly lack guarantees for ﬁnding the correct causal graph, often converging to suboptimal solutions.
Revised: While simpler to optimize, methods relying on acyclicity regularizers commonly lack guarantees for ﬁnding the correct causal graph, often converging to suboptimal solutions.

Original: Despite the advances, beyond linear, continuous settings (Ng et al., 2020; Varando, 2020) continuous optimization methods still cannot scale to more than 100 variables, often due to difﬁculties in enforcing acyclicity.
Revised: Despite the advances, beyond linear, continuous settings (Ng et al., 2020; Varando, 2020) continuous optimization methods still cannot scale to more than 100 variables, often due to difﬁculties in enforcing acyclicity.

Original: In this work, we address both problems.
Revised: In this work, we address both problems.

Original: By modeling the orientation of an edge as a separate parameter, we can deﬁne the score function without any acyclicity constraints or regularizers.
Revised: By modeling the orientation of an edge as a separate parameter, we can deﬁne the score function without any acyclicity constraints or regularizers.

Original: This allows for unbiased low-variance gradient estimators that scale learning to much larger graphs.
Revised: This allows for unbiased low-variance gradient estimators that scale learning to much larger graphs.

Original: Yet, if we are able to intervene on all variables, the proposed optimization is guaranteed to converge to the correct, acyclic graph.
Revised: Yet, if we are able to intervene on all variables, the proposed optimization is guaranteed to converge to the correct, acyclic graph.

Original: Importantly, since such interventions might not always be available, we show that our algorithm performs robustly even when intervening on fewer variables and having small sample sizes.
Revised: Importantly, since such interventions might not always be available, we show that our algorithm performs robustly even when intervening on fewer variables and having small sample sizes.

Original: We call our method ENCO for E fﬁcient N eural C ausal Disc o very .
Revised: We call our method ENCO for E fﬁcient N eural C ausal Disc o very .

Original: We make the following four contributions.
Revised: We make the following four contributions.

Original: Firstly, we propose ENCO, a causal structure learning method for observational and interventional data using continuous optimization.
Revised: Firstly, we propose ENCO, a causal structure learning method for observational and interventional data using continuous optimization.

Original: Different from recent methods, ENCO models the edge orientation as a separate parameter.
Revised: Different from recent methods, ENCO models the edge orientation as a separate parameter.

Original: Secondly, we derive unbiased, low-variance gradient estimators, which is crucial for scaling up the model to large numbers of variables.
Revised: Secondly, we derive unbiased, low-variance gradient estimators, which is crucial for scaling up the model to large numbers of variables.

Original: Thirdly, we show that ENCO is guaranteed to converge to the correct causal graph if interventions on all variables are available, despite not having any acyclicity constraints.
Revised: Thirdly, we show that ENCO is guaranteed to converge to the correct causal graph if interventions on all variables are available, despite not having any acyclicity constraints.

Original: That said, we show in practice that the algorithm works on partial intervention sets as well.
Revised: Yet, we show in practice that the algorithm works on partial intervention sets as well.

Original: Fourthly, we extend ENCO to detecting latent confounders.
Revised: Fourthly, we extend ENCO to detecting latent confounders.

Original: In various experimental settings,data set sizes, and real-world datasets, we show that ENCO recovers graphs accurately, making less than one mistake on average on graphs with 1,000 variables in less than nine hours of computation.
Revised: In various experimental settings, ENCO recovers graphs accurately, making less than one error on graphs with 1,000 variables in less than nine hours of computation.

Original:  B ACKGROUND AND R ELATED W ORK 2. C AUSAL GRAPHICAL MODELS
Revised:  B ACKGROUND AND R ELATED W ORK 2. C AUSAL GRAPHICAL MODELS

Original: A causal graphical model (CGM) is deﬁned by a distribution P over a set of random variables
Revised: A causal graphical model (CGM) is deﬁned by a distribution P over a set of random variables

Original: X =
Revised: X =

Original: { X 1 , ..., X N } and a directed acyclic graph (DAG)
Revised: { X 1 , ..., X N } and a directed acyclic graph (DAG)

Original: G =
Revised: G =

Original: ( V, E ) .
Revised: ( V, E ) .

Original: Each node i ∈ V corresponds to the random variable X i and an edge ( i,
Revised: Each node i ∈ V corresponds to the random variable X i and an edge ( i,

Original: j ) ∈
Revised: j ) ∈

Original: E represents a direct causal relation from variable
Revised: E represents a direct causal relation from variable

Original: X i to X j : X i → X j .
Revised: X i to X j : X i → X j .

Original: A joint distribution P is faithful to a graph G if all and only the conditional independencies present in P are entailed by G (Pearl, 1988).
Revised: A joint distribution P is faithful to a graph G if all and only the conditional independencies present in P are entailed by G (Pearl, 1988).

Original: Vice versa, a distribution P is Markov to a graph G if the joint distribution can be factorized as p ( X ) =
Revised: Vice versa, a distribution P is Markov to a graph G if the joint distribution can be factorized as p ( X ) =

Original: (cid:81)
Revised: (cid:81)

Original: Ni =1 p
Revised: Ni =1 p

Original: i
Revised: i

Original: X i | pa ( X i )) where pa ( X i ) is the set of parents of the node i in G .
Revised: X i | pa ( X i )) where pa ( X i ) is the set of parents of the node i in G .

Original: An important concept which distinguishes CGMs from standard Bayesian Networks are interventions.
Revised: An important concept which distinguishes CGMs from standard Bayesian Networks are interventions.

Original: An intervention on a variable X i describes the local replacement of its conditional distribution p
Revised: An intervention on a variable X i describes the local replacement of its conditional distribution p

Original: i ( X i | pa ( X i )) by a new distribution ˜ p ( X i | pa ( X i )) .
Revised: i ( X i | pa ( X i )) by a new distribution ˜ p ( X i | pa ( X i )) .

Original: X i is thereby referred to as the intervention target.
Revised: X i is thereby referred to as the intervention target.

Original: An intervention is “perfect” when the new distribution is independent of the original parents, i.e. ˜ p
Revised: An intervention is “perfect” when the new distribution is independent of the original parents, i.e. ˜ p

Original: ( X i | pa ( X i )) =
Revised: ( X i | pa ( X i )) =

Original: ˜ p
Revised: ˜ p

Original: ( X i ) .
Revised: ( X i ) .

Original: 2. C AUSAL STRUCTURE LEARNING
Revised: 2. C AUSAL STRUCTURE LEARNING

Original: Discovering the graph G from samples of a joint distribution P is called causal structure learning or causal discovery , a fundamental problem in causality (Pearl, 2009; Peters et al., 2017).
Revised: Discovering the graph G from samples of a joint distribution P is called causal structure learning or causal discovery , a fundamental problem in causality (Pearl, 2009; Peters et al., 2017).

Original: While often performed from observational data, i.e. samples from P (see Glymour et al.
Revised: While often performed from observational data, i.e. samples from P (see Glymour et al.

Original: for an overview), we focus in this paper on algorithms that recover graphs from joint observational and interventional data.
Revised: for an overview), we focus in this paper on algorithms that recover graphs from joint observational and interventional data.

Original: Commonly, such methods are grouped into constraint-based and score-based approaches.
Revised: Commonly, such methods are grouped into constraint-based and score-based approaches.

Original: Constraint-based methods use conditional independence tests to identify causal relations (Hyttinen et al., 2014; Jaber et al., 2020; Kocaoglu et al., 2019; Monti et al., 2019; Spirtes et al., 2000; Sun et al., 2007).
Revised: Constraint-based methods use conditional independence tests to identify causal relations (Monti et al., 2019; Spirtes et al., 2000; Kocaoglu et al., 2019; Jaber et al., 2020; Sun et al., 2007; Hyttinen et al., 2014).

Original: For instance, the Invariant Causal Prediction (ICP) algorithm (Christina et al., 2018;
Revised: For instance, the Invariant Causal Prediction (ICP) algorithm (Peters et al., 2016;

Original: Peters et al., 2016) exploits that causal mechanisms remain unchanged under an intervention except the one intervened on (Pearl, 2009; Schölkopf et al., 2012).
Revised: Christina et al., 2018) exploits that causal mechanisms remain unchanged under an intervention except the one intervened on (Pearl, 2009; Schölkopf et al., 2012).

Original: We rely on a similar idea by testing for mechanisms that generalize from the observational to the interventional setting.
Revised: We rely on a similar idea by testing for mechanisms that generalize from the observational to the interventional setting.

Original: Another line of work is to extend methods working on observations only to interventions by incorporating those as additional constraints in the structure learning process (Jaber et al., 2020; Mooij et al., 2020).
Revised: Another line of work is to extend methods working on observations only to interventions by incorporating those as additional constraints in the structure learning process (Mooij et al., 2020; Jaber et al., 2020).

Original: Score-based methods , on the other hand, search through the space of all possible causal structures with the goal of optimizing a speciﬁed metric (Goudet et al., 2017; Ke et al., 2019; Tsamardinos et al., 2006; Zhu et al., 2020).
Revised: Score-based methods , on the other hand, search through the space of all possible causal structures with the goal of optimizing a speciﬁed metric (Tsamardinos et al., 2006; Ke et al., 2019; Goudet et al., 2017; Zhu et al., 2020).

Original: This metric, also referred to as score function, is usually a combination of how well the structure ﬁts the data, for instance in terms of log-likelihood, as well as regularizers for encouraging sparsity.
Revised: This metric, also referred to as score function, is usually a combination of how well the structure ﬁts the data, for instance in terms of log-likelihood, as well as regularizers for encouraging sparsity.

Original: Since the search space of DAGs is super-exponential in the number of nodes, many methods rely on a greedy search, yet returning graphs in the true equivalence class (Hauser & 
Revised: Since the search space of DAGs is super-exponential in the number of nodes, many methods rely on a greedy search, yet returning graphs in the true equivalence class (Meek, 1997; Hauser & Bühlmann, 2012;

Original: Bühlmann, 2012; Meek, 1997; Wang et al., 2017; Yang et al., 2018).
Revised: Wang et al., 2017; Yang et al., 2018).

Original: For instance, GIES (Hauser & Bühlmann, 2012) repeatedly adds, removes, and ﬂips the directions of edges in a proposal graphuntil
Revised: For instance, GIES (Hauser & Bühlmann, 2012) repeatedly adds, removes, and ﬂips the directions of edges in a proposal graph

Original: [Table]
Revised: [Table]

Original: ﬁtting

Original: Graphﬁtting
Revised: Graph

Original: [Table]
Revised: [Table]

Original: ﬁtting
Revised: @ @ ij L = ↵ · E X ,C  ij ⇥ L X i ! X j

Original: Graph ﬁtting
Revised: ( X j )  L X i 6! X j ( X j ) +  sparse

Original: [Table]
Revised: [Table]

Original: 
Revised: C (1) C (2) C (3) L X 1 ! X 2 ( X 2 ) L X 1 6! X 2 ( X 2 ) L

Original: [Table]
Revised: [Table]

Revised: X 2 ( X 2 )

Revised: L X 3 6!

Original:  L 
Revised: X 2 ( X 2 ) L X 1 ! X 3 ( X 3 ) L X 1 6! X

Original: [Table]
Revised: [Table]

Revised: X 3 )

Revised: L X 2 !

Revised: X

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: X 3 ) L X 2 6! X 3 ( X 3 ) @ @ 12 ˜ L =  0 (  12 ) ·  ( ✓ 12 ) ·

Original: [Table]
Revised: [ L X 1 !

Original: [Table]
Revised: X 2 ( X 2 )  L X 1 6! X 2 ( X 2 ) +  sparse ] @ @ 32 ˜ L =  0 (  32 ) ·  (

Original: [Table]
Revised: [Table]

Revised: [ L X 3 !

Original: [Table]
Revised: [Table]

Original: @
Revised: 2 ( X 2 )  L X

Original:  @23 ˜
Revised: [Table] X 2 ( X 2 ) +  sparse ] @

Original: = 
Revised: [Table] [Table] [Table]

Original: ·  ( ✓
Revised: [Table] [Table] [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table][Table]
Revised: [Table] [Table]

Original: [Table] [Table][Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Figure]
Revised: [Figure]

Original:  no higher-scoring graph 
Revised: Graph ﬁtting until no higher-scoring graph can be

Original: The Interventional Greedy SP (IGSP) algorithm (Wang et al., 2017)  independence 
Revised: The Interventional Greedy SP (IGSP) algorithm (Wang et al., 2017) is a hybrid method using conditional independence tests in its score function.

Original: Continuous-optimization methods are score-based methods that avoid the combinatorial greedy search over DAGs by using gradient-based methods (Brouillard et al., 2020; Ke et al., 2019; Lachapelle et al., 2020; Yu et al., 2019; Zheng et al.,2018; 2020; Zhu et al.,  2020).
Revised: Continuous-optimization methods are score-based methods that avoid the combinatorial greedy search over DAGs by using gradient-based methods (Zheng et al., 2018; Ke et al., 2019; Lachapelle et al., 2020; Yu et al., 2019; Zheng et al., 2020; Zhu et al., 2020; Brouillard et al., 2020).

Original: Thereby, the adjacency matrix is parameterized by weights that represent linear factors or probabilities of having an edge between a pair of nodes.
Revised: Thereby, the adjacency matrix is parameterized by weights that represent linear factors or probabilities of having an edge between a pair of nodes.

Original: The main challenge of such methods is how to limit the search space to acyclic graphs.
Revised: The main challenge of such methods is how to limit the search space to acyclic graphs.

Original: One common approach is to view the search as a constrained optimization problem and deploy an augmented Lagrangian procedure to solve it (Brouillard et al.,  2020; Yu et al., 2019; Zheng et al.,2018; 2020), including NOTEARS (Zheng et al., 2018) and DCDI (Brouillard et al., 2020).
Revised: One common approach is to view the search as a constrained optimization problem and deploy an augmented Lagrangian procedure to solve it (Zheng et al., 2018; 2020; Yu et al., 2019; Brouillard et al., 2020), including NOTEARS (Zheng et al., 2018) and DCDI (Brouillard et al., 2020).

Original: Alternatively, Ke et al.
Revised: Alternatively, Ke et al.

Original: (2019) propose to use a regularization term penalizing cyclic graphs while allowing unconstrained optimization.
Revised: (2019) propose to use a regularization term penalizing cyclic graphs while allowing unconstrained optimization.

Original: However, the regularizer must be designed and weighted such that the correct, acyclic causal graph is the global optimum of the score function.
Revised: However, the regularizer must be designed and weighted such that the correct, acyclic causal graph is the global optimum of the score function.

Original:  E FFICIENT N EURAL C AUSAL D ISCOVERY 3. S COPE AND A SSUMPTIONS
Revised:  E FFICIENT N EURAL C AUSAL D ISCOVERY 3. S COPE AND A SSUMPTIONS

Original: We consider the task of ﬁnding a directed acyclic graph G =
Revised: We consider the task of ﬁnding a directed acyclic graph G =

Original: ( V, E ) with N variables of an unknown
Revised: ( V, E ) with N variables of an unknown

Original: CGM given observational and interventional samples.
Revised: CGM given observational and interventional samples.

Original: Firstly, we assume that: (1) The CGM is causally sufﬁcient, i.e. , all common causes of variables are included and observable; (2) We have N interventional datasets, each sparsely intervening on a different variable; (3) The interventions are “perfect” and “stochastic”, meaning the intervention does not set the variable necessarily to a single value.
Revised: Firstly, we assume that: (1) The CGM is causally sufﬁcient, i.e. , all common causes of variables are included and observable; (2) We have N interventional datasets, each sparsely intervening on a different variable; (3) The interventions are “perfect” and “stochastic”, meaning the intervention does not set the variable necessarily to a single value.

Original: Thereby, we do not assume faithfulness, thuswe can also recover graphs with deterministic nodes.
Revised: Thereby, we do not strictly require faithfulness, thus also recovering some graphs violating faithfulness.

Original: We emphasize that we place no constraints on the domains of the variables (they can be discrete, continuous, or mixed) or the distributions of the interventions.
Revised: We emphasize that we place no constraints on the domains of the variables (they can be discrete, continuous, or mixed) or the distributions of the interventions.

Original: We discuss later how to extend the algorithm to infer causal mechanisms in graphs with latent confounding causal variables.
Revised: We discuss later how to extend the algorithm to infer causal mechanisms in graphs with latent confounding causal variables.

Original: Further, we discuss how to extend the algorithm to support interventions to subsets of variables only.
Revised: Further, we discuss how to extend the algorithm to support interventions to subsets of variables only.

Original: 3. O VERVIEW
Revised: 3. O VERVIEW

Original: ENCO learns a causal graph from observational and interventional data by modelling a probability for every possible directed edge between pairs of variables.
Revised: ENCO learns a causal graph from observational and interventional data by modelling a probability for every possible directed edge between pairs of variables.

Original: The goal is that the probabilities corresponding to the edges of the ground truth graph converge to one, while the probabilities of all other edges converge to zero.
Revised: The goal is that the probabilities corresponding to the edges of the ground truth graph converge to one, while the probabilities of all other edges converge to zero.

Original: For this to happen, we exploit the idea of independent causal mechanisms (Pearl, 2009; Peters et al., 2016), according to which the conditional distributions for all variables in the ground-truth CGM stay invariant under an intervention, except for the intervened ones.
Revised: For this to happen, we exploit the idea of independent causal mechanisms (Pearl, 2009; Peters et al., 2016), according to which the conditional distributions for all variables in the ground-truth CGM stay invariant under an intervention, except for the intervened ones.

Original: By contrast, for graphs modelling the same joint distribution but with a ﬂipped or additional edge, this does not hold (Peters et al., 2016).
Revised: By contrast, for graphs modelling the same joint distribution but with a ﬂipped or additional edge, this does not hold (Peters et al., 2016).

Original: In short, we search for the graph which generalizes best from observational to interventional data.
Revised: In short, we search for the graph which generalizes best from observational to interventional data.

Original: To implement the optimization, we alternate between two learning stages, that is distribution ﬁtting and graph ﬁtting, visually summarized in Figure 1.
Revised: To implement the optimization, we alternate between two learning stages, that is distribution ﬁtting and graph ﬁtting, visually summarized in Figure 1.

Original: Distribution ﬁtting trains a neural network f φ i per variable X i parameterized by φ i to model its observational, conditional data distribution p ( X i | ... ) .
Revised: Distribution ﬁtting trains a neural network f φ i per variable X i parameterized by φ i to model its observational, conditional data distribution p ( X i | ... ) .

Original: The input to the network are all other variables,
Revised: The input to the network are all other variables,

Original: X − i .
Revised: X − i .

Original: For simplicity, we want this neural network to model the conditional of the variable X i with respect to any possible set of parent variables.
Revised: For simplicity, we want this neural network to model the conditional of the variable X i with respect to any possible set of parent variables.

Original: We, therefore, apply a dropout-like scheme to the input to simulate different sets of parents, similar as (Brouillard et al., 2020; Ivanov et al., 2019; Ke et al., 2019; Li et al., 2020).
Revised: We, therefore, apply a dropout-like scheme to the input to simulate different sets of parents, similar as (Ke et al., 2019; Ivanov et al., 2019; Li et al., 2020; Brouillard et al., 2020).

Original: In that case, during training, we randomly set an input variable
Revised: In that case, during training, we randomly set an input variable

Original: X j to zero based on the probability of its corresponding edge X j
Revised: X j to zero based on the probability of its corresponding edge X j

Original: → X i , and minimize
Revised: → X i , and minimize

Original: [Equation]
Revised: [Equation]

Original: where M j ∼ Ber ( p ( X j → X i )) .
Revised: where M j ∼ Ber ( p ( X j → X i )) .

Original: For categorical random variables
Revised: For categorical random variables

Original: X i , we apply a softmax output activation for f φ i , and for continuous ones, we use Normalizing Flows (Rezende & Mohamed, 2015).
Revised: X i , we apply a softmax output activation for f φ i , and for continuous ones, we use Normalizing Flows (Rezende & Mohamed, 2015).

Original: Graph ﬁtting uses the learned networks to score and compare different graphs on interventional data.
Revised: Graph ﬁtting uses the learned networks to score and compare different graphs on interventional data.

Original: For parameterizing the edge probabilities, we use two sets of parameters: γ ∈ R N × N represents the existence of edges in a graph, and θ ∈ R N × N the orientation of the edges.
Revised: For parameterizing the edge probabilities, we use two sets of parameters: γ ∈ R N × N represents the existence of edges in a graph, and θ ∈ R N × N the orientation of the edges.

Original: The likelihood of an edge is determined by p ( X i → X j ) =
Revised: The likelihood of an edge is determined by p ( X i → X j ) =

Original: σ ( γ ij ) · σ ( θ ij ) , with σ ( ... ) being the sigmoid function and θ ij = − θ ji .
Revised: σ ( γ ij ) · σ ( θ ij ) , with σ ( ... ) being the sigmoid function and θ ij = − θ ji .

Original: The probability of the two orientations always sum to one.
Revised: The probability of the two orientations always sum to one.

Original: The beneﬁt of separating the edge probabilities into two independent parameters γ and θ is that it gives us more control over the gradient updates.
Revised: The beneﬁt of separating the edge probabilities into two independent parameters γ and θ is that it gives us more control over the gradient updates.

Original: The existence of an (undirected) edge can usually be already learned from observational or arbitrary interventional data alone, excluding deterministic variables (Pearl, 2009).
Revised: The existence of an (undirected) edge can usually be already learned from observational or arbitrary interventional data alone, excluding deterministic variables (Pearl, 2009).

Original: In contrast, the orientation can only be reliably detected from data for which an intervention is performed on its adjacent nodes, i.e. , X i or X j for learning θ ij .
Revised: In contrast, the orientation can only be reliably detected from data for which an intervention is performed on its adjacent nodes, i.e. , X i or X j for learning θ ij .

Original: While other interventions eventually provide information on the edge direction, e.g. , intervening on a node X k which is a child of X i and a parent of X j , we do not know the relation of X k to X i and X j at this stage, as we are in the process of learning the structure.
Revised: While other interventions eventually provide information on the edge direction, e.g. , intervening on a node X k which is a child of X i and a parent of X j , we do not know the relation of X k to X i and X j at this stage, as we are in the process of learning the structure.

Original: Despite having just one variable for the orientation, γ ij and γ ji are learned as two separate parameters.
Revised: Despite having just one variable for the orientation, γ ij and γ ji are learned as two separate parameters.

Original: One reason is that on interventional data, an edge can improve the log-likelihood estimate in one direction, but not necessarily the other, leading to conﬂicting gradients.
Revised: One reason is that on interventional data, an edge can improve the log-likelihood estimate in one direction, but not necessarily the other, leading to conﬂicting gradients.

Original: We optimize the graph parameters γ and θ by minimizing
Revised: We optimize the graph parameters γ and θ by minimizing

Original: [Equation]
Revised: [Equation]

Original: where p I ( I ) is the distribution over which variable to intervene on (usually uniform), and
Revised: where p I ( I ) is the distribution over which variable to intervene on (usually uniform), and

Original: ˜ p
Revised: ˜ p

Original: ˆ I ( X ) the joint distribution of all variables under the intervention ˆ I .
Revised: ˆ I ( X ) the joint distribution of all variables under the intervention ˆ I .

Original: In other words, these two distributions represent our interventional data distribution.
Revised: In other words, these two distributions represent our interventional data distribution.

Original: With p γ , θ ( C ) , we denote the distribution over adjacency matrices C under γ , θ , where C ij ∼ Ber ( σ ( γ ij ) σ ( θ ij )) .
Revised: With p γ , θ ( C ) , we denote the distribution over adjacency matrices C under γ , θ , where C ij ∼ Ber ( σ ( γ ij ) σ ( θ ij )) .

Original: L C ( X i ) is the negative log-likelihood estimate of variable X i conditioned on the parents according to C :
Revised: L C ( X i ) is the negative log-likelihood estimate of variable X i conditioned on the parents according to C :

Original: L C ( X i )
Revised: L C ( X i )

Original: = − log f
Revised: = − log f

Original: φ
Revised: φ

Original: i
Revised: i

Original: ( X i ; C ·
Revised: ( X i ; C ·

Original: [Equation]
Revised: [Equation]

Original: i (cid:12) X − i ) .
Revised: i (cid:12) X − i ) .

Original: The second term of Equation 2 is an (cid:96) 1 -regularizer on the edge probabilities.
Revised: The second term of Equation 2 is an (cid:96) 1 -regularizer on the edge probabilities.

Original: It acts as a prior, selecting the sparsest graph of those with similar likelihood estimates by removing redundant edges.
Revised: It acts as a prior, selecting the sparsest graph of those with similar likelihood estimates by removing redundant edges.

Original: Prediction.
Revised: Prediction.

Original: Alternating between the distribution and graph ﬁtting stages allows us to ﬁne-tune the neural networks to the most probable parent sets along the training.
Revised: Alternating between the distribution and graph ﬁtting stages allows us to ﬁne-tune the neural networks to the most probable parent sets along the training.

Original: After training, we obtain a graph prediction by selecting the edges for which σ ( γ ij ) and σ ( θ ij ) are greater than 0.5.
Revised: After training, we obtain a graph prediction by selecting the edges for which σ ( γ ij ) and σ ( θ ij ) are greater than 0.5.

Original: The orientation parameters prevent loops between any two variables, since σ ( θ ij ) can only be greater than 0.5 in one direction.
Revised: The orientation parameters prevent loops between any two variables, since σ ( θ ij ) can only be greater than 0.5 in one direction.

Original: Although the orientation parameters do not guarantee the absence of loops with more variable, we show that under certain conditions ENCO yet converges to the correct, acyclic graph.
Revised: Although the orientation parameters do not guarantee the absence of loops with more variable, we show that under certain conditions ENCO yet converges to the correct, acyclic graph.

Original: 3. L OW - VARIANCE GRADIENT ESTIMATORS FOR EDGE PARAMETERS
Revised: 3. L OW - VARIANCE GRADIENT ESTIMATORS FOR EDGE PARAMETERS

Original: To update γ and θ based on Equation 2, we need to determine their gradients through the expectation
Revised: To update γ and θ based on Equation 2, we need to determine their gradients through the expectation

Original: E p γ , θ ( C ) , where C is a discrete variable.
Revised: E p γ , θ ( C ) , where C is a discrete variable.

Original: For this, we apply REINFORCE (Williams, 1992).
Revised: For this, we apply REINFORCE (Williams, 1992).

Original: For clarity of exposition, we limit the discussion here to the ﬁnal results and provide the detailed derivations in Appendix A. For parameter γ ij , we obtain the following gradient:
Revised: For clarity of exposition, we limit the discussion here to the ﬁnal results and provide the detailed derivations in Appendix A. For parameter γ ij , we obtain the following gradient:

Original: [Equation]
Revised: [Equation]

Original: where E X ,C − ij summarizes for brevity the three expectations in Equation 2, excluding the edge
Revised: where E X ,C − ij summarizes for brevity the three expectations in Equation 2, excluding the edge

Original: X i → X j from C .
Revised: X i → X j from C .

Original: Further, L X i → X j ( X j ) denotes the negative log-likelihood for X j , if we include the edge X i → X j to the adjacency matrix C − ij , i.e. , C ij = 1 , and L X i (cid:54)→ X j ( X j ) if we set C ij = 0 .
Revised: Further, L X i → X j ( X j ) denotes the negative log-likelihood for X j , if we include the edge X i → X j to the adjacency matrix C − ij , i.e. , C ij = 1 , and L X i (cid:54)→ X j ( X j ) if we set C ij = 0 .

Original: The gradient in Equation 3 can be intuitively explained: if by the addition of the edge X i → X j , the log-likelihood estimate of X j is improved by more than λ sparse , we increase the corresponding edge parameter γ ij ; otherwise, we decrease it.
Revised: The gradient in Equation 3 can be intuitively explained: if by the addition of the edge X i → X j , the log-likelihood estimate of X j is improved by more than λ sparse , we increase the corresponding edge parameter γ ij ; otherwise, we decrease it.

Original: We derive the gradients for the orientation parameters θ similarly.
Revised: We derive the gradients for the orientation parameters θ similarly.

Original: As mentioned before, we only take the gradients for θ ij when we perform an intervention on either X i or X j .
Revised: As mentioned before, we only take the gradients for θ ij when we perform an intervention on either X i or X j .

Original: This leads us to:
Revised: This leads us to:

Original: [Equation]
Revised: [Equation]

Original: The probability of taking an intervention on X i is represented by p ( I X i ) (usually uniform across variables), and E I Xi , X ,C − ij the same expectation as before under the intervention on X i .
Revised: The probability of taking an intervention on X i is represented by p ( I X i ) (usually uniform across variables), and E I Xi , X ,C − ij the same expectation as before under the intervention on X i .

Original: When the oriented edge X i → X j improves the log-likelihood of X j under intervention on X i , then the ﬁrst part of the gradient increases θ ij .
Revised: When the oriented edge X i → X j improves the log-likelihood of X j under intervention on X i , then the ﬁrst part of the gradient increases θ ij .

Original: In contrast, when the true edge is X j → X i , the correlation between X i and X j learned from observational data would yield a worse likelihood estimate of X j on interventional data on X i than without the edge X j → X i .
Revised: In contrast, when the true edge is X j → X i , the correlation between X i and X j learned from observational data would yield a worse likelihood estimate of X j on interventional data on X i than without the edge X j → X i .

Original: This is because p ( X j | X i , ... ) does not stay invariant under intervening on X i .
Revised: This is because p ( X j | X i , ... ) does not stay invariant under intervening on X i .

Original: The same dynamic holds for interventions on X j .
Revised: The same dynamic holds for interventions on X j .

Original: Lastly, for independent nodes, the expectation of the gradient is zero.
Revised: Lastly, for independent nodes, the expectation of the gradient is zero.

Original: Based on Equations 3 and 4, we obtain a tractable, unbiased gradient estimator by using Monte-Carlo sampling.
Revised: Based on Equations 3 and 4, we obtain a tractable, unbiased gradient estimator by using Monte-Carlo sampling.

Original: Luckily, samples can be shared across variables, making training efﬁcient.
Revised: Luckily, samples can be shared across variables, making training efﬁcient.

Original: We ﬁrst sample an intervention, a corresponding data batch, and K graphs from p γ , θ ( C ) ( K usually between 20 and 100).
Revised: We ﬁrst sample an intervention, a corresponding data batch, and K graphs from p γ , θ ( C ) ( K usually between 20 and 100).

Original: We then evaluate the log likelihoods of all variables for these graphs on the batch, and estimate
Revised: We then evaluate the log likelihoods of all variables for these graphs on the batch, and estimate

Original: L X i → X j ( X j ) and L X i (cid:54)→ X j ( X j ) for all pairs of variables
Revised: L X i → X j ( X j ) and L X i (cid:54)→ X j ( X j ) for all pairs of variables

Original: X i and X j by simply averaging the results for the two cases separately.
Revised: X i and X j by simply averaging the results for the two cases separately.

Original: Finally, the estimates are used to determine the gradients for γ and θ .
Revised: Finally, the estimates are used to determine the gradients for γ and θ .

Original: Low variance.
Revised: Low variance.

Original: Previous methods (Bengio et al., 2020; Ke et al., 2019) relied on a different
Revised: Previous methods (Ke et al., 2019; Bengio et al., 2020) relied on a different

Original: REINFORCE-like estimator proposed by Bengio
Revised: REINFORCE-like estimator proposed by Bengio

Original: et al.
Revised: et al.

Original: Adjusting their estimator to our setting of the parameter γ ij , for instance, the gradient looks as follows:
Revised: Adjusting their estimator to our setting of the parameter γ ij , for instance, the gradient looks as follows:

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: [Equation] 
Revised: [Equation] [Figure] [Equation]

Original: [Figure]
Revised: [Figure]

Original: where g ij represents the gradient of γ ij .
Revised: where g ij represents the gradient of γ ij .

Original: Performing MonteCarlo sampling for estimating the gradient leads to a biased estimate which becomes asymptotically unbiased with increasing number of samples (Bengio et al., 2020).
Revised: Performing MonteCarlo sampling for estimating the gradient leads to a biased estimate which becomes asymptotically unbiased with increasing number of samples (Bengio et al., 2020).

Original: The division by the expectation of L C ( X j ) is done for variance reduction (Mahmood et al., 2014).
Revised: The division by the expectation of L C ( X j ) is done for variance reduction (Mahmood et al., 2014).

Original: Equation 5, however, is still sensitive to the proportion of sampled C ij being one or zero.
Revised: Equation 5, however, is still sensitive to the proportion of sampled C ij being one or zero.

Original: A major beneﬁt of our gradient formulation in
Revised: A major beneﬁt of our gradient formulation in

Original: Equation 3, instead, is that it removes this noise by considering the difference of the two independent Monte-Carlo estimates L X
Revised: Equation 3, instead, is that it removes this noise by considering the difference of the two independent Monte-Carlo estimates L X

Original: i → X j ( X j ) and L X i (cid:54)→ X j ( X j ) .
Revised: i → X j ( X j ) and L X i (cid:54)→ X j ( X j ) .

Original: Hence, we can use a smaller sample size than previous methods and attain 10 times lower standard deviation, as visualized in Figure 2.
Revised: Hence, we can use a smaller sample size than previous methods and attain 10 times lower standard deviation, as visualized in Figure 2.

Original: 3. C ONVERGENCE GUARANTEES
Revised: 3. C ONVERGENCE GUARANTEES

Original: Next, we discuss the conditions under which ENCO convergences to the correct causal graph.
Revised: Next, we discuss the conditions under which ENCO convergences to the correct causal graph.

Original: We show that not only does the global optimum of Equation 2 correspond to the true graph, but also that there exist no other local minima ENCO can converge to.
Revised: We show that not only does the global optimum of Equation 2 correspond to the true graph, but also that there exist no other local minima ENCO can converge to.

Original: We outline the derivation and proof of these conditions in Appendix B, and limit our discussion here to the main assumptions and implications.
Revised: We outline the derivation and proof of these conditions in Appendix B, and limit our discussion here to the main assumptions and implications.

Original: To construct a theoretical argument, we make the following assumptions.
Revised: To construct a theoretical argument, we make the following assumptions.

Original: First, we assume that sparse interventions have been performed on all variables.
Revised: First, we assume that sparse interventions have been performed on all variables.

Original: Later, we show how to extend the algorithm to avoid this strong assumption.
Revised: Later, we show how to extend the algorithm to avoid this strong assumption.

Original: Further, given a CGM, we assume that its joint distribution p ( X ) is
Revised: Further, given a CGM, we assume that its joint distribution p ( X ) is

Original: Markovian with respect to the true graph G .
Revised: Markovian with respect to the true graph G .

Original: In other words, the parent set pa ( X i ) reﬂects the inputs to the causal generation mechanism of X i .
Revised: In other words, the parent set pa ( X i ) reﬂects the inputs to the causal generation mechanism of X i .

Original: We assume that there exist no latent confounders in G .
Revised: We assume that there exist no latent confounders in G .

Original: Also, we assume the neural networks in ENCO are sufﬁciently large and sufﬁcient observational data is provided to model the conditional distributions of the CGM up to an arbitrary small error.
Revised: Also, we assume the neural networks in ENCO are sufﬁciently large and sufﬁcient observational data is provided to model the conditional distributions of the CGM up to an arbitrary small error.

Original: Under these assumptions, we produce the following conditions for convergence:
Revised: Under these assumptions, we produce the following conditions for convergence:

Original: Theorem 3.1.
Revised: Theorem 3.1.

Original: Given a causal graph G with variables X 1 , ..., X N and conditional observational distributions p ( X i | ... ) , the proposed method ENCO will converge to the true, causal graph G , if the following conditions hold for all edges X i → X j in G :
Revised: Given a causal graph G with variables X 1 , ..., X N and conditional observational distributions p ( X i | ... ) , the proposed method ENCO will converge to the true, causal graph G , if the following conditions hold for all edges X i → X j in G :

Original: For all possible sets of parents of X j excluding X i , by adding X i the log-likelihood estimate of X j is improved or unchanged under the intervention on X i :
Revised: For all possible sets of parents of X j excluding X i , by adding X i the log-likelihood estimate of X j is improved or unchanged under the intervention on X i :

Original: [Equation]
Revised: [Equation]

Original: For at least one set of nodes
Revised: For at least one set of nodes

Original: [Equation]
Revised: [Equation]

Original: pa ( X j ) , for which the probability to be sampled as parents of X j is greater than 0, Equation 6 must be strictly greater than zero.
Revised: pa ( X j ) , for which the probability to be sampled as parents of X j is greater than 0, Equation 6 must be strictly greater than zero.

Original: The effect of X i on X j cannot be described by other variables up to λ sparse :
Revised: The effect of X i on X j cannot be described by other variables up to λ sparse :

Original: [Equation]
Revised: [Equation]

Original: where gpa i ( X j ) is the set of nodes excluding X i which , according to the ground truth graph, 
Revised: where gpa i ( X j ) is the set of nodes excluding X i which, according to the ground truth graph, could have an edge to X j without introducing a cycle, and p I − j

Original: could have an edge to X j without introducing a
Revised: ( I )

Original: cycle and
Revised: refers

Original: p I − j ( I ) refers to the distribution over interventions p
Revised: to the distribution over interventions p

Original: I  I  excluding the intervention on variable X j 
Revised: I ( I ) excluding the intervention on variable X j .

Original: Further, for all other pairs X i , X j for which X j is a descendant of X i , condition 1 and 2 must hold.
Revised: Further, for all other pairs X i , X j for which X j is a descendant of X i , condition 1 and 2 must hold.

Original: Condition 1 and 2 ensure that the orientations can be learned from interventions.
Revised: Condition 1 and 2 ensure that the orientations can be learned from interventions.

Original: Intuitively, ancestors and descendants in the graph have to be dependent when intervening on the ancestors.
Revised: Intuitively, ancestors and descendants in the graph have to be dependent when intervening on the ancestors.

Original: This aligns with the technical interpretation in Theorem 3.1 that the likelihood estimate of the child variable must improve when intervening and conditioning on its ancestor variables.
Revised: This aligns with the technical interpretation in Theorem 3.1 that the likelihood estimate of the child variable must improve when intervening and conditioning on its ancestor variables.

Original: Condition 3 states intuitively that the sparsity regularizer needs to be selected such that it chooses the sparsest graph among those graphs with equal joint distributions as the ground truth graph, without trading sparsity for worse distribution estimates.
Revised: Condition 3 states intuitively that the sparsity regularizer needs to be selected such that it chooses the sparsest graph among those graphs with equal joint distributions as the ground truth graph, without trading sparsity for worse distribution estimates.

Original: The speciﬁc condition in Theorem 3.1 ensures thereby that the set can be learned with a gradient-based algorithm.
Revised: The speciﬁc condition in Theorem 3.1 ensures thereby that the set can be learned with a gradient-based algorithm.

Original: We emphasize that this condition only gives an upper bound for λ sparse when sufﬁciently large datasets are available.
Revised: We emphasize that this condition only gives an upper bound for λ sparse when sufﬁciently large datasets are available.

Original: In practice, the graph can thus be recovered with a sufﬁciently small sparsity regularizer and dependencies among variables under interventions.
Revised: In practice, the graph can thus be recovered with a sufﬁciently small sparsity regularizer and dependencies among variables under interventions.

Original: We provide more details for various settings  in Appendix B.
Revised: We provide more details for various settings and further intuition in Appendix B.

Original: Interventions on fewer variables.
Revised: Interventions on fewer variables.

Original: It is straightforward to extend ENCO to support interventions on fewer variables.
Revised: It is straightforward to extend ENCO to support interventions on fewer variables.

Original: Normally, in the graph ﬁtting stage, we sample one intervention at a time.
Revised: Normally, in the graph ﬁtting stage, we sample one intervention at a time.

Original: We can, thus, simply restrict the sampling only to the interventions that are possible (or provided in the dataset).
Revised: We can, thus, simply restrict the sampling only to the interventions that are possible (or provided in the dataset).

Original: In this case, we update the orientation parameters θ ij of only those edges that connect to an intervened variable, either X i or X j , as before.
Revised: In this case, we update the orientation parameters θ ij of only those edges that connect to an intervened variable, either X i or X j , as before.

Original: For all other orientation parameters, we extend the gradient estimator to include interventions on all variables.
Revised: For all other orientation parameters, we extend the gradient estimator to include interventions on all variables.

Original: Although this estimate is more noisy and does not have convergence guarantees, it can still be informative about the edge orientations.
Revised: Although this estimate is more noisy and does not have convergence guarantees, it can still be informative about the edge orientations.

Original: Enforcing acyclicity When the conditions are violated, e.g. by limited data, cycles can occur in the prediction.
Revised: Enforcing acyclicity When the conditions are violated, e.g. by limited data, cycles can occur in the prediction.

Original: Since ENCO learns the orientations as a separate parameter, we can remove cycles by ﬁnding the global order of variables O ∈ S N , with S N being the set of permutations, that maximizes the pairwise orientation probabilities: arg[Equation] O
Revised: Since ENCO learns the orientations as a separate parameter, we can remove cycles by ﬁnding the global order of variables O ∈ S N , with S N being the set of permutations, that maximizes the pairwise orientation probabilities: arg

Original: [Equation]
Revised: [Equation]

Original: Nj=
Revised: Nj

Original: [Equation]
Revised: [Equation]

Original: σ
Revised: θ

Original: [Equation]
Revised: [Equation]

Original: θ O i ,O j ) .
Revised: i ,O j ) .

Original: This utilizes the learned ancestor-descendant relations, making the algorithm more robust to noise in single interventions.
Revised: This utilizes the learned ancestor-descendant relations, making the algorithm more robust to noise in single interventions.

Original: 3. H ANDLING LATENT CONFOUNDERS
Revised: 3. H ANDLING LATENT CONFOUNDERS

Original: So far, we have assumed that all variables of the graph are observable and can be intervened on.
Revised: So far, we have assumed that all variables of the graph are observable and can be intervened on.

Original: A common issue in causal discovery is the existence of latent confounders, i.e. , an unobserved common cause of two or more variables introducing dependencies between each other.
Revised: A common issue in causal discovery is the existence of latent confounders, i.e. , an unobserved common cause of two or more variables introducing dependencies between each other.

Original: In the presence of latent confounders, structure learning methods may predict false positive edges.
Revised: In the presence of latent confounders, structure learning methods may predict false positive edges.

Original: Interestingly, in the context of ENCO latent confounders for two variables X i , X j cause a unique pattern of learned parameters.
Revised: Interestingly, in the context of ENCO latent confounders for two variables X i , X j cause a unique pattern of learned parameters.

Original: When intervening on X i or X j , having an edge between the two variables is disadvantageous, as in the intervened graph X i and X j are (conditionally) independent.
Revised: When intervening on X i or X j , having an edge between the two variables is disadvantageous, as in the intervened graph X i and X j are (conditionally) independent.

Original: For interventions on all other variables, however, an edge can be beneﬁcial as X i and X j are correlated.
Revised: For interventions on all other variables, however, an edge can be beneﬁcial as X i and X j are correlated.

Original: Exploiting this, we extend ENCO to detect latent confounders.
Revised: Exploiting this, we extend ENCO to detect latent confounders.

Original: We focus on latent confounders between two variables that do not have any direct edges with each other, and assume that the confounder is not a child of any other observed variable.
Revised: We focus on latent confounders between two variables that do not have any direct edges with each other, and assume that the confounder is not a child of any other observed variable.

Original: For all other edges besides between X i
Revised: For all other edges besides between X i

Original: [Table]
Revised: [Table]

Original:  and X j , we can still rely on the guarantees in Section 3.4 since Equation 7 already includes the possibility of additional edges in such cases.
Revised: ± 1 . 6 ) 5 . 3 ( ± 2 . 3 ) 0 . 6 ( ± 1 . 1 ) 0 . 2 ( ± 0 . 5 ) and X j , we can still rely on the guarantees in Section 3.4 since Equation 7 already includes the possibility of additional edges in such cases.

Original: After convergence, we score every pair of variables on how likely they share a latent confounder using a function lc ( · ) that is maximized in the scenario mentioned above.
Revised: After convergence, we score every pair of variables on how likely they share a latent confounder using a function lc ( · ) that is maximized in the scenario mentioned above.

Original: For this, we deﬁne γ ij = γ ( I ) ij + γ ( O ) ij where γ ( I ) ij is only updated with gradients from Equation 3 under interventions on X i , and γ ( O ) ij on all others.
Revised: For this, we deﬁne γ ij = γ ( I ) ij + γ ( O ) ij where γ ( I ) ij is only updated with gradients from Equation 3 under interventions on X i , and γ ( O ) ij on all others.

Original: With this separation, we deﬁne the following score function which is maximized by latent confounders:
Revised: With this separation, we deﬁne the following score function which is maximized by latent confounders:

Original: [Equation]
Revised: [Equation]

Original: To converge to the mentioned values, especially of γ ( O ) ij , we need a similar condition as in Equation 7: the improvement on the log-likelihood estimate gained by the edge X i → X j and conditioned on all other parents of X j needs to be larger than λ sparse on interventional data excluding X i and X j .
Revised: To converge to the mentioned values, especially of γ ( O ) ij , we need a similar condition as in Equation 7: the improvement on the log-likelihood estimate gained by the edge X i → X j and conditioned on all other parents of X j needs to be larger than λ sparse on interventional data excluding X i and X j .

Original: If this is not the case, the sparsity regularizer will instead remove the edge between X i and X j preventing any false predictions among observed variables.
Revised: If this is not the case, the sparsity regularizer will instead remove the edge between X i and X j preventing any false predictions among observed variables.

Original: For all other pairs of variables, at least one of the terms in Equation 8 converges to zero.
Revised: For all other pairs of variables, at least one of the terms in Equation 8 converges to zero.

Original: Thus, we can detect latent confounders by checking whether the score function lc
Revised: Thus, we can detect latent confounders by checking whether the score function lc

Original: ( X i , X j ) is greater than a threshold hyperparameter τ ∈ (0 . 0 , 1 . 0) .
Revised: ( X i , X j ) is greater than a threshold hyperparameter τ ∈ (0 . 0 , 1 . 0) .

Original: We discuss possible guarantees in Appendix B, and experimentally verify this approach in Section 4.5.
Revised: We discuss possible guarantees in Appendix B, and experimentally verify this approach in Section 4.5.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We evaluate ENCO on structure learning on synthetic datasets for systematic comparisons and realworld datasets for benchmarking against other methods in the literature.
Revised: We evaluate ENCO on structure learning on synthetic datasets for systematic comparisons and realworld datasets for benchmarking against other methods in the literature.

Original: The experiments focus on graphs with categorical variables, and experiments on continuous data are included in Appendix D.5.
Revised: The experiments focus on graphs with categorical variables, and experiments on continuous data are included in Appendix D.5.

Revised: Our code is publicly available at https://github.com/phlippe/ENCO .

Original: 4.  E XPERIMENTAL SETUP
Revised:  E XPERIMENTAL SETUP

Original: Graphs and datasets.
Revised: Graphs and datasets.

Original: Given a ground-truth causal graphical model, all methods are tasked to recover the original DAG from a set of observational and interventional data points for each variable.
Revised: Given a ground-truth causal graphical model, all methods are tasked to recover the original DAG from a set of observational and interventional data points for each variable.

Original: In case of synthetic graphs, we follow the setup of Ke et al.
Revised: In case of synthetic graphs, we follow the setup of Ke et al.

Original: and create the conditional distributions from neural networks.
Revised: and create the conditional distributions from neural networks.

Original: These networks take as input the categorical values of its variable’s parents, and are initialized orthogonally to output a non-trivial distribution.
Revised: These networks take as input the categorical values of its variable’s parents, and are initialized orthogonally to output a non-trivial distribution.

Original: Baselines.
Revised: Baselines.

Original: We compare ENCO to GIES (Hauser & Bühlmann, 2012) and IGSP (Wang et al., 2017;
Revised: We compare ENCO to GIES (Hauser & Bühlmann, 2012) and IGSP (Wang et al., 2017;

Original: Yang et al., 2018) as greedy score-based approaches, and DCDI (Brouillard et al., 2020) and SDI (Ke et al., 2019) as continuous optimization methods.
Revised: Yang et al., 2018) as greedy score-based approaches, and DCDI (Brouillard et al., 2020) and SDI (Ke et al., 2019) as continuous optimization methods.

Original: Pure constraint-based methods do not scale well to thegiven graph and dataset sizes (Guo et al., 2020; Peters et al., 2017). We do not compare to methods with observational dataonly, since those can recover the graph up to its Markov equivalence class only.
Revised: Further, as a common observational baseline, we apply GES (Chickering, 2002) on the observational data to obtain a graph skeleton, and orient each edge by learning the skeleton on the corresponding interventional distribution.

Original: We perform a separate hyperparameter search for all baselines, and use the same neural network setup for SDI, DCDI, andENCO.
Revised: We perform a separate hyperparameter search for all baselines, and use the same neural network setup for SDI, DCDI, and

Original: All methods were executed on the same hardware, namely a 24-core CPU with a single NVIDIA RTX3090 GPU.
Revised: ENCO.

Original: Appendix C provides a detailed overview of the hyperparameters for all experiments and methods.
Revised: Appendix C provides a detailed overview of the hyperparameters for all experiments.

Original: 4. C AUSAL STRUCTURE LEARNING ON COMMON GRAPH STRUCTURES
Revised: 4. C AUSAL STRUCTURE LEARNING ON COMMON GRAPH STRUCTURES

Original: We ﬁrst experiment on synthetic graphs.
Revised: We ﬁrst experiment on synthetic graphs.

Original: We pick six common graph structures and sample 5k observational data points and 200 per intervention.
Revised: We pick six common graph structures and sample 5, observational data points and 200 per intervention.

Original: The graphs chain and full represent the
Revised: The graphs chain and full represent the

Original: [Figure]
Revised: [Figure]

Original: minimally- and maximally- bidiag connections,is
Revised: bidiag connections,

Original: In
Revised: In

Original: , one hasas
Revised: , one has

Original: Finally, random has a randomly sampled graph structure with a likelihood of 0 .
Revised: Finally, random has a randomly sampled graph structure with a likelihood of 0 .

Original: 3 of two nodes being connected by a direct edge.
Revised: 3 of two nodes being connected by a direct edge.

Original: For each graph structure, we generate 25 graphs with 25 nodes each, on which we report the average performance and standard deviation.
Revised: For each graph structure, we generate 25 graphs with 25 nodes each, on which we report the average performance and standard deviation.

Original: Following common practice, we use structural hamming distance (SHD) as evaluation metric.
Revised: Following common practice, we use structural hamming distance (SHD) as evaluation metric.

Original: SHD counts the number of edges that need to be removed, added, or ﬂipped in order to obtain the ground truth graph.
Revised: SHD counts the number of edges that need to be removed, added, or ﬂipped in order to obtain the ground truth graph.

Original: Table 1 shows that the continuous optimization methods outperform the greedy search approaches on categorical variables.
Revised: Table 1 shows that the continuous optimization methods outperform the greedy search approaches on categorical variables.

Original: SDI works reasonably well on sparse graphs, but struggles with nodes that have many parents.
Revised: SDI works reasonably well on sparse graphs, but struggles with nodes that have many parents.

Original: DCDI can recover the collider and full graph to a better degree, yet degrades for sparse graphs.
Revised: DCDI can recover the collider and full graph to a better degree, yet degrades for sparse graphs.

Original: ENCO performs well on all graph structures, outperforming all baselines.
Revised: ENCO performs well on all graph structures, outperforming all baselines.

Original: For sparse graphs, cycles can occur due to limited sample size.
Revised: For sparse graphs, cycles can occur due to limited sample size.

Original: However, with enforcing acyclicity, ENCO-acyclic is able to recover four out of six graphs with less than one error on average.
Revised: However, with enforcing acyclicity, ENCO-acyclic is able to recover four out of six graphs with less than one error on average.

Original: We further include experiments with various sample sizes in Appendix D.1.
Revised: We further include experiments with various sample sizes in Appendix D.1.

Original: While other methods do not reliably recover the causal graph even for large sample sizes, ENCO attains low errors even with smaller sample sizes.
Revised: While other methods do not reliably recover the causal graph even for large sample sizes, ENCO attains low errors even with smaller sample sizes.

Original: 4. S CALABILITY
Revised: 4. S CALABILITY

Original: [Figure]
Revised: [Figure]

Original: Next, we test ENCO on graphs with large sets of variables.
Revised: Next, we test ENCO on graphs with large sets of variables.

Original: We create random graphs ranging from N = 100 to N = 1 , 000 nodes with larger sample sizes.
Revised: We create random graphs ranging from N = 100 to N = 1 , 000 nodes with larger sample sizes.

Original: Every node has on average 8 edges and a maximum of 10 parents.
Revised: Every node has on average 8 edges and a maximum of 10 parents.

Original: The challenge of large graphs is that the number of possible edges grows quadratically and the number of DAGs super-exponentially, requiring efﬁcient methods.
Revised: The challenge of large graphs is that the number of possible edges grows quadratically and the number of DAGs super-exponentially, requiring efﬁcient methods.

Original: We compare ENCO to the two best performing baselines from Table 1, SDI and DCDI.
Revised: We compare ENCO to the two best performing baselines from Table 1, SDI and DCDI.

Original: All methods were given the same setup of neural networks and a maximum runtime which corresponds toepochs for ENCO.
Revised: All methods were given the same setup of neural networks and a maximum runtime which corresponds toepochs for ENCO.

Original: We plot the SHD over graph size and runtime in Figure 3.
Revised: We plot the SHD over graph size and runtime in Figure 3.

Original: ENCO recovers the causal graphs perfectly with no errors except for the 1 , 000 -node graph, for which it misses only one out of 1 million edges in 2 out of 10 experiments.
Revised: ENCO recovers the causal graphs perfectly with no errors except for the 1 , 000 -node graph, for which it misses only one out of 1 million edges in 2 out of 10 experiments.

Original: SDI and DCDI achieve considerably worse performance.
Revised: SDI and DCDI achieve considerably worse performance.

Original: This shows that ENCO can efﬁciently be applied to 1 , 000 variables while maintaining its convergence guarantees, underlining the beneﬁt of its low-variance gradient estimators.
Revised: This shows that ENCO can efﬁciently be applied to 1 , 000 variables while maintaining its convergence guarantees, underlining the beneﬁt of its low-variance gradient estimators.

Original: 4. I NTERVENTIONS ON FEWER VARIABLES
Revised: 4. I NTERVENTIONS ON FEWER VARIABLES

Original: We perform experiments on the same datasets as in Section 4.2, but provide interventional data only for a randomly sampled subset of the 25 variables of each graph.
Revised: We perform experiments on the same datasets as in Section 4.2, but provide interventional data only for a randomly sampled subset of the 25 variables of each graph.

Original: We compare ENCO to DCDI, which supports partial intervention sets, and plot the SHD over the number of intervened variables in
Revised: We compare ENCO to DCDI, which supports partial intervention sets, and plot the SHD over the number of intervened variables in

Original: Figure 4.
Revised: Figure 4.

Original: Despite ENCO’s guarantees only holding for full interventions, it is still competitive and outperforms DCDI in most settings.
Revised: Despite ENCO’s guarantees only holding for full interventions, it is still competitive and outperforms DCDI in most settings.

Original: Importantly, enforcing acyclicity has an even greater impact on fewer interventions as more orientations are trainedon non-adjacent interventions (see Appendix B. for detailed discussion).
Revised: Importantly, enforcing acyclicity has an even greater impact on fewer interventions as more orientations are trained

Original: We conclude that ENCO works competitively with partial interventions too.

Original: [Table]
Revised: [Table]

Original:  D ETECTING LATENT CONFOUNDERS
Revised: 4. D ETECTING LATENT CONFOUNDERS

Original: [Table]
Revised: [Table]

Original: To test the detection of latent confounders
Revised: To test the detection of latent confounders

Original: [Table]
Revised: [Table]

Original: we create a set of 25 random graphs with 5 additional latent confounders.
Revised: we create a set of 25 random graphs with 5 additional latent confounders.

Original: The dataset is generated in the same way as before, except that we remove the latent variable from the input data and increase the observational and interventional sample size (see Appendix C.3 for ablation studies).
Revised: The dataset is generated in the same way as before [Table] except that we remove the latent variable from the input data and increase the observational and interventional sample size (see Appendix C.3 for ablation studies).

Original: After training, we predict the existence of a latent confounder on any pair of variables
Revised: After training, we predict the existence of a latent confounder on any pair of variables

Original: X i and X j if lc
Revised: X i and X j if lc

Original: ( X i , X j ) is greater than τ .
Revised: ( X i , X j ) is greater than τ .

Original: We choose τ = 0 .
Revised: We choose τ = 0 .

Original: 4 but verify in Appendix C.3 that the method is not sensitive to the speciﬁc value of τ .
Revised: 4 but verify in Appendix C.3 that the method is not sensitive to the speciﬁc value of τ .

Original: As shown in Table 2,
Revised: As shown in Table 2,

Original: ENCO detects more than 95% of the latent confounders without any false positives.
Revised: ENCO detects more than 95% of the latent confounders without any false positives.

Original: What is more, the few mistakes do not affect the detection of all other edges, which are recovered perfectly.
Revised: What is more, the few mistakes do not affect the detection of all other edges, which are recovered perfectly.

Original: 4. R EAL - WORLD INSPIRED DATA
Revised: 4. R EAL - WORLD INSPIRED DATA

Original: Finally, we evaluate ENCO on causal graphs from the Bayesian Network Repository (BnLearn) (Scutari, 2010).
Revised: Finally, we evaluate ENCO on causal graphs from the Bayesian Network Repository (BnLearn) (Scutari, 2010).

Original: The repository contains graphs inspired by real-world applications that are used as benchmarks in literature.
Revised: The repository contains graphs inspired by real-world applications that are used as benchmarks in literature.

Original: In comparison to the synthetic graphs, the real-world graphs are sparser with a maximum of 6 parents per node and contain nodes with strongly peaked marginal distributions.
Revised: In comparison to the synthetic graphs, the real-world graphs are sparser with a maximum of 6 parents per node and contain nodes with strongly peaked marginal distributions.

Original: They also include deterministic variables, making the task challenging even for small graphs.
Revised: They also include deterministic variables, making the task challenging even for small graphs.

Original: We evaluate ENCO, SDI, and DCDI on 7 graphs with increasing sizes, see Table 3.
Revised: We evaluate ENCO, SDI, and DCDI on 7 graphs with increasing sizes, see Table 3.

Original: We observe that ENCO recovers almost all real-world causal graphs without errors, independent of their size.
Revised: We observe that ENCO recovers almost all real-world causal graphs without errors, independent of their size.

Original: In contrast, SDI suffers from more mistakes as the graphs become larger.
Revised: In contrast, SDI suffers from more mistakes as the graphs become larger.

Original: An exception is pigs (Scutari, 2010), which has a maximum of 2 parents per node, and hence is easier to learn.
Revised: An exception is pigs (Scutari, 2010), which has a maximum of 2 parents per node, and hence is easier to learn.

Original: The most challenging graph is diabetes (Andreassen et al., 1991) due to its large size and many deterministic variables.
Revised: The most challenging graph is diabetes (Andreassen et al., 1991) due to its large size and many deterministic variables.

Original: ENCO makes only two mistakes, showing that it can handle deterministic variables well.
Revised: ENCO makes only two mistakes, showing that it can handle deterministic variables well.

Original: We discuss results on small sample sizes in Appendix C.5, observing similar trends.
Revised: We discuss results on small sample sizes in Appendix C.5, observing similar trends.

Original: We conclude that ENCO can reliably perform structure learning on a wide variety of settings, including deterministic variables.
Revised: We conclude that ENCO can reliably perform structure learning on a wide variety of settings, including deterministic variables.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We propose ENCO, an efﬁcient causal structure learning method leveraging observational and interventional data.
Revised: We propose ENCO, an efﬁcient causal structure learning method leveraging observational and interventional data.

Original: Compared to previous work, ENCO models the edge orientations as separate parameters and uses an objective unconstrained with respect to acyclicity.
Revised: Compared to previous work, ENCO models the edge orientations as separate parameters and uses an objective unconstrained with respect to acyclicity.

Original: This allows for easier optimization and low-variance gradient estimators while having convergence guarantees.
Revised: This allows for easier optimization and low-variance gradient estimators while having convergence guarantees.

Original: As a consequence, the algorithm can efﬁciently scale to graphs that are at least one order of magnitude larger graphs than what was possible.
Revised: As a consequence, the algorithm can efﬁciently scale to graphs that are at least one order of magnitude larger graphs than what was possible.

Original: Experiments corroborate the capabilities of ENCO compared to the state-of-the-art on an extensive array of settings on graph sizes, sizes of observational and interventional data, latent confounding, as well as on both partial and full intervention sets.
Revised: Experiments corroborate the capabilities of ENCO compared to the state-of-the-art on an extensive array of settings on graph sizes, sizes of observational and interventional data, latent confounding, as well as on both partial and full intervention sets.

Original: Limitations.
Revised: Limitations.

Original: The convergence guarantees of ENCO require interventions on all variables, although experiments on fewer interventions have shown promising results.
Revised: The convergence guarantees of ENCO require interventions on all variables, although experiments on fewer interventions have shown promising results.

Original: Future work includes investigating guarantee extensions of ENCO to this setting.
Revised: Future work includes investigating guarantee extensions of ENCO to this setting.

Original: A second limitation is that the orientations are missing transitivity: if X 1 (cid:31) X 2 and X 2 (cid:31) X 3 , then X 1 (cid:31) X 3 must hold.
Revised: A second limitation is that the orientations are missing transitivity: if X 1 (cid:31) X 2 and X 2 (cid:31) X 3 , then X 1 (cid:31) X 3 must hold.

Original: A potential direction is incorporating transitive relations for improving convergence speed and results on fewer interventions.
Revised: A potential direction is incorporating transitive relations for improving convergence speed and results on fewer interventions.

Original: E THICS S TATEMENT
Revised: E THICS S TATEMENT

Original: Causal structure learning algorithms such as the proposed method are mainly used to uncover and understand causal mechanisms from data.
Revised: Causal structure learning algorithms such as the proposed method are mainly used to uncover and understand causal mechanisms from data.

Original: The knowledge of the underlying causal mechanisms can then be applied to decide on speciﬁc actions that inﬂuence variables or factors in a desired way.
Revised: The knowledge of the underlying causal mechanisms can then be applied to decide on speciﬁc actions that inﬂuence variables or factors in a desired way.

Original: For instance, by knowing that the environmental pollution in a city has an impact on the risk of cancer of its residents, one can try to reduce the pollution to decrease the risk of cancer.
Revised: For instance, by knowing that the environmental pollution in a city has an impact on the risk of cancer of its residents, one can try to reduce the pollution to decrease the risk of cancer.

Original: The applications of causal structure learning are ranging across many scientiﬁc disciplines, including computational biology (Friedman et al., 2000;Opgen-Rhein & Strimmer, 2007; Sachs et al., 2005), epidemiology (Robins et al., 2000; Vandenbroucke et al., 2016), and economics (Hicks et al., 1980; Pearl, 2009).
Revised: The applications of causal structure learning are ranging across many scientiﬁc disciplines, including computational biology (Friedman et al., 2000; Sachs et al., 2005; Opgen-Rhein & Strimmer, 2007), epidemiology (Robins et al., 2000; Vandenbroucke et al., 2016), and economics (Pearl, 2009; Hicks et al., 1980).

Original: We envision that our work can have positive impacts on those ﬁelds.
Revised: We envision that our work can have positive impacts on those ﬁelds.

Original: One example we want to highlight is the ﬁeld of genomics.
Revised: One example we want to highlight is the ﬁeld of genomics.

Original: Recent advances have enabled to perform gene knockdown experiments in a large scale, providing large amounts of interventional data (Dixit et al., 2016; Macosko et al., 2015).
Revised: Recent advances have enabled to perform gene knockdown experiments in a large scale, providing large amounts of interventional data (Dixit et al., 2016; Macosko et al., 2015).

Original: Gaining insights into how speciﬁc genes and diseases interact can lead to the development of novel pharmaceutic methods for treating current diseases.
Revised: Gaining insights into how speciﬁc genes and diseases interact can lead to the development of novel pharmaceutic methods for treating current diseases.

Original: Since the number of variables in those experiments is tremendous, efﬁcient causal structure learning algorithms are needed.
Revised: Since the number of variables in those experiments is tremendous, efﬁcient causal structure learning algorithms are needed.

Original: The proposed method constitutes a ﬁrst step towards this goal, and our work can foster future work for creating algorithms scaling beyond 10 , 000 variables.
Revised: The proposed method constitutes a ﬁrst step towards this goal, and our work can foster future work for creating algorithms scaling beyond 10 , 000 variables.

Original: Since the possible applications are fairly wide-ranging, there might be potential impacts we cannot forecast at the current time.
Revised: Since the possible applications are fairly wide-ranging, there might be potential impacts we cannot forecast at the current time.

Original: This includes misuses of the method for unethical purposes.
Revised: This includes misuses of the method for unethical purposes.

Original: For instance, the method can be used to justify gender and race as causes for irrelevant variables if the output is misinterpreted, initial assumptions of the model are ignored, or the input data has been manipulated.
Revised: For instance, the method can be used to justify gender and race as causes for irrelevant variables if the output is misinterpreted, initial assumptions of the model are ignored, or the input data has been manipulated.

Original: Hence, the obligation to use this method in a correct way within ethical boundaries lies on the user.
Revised: Hence, the obligation to use this method in a correct way within ethical boundaries lies on the user.

Original: We emphasize this responsibility of the user in the license of our code.
Revised: We emphasize this responsibility of the user in the license of our code.

Original: R EPRODUCIBILITY S TATEMENT
Revised: R EPRODUCIBILITY S TATEMENT

Original: To ensure reproducibility, we have published theanonymized source code of the proposed method ENCO at https://github.com/anony192847385/ICLR2022_Paper657 .
Revised: To ensure reproducibility, we have published the source code of the proposed method ENCO at https://github.com/phlippe/ENCO .

Original: The code includes instructions on how to download the datasets, and reproduce the experiments in Sectionand additional experiments in Appendix D. Further, for all experiments of Section 4, we have included a detailed overview in Appendix C of (a) the used data and its generation process, (b) all hyperparameters used for all methods, and (c) additional details on the results.
Revised: The code includes instructions on how to download the datasets, and reproduce the experiments in Section 4 and additional experiments in Appendix D. Further, for all experiments of Section 4, we have included a detailed overview in Appendix C of (a) the used data and its generation process, (b) all hyperparameters used for all methods, and (c) additional details on the results.

Original: All experiments have been repeated with 5 to 25 seeds to obtain stable, reproducible results.
Revised: All experiments have been repeated with 5 to 25 seeds to obtain stable, reproducible results.

Original: Appendix C.1.2 outlines the packages that have been used for running the baselines.
Revised: Appendix C.1.2 outlines the packages that have been used for running the baselines.

Original: We have given a description of the used computation resources in Section 4.1, namely a 24-core CPU with a single NVIDIA RTX3090 GPU.
Revised: The computation resources deployed for all experiments are a 24-core CPU with a single NVIDIA RTX3090 GPU.

Original: All experiments can be reproduced on a computer with a single GPU, and only the experiments on graphs larger than 100 variables require a GPU memory of about 24GB.
Revised: All experiments can be reproduced on a computer with a single GPU, and only the experiments on graphs larger than 100 variables require a GPU memory of about 24GB.

Original: The other experiments can be performed on smaller GPUs as well.
Revised: The other experiments can be performed on smaller GPUs as well.

Revised: A CKNOWLEDGEMENTS

Revised: This work is ﬁnancially supported by Qualcomm Technologies Inc., the University of Amsterdam and the allowance Top consortia for Knowledge and Innovation (TKIs) from the Netherlands Ministry of Economic Affairs and Climate Policy.

Revised: We thank Pascal Mettes, Christina Winkler, and Sara

Original:  S UPPLEMENTARY MATERIAL E FFICIENT N EURAL C AUSAL D ISCOVERY WITHOUT A CYCLICITY C ONSTRAINTS T ABLE OF C ONTENTS
Revised: Magliacane for their valuable comments and feedback on an initial draft of this work. We thank the anonymous reviewers for the reviews, suggestions, and engaging discussion which helped to improve this work further. Finally, we thank SURFsara for the support in using the Lisa Compute Cluster. S UPPLEMENTARY MATERIAL E FFICIENT N EURAL C AUSAL D ISCOVERY WITHOUT A CYCLICITY C ONSTRAINTS T ABLE OF C ONTENTS

Original: A Gradient estimators
Revised: A Gradient estimators

Original: A.1 Low-variance gradient estimator for edge parameters 
Revised: A.1 Low-variance gradient estimator for edge parameters . . . . . . . . . . . . . . . . . .

Original: A.2 Low-variance gradient estimator for orientation parameters 
Revised: A.2 Low-variance gradient estimator for orientation parameters . . . . . . . . . . . . . . .

Original: A.3 Training loop 
Revised: A.3 Training loop . . .

Original: B Conditions for converging to the true causal graph
Revised: B Conditions for converging to the true causal graph

Original: B.1 Assumptions . . .
Revised: B.1 Assumptions . . .

Original: B. Convergence conditions 
Revised: B.2 Convergence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: B 3 Convergence conditions for latent confounder detection 
Revised: B.3 Convergence conditions for latent confounder detection . . . . . . . . . . . . . . . . .

Original: B 4 Convergence for partial intervention sets 
Revised: B.4 Convergence for partial intervention sets . . . . . . . . . . . . . . . . . . . . . . . . .

Original: B 5 Example for non-faithful graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: B.5 Example for non-faithful graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: C Experimental details
Revised: C Experimental details

Original: C.1 Common graph structure experiments
Revised: C.1 Common graph structure experiments

Original: C.2 Scalability experiments
Revised: C.2 Scalability experiments

Original: C.3 Latent confounder experiments
Revised: C.3 Latent confounder experiments

Original: C.4 Interventions on fewer variables
Revised: C. 4 Interventions on fewer variables

Original: C.5 Real-world inspired experiments
Revised: C.5 Real-world inspired experiments

Original: D Additional experiments
Revised: D Additional experiments

Original: D. 1 Effect of the sample 
Revised: D.1 Effect of the sample size

Original: size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: D. 2 Interventions on fewer variables. .
Revised: D.2 Interventions on fewer variables

Original: D Ablation study on gradient estimators
Revised: D.3 Ablation study on gradient estimators

Original: D. 4 Deterministic variables
Revised: D.4 Deterministic variables

Original: D.5 Continuous data
Revised: D.5 Continuous data

Revised: D.6 Skeleton learning with observational baseline

Original: D.6 Non-neural based data simulators
Revised: D.7 Non-neural based data simulators

Original: A G RADIENT ESTIMATORS
Revised: A G RADIENT ESTIMATORS

Original: The following section describes in detail the derivation of the gradient estimators discussed in
Revised: The following section describes in detail the derivation of the gradient estimators discussed in

Original: Section 3.3.
Revised: Section 3.3.

Original: We consider the problem of causal structure learning where we parameterize the graph by edge existence parameters γ and orientation parameters θ .
Revised: We consider the problem of causal structure learning where we parameterize the graph by edge existence parameters γ and orientation parameters θ .

Original: Our objective is to optimize γ and θ such that we maximize the probability of interventional data, i.e. , data generated from the true graphs under (arbitrary) interventions on single variables.
Revised: Our objective is to optimize γ and θ such that we maximize the probability of interventional data, i.e. , data generated from the true graphs under (arbitrary) interventions on single variables.

Original: Thereby, the likelihood estimates have been trained on observational data only.
Revised: Thereby, the likelihood estimates have been trained on observational data only.

Original: Additionally, we want to ensure that the graph is as sparse as possible to prevent unnecessary connections.
Revised: Additionally, we want to ensure that the graph is as sparse as possible to prevent unnecessary connections.

Original: Thus, an (cid:96) 1 regularizer is added on top of the edge probabilities.
Revised: Thus, an (cid:96) 1 regularizer is added on top of the edge probabilities.

Original: The full objective can be written as follows:
Revised: The full objective can be written as follows:

Original: [Equation]
Revised: [Equation]

Original: where:
Revised: where:

Original: • N is the number of variables in the causal graph ( X 1 , ..., X N ); • p
Revised: • N is the number of variables in the causal graph ( X 1 , ..., X N ); • p

Original: I ( I ) is the distribution over interventions that are performed.
Revised: I ( I ) is the distribution over interventions that are performed.

Original: This distribution can be set as a hyperparameter to weight certain interventions higher than others.
Revised: This distribution can be set as a hyperparameter to weight certain interventions higher than others.

Original: In our experiments, we assume it to be uniform across interventions on variables; • ˜ p
Revised: In our experiments, we assume it to be uniform across interventions on variables; • ˜ p

Original: ˆ I ( X ) is the joint distribution of all variables under the intervention ˆ
Revised: ˆ I ( X ) is the joint distribution of all variables under the intervention ˆ

Original: I ; • p γ , θ ( C ) is the distribution over adjacency matrices C , which we model as a product of independent edge probabilities: p γ , θ ( C ) = 
Revised: I ; • p γ , θ ( C ) is the distribution over adjacency matrices C , which we model as a product of independent edge probabilities: p γ , θ ( C ) = (cid:81) N i = [Equation] [Equation]

Revised: (cid:54)

Original: [Equation]
Revised: [Equation]

Original:  ) ; • L C ( X i ) is the negative log-likelihood estimate of variable X
Revised: σ ( γ ij ) · σ ( θ ij ) ; • L C ( X i ) is the negative log-likelihood estimate of variable X

Original: i under sampled adjacency matrix C :
Revised: i under sampled adjacency matrix C :

Original: L C ( X i )
Revised: L C ( X i )

Original: = − log f
Revised: = − log f

Original: φ
Revised: φ

Original: i
Revised: i

Original: ( X i ; C · ,i (cid:12) X − i ) ; • λ sparse is a hyperparameter representing the regularization weight.
Revised: ( X i ; C · ,i (cid:12) X − i ) ; • λ sparse is a hyperparameter representing the regularization weight.

Original: Based on this objective, we derive the gradient estimators for optimizing both edge existence and orientation parameters.
Revised: Based on this objective, we derive the gradient estimators for optimizing both edge existence and orientation parameters.

Original: A. L OW - VARIANCE GRADIENT ESTIMATOR FOR EDGE PARAMETERS
Revised: A. L OW - VARIANCE GRADIENT ESTIMATOR FOR EDGE PARAMETERS

Original: In order to optimize the edge parameters via SGD, we need to determine the gradient ∂∂γ ij ˜ L .
Revised: In order to optimize the edge parameters via SGD, we need to determine the gradient ∂∂γ ij ˜ L .

Original: Since L consists of a sum of two terms, i.e. , the log-likelihood estimate and the regularization, we can look at both parts separately.
Revised: Since L consists of a sum of two terms, i.e. , the log-likelihood estimate and the regularization, we can look at both parts separately.

Original: To prevent any confusion of index variables, we will use k, l as indices for the parameter γ kl for which we determine the gradient, i.e. , ∂∂γ kl ˜ L , and i, j as indices for sums.
Revised: To prevent any confusion of index variables, we will use k, l as indices for the parameter γ kl for which we determine the gradient, i.e. , ∂∂γ kl ˜ L , and i, j as indices for sums.

Original: As a ﬁrst step, we determine the gradients for the regularization term.
Revised: As a ﬁrst step, we determine the gradients for the regularization term.

Original: Those can be found by taking the derivative of the sigmoid:
Revised: Those can be found by taking the derivative of the sigmoid:

Original: [Equation]
Revised: [Equation]

Original: Thus, it is straight-forward to calculate for any edge parameter.
Revised: Thus, it is straight-forward to calculate for any edge parameter.

Original: In the following, we use σ (cid:48) ( ... ) to abbreviate the derivate of the sigmoid: σ(cid:48)
Revised: In the following, we use σ (cid:48) ( ... ) to abbreviate the derivate of the sigmoid: σ

Original: [Equation]
Revised: [Equation]

Original: For the log-likelihood term, we start by reorganizing the expectations to simplify the gradient expression.
Revised: For the log-likelihood term, we start by reorganizing the expectations to simplify the gradient expression.

Original: The derivate term ∂∂γ kl can be moved inside the two expectations over interventional data since those are independent of the graph parameters.
Revised: The derivate term ∂∂γ kl can be moved inside the two expectations over interventional data since those are independent of the graph parameters.

Original: Thus, we can write:
Revised: Thus, we can write:

Original: [Equation]
Revised: [Equation]

Original: For readability, we denote ˜ L (cid:48) to be the objective in Equation 9 without the regularizer.
Revised: For readability, we denote ˜ L (cid:48) to be the objective in Equation 9 without the regularizer.

Original: Next, we take a closer look at the derivate of the expectation over adjacency matrices.
Revised: Next, we take a closer look at the derivate of the expectation over adjacency matrices.

Original: Note that we have deﬁned the adjacency matrix distribution as p γ , θ ( C ) =
Revised: Note that we have deﬁned the adjacency matrix distribution as p γ , θ ( C ) =

Original: (cid:81)
Revised: (cid:81)

Original: Ni = (cid:81)
Revised: [Equation]

Original: Nj =1 ,j (cid:54) =
Revised: 1 ,j (cid:54) =

Original: i σ ( γ ij ) σ ( θ ij ) , with
Revised: i σ ( γ ij ) σ ( θ ij ) , with

Original: C ij = 1 representing the edge
Revised: C ij = 1 representing the edge

Original: X i → X j .
Revised: X i → X j .

Original: Since a parameter γ ij only inﬂuences the likelihood of the edge X i → X j and no other edges, we can reduce the expectation to a single binary variable over which we need to differentiate the expectation:
Revised: Since a parameter γ ij only inﬂuences the likelihood of the edge X i → X j and no other edges, we can reduce the expectation to a single binary variable over which we need to differentiate the expectation:

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: X 3 ( X 3 )
Revised: X 3 ( X 3 )

Original: L X 2 6!
Revised: L X 2 6!

Original: X 3 ( X 3 ) @
Revised: X 3 ( X 3 ) @

Original: @ ij L = ↵ · E X ,C  ij ⇥ 
Revised: @ ij L = ↵ · E X ,C  ij ⇥ L

Original: [Table]
Revised: [Table]

Original: j ( X
Revised: i

Original: [Table]
Revised: [Table]

Original:  +  sparse 
Revised: j ( X j ) [Table] [Table] j ( X j ) +  sparse ⇤ C

Original: [Table]
Revised: [Table]

Original: C
Revised: C

Original: [Table]
Revised: [Table]

Original: @ @ ij L = ↵ · E X ,C  ij ⇥
Revised: @ @ ij L = ↵ · E X ,C  ij ⇥

Original: L X i !
Revised: L X i !

Original: X j
Revised: X j

Original: ( X j )  L X
Revised: ( X j )  L X

Original: i 6!
Revised: i 6!

Original: X j
Revised: X j

Original: ( X j ) +  sparse ⇤
Revised: ( X j ) +  sparse ⇤

Original: C (1)
Revised: C (1)

Original: [Table]
Revised: [Table]

Original: @ @ ij L = ↵ · E X ,C  ij ⇥
Revised: @ @ ij L = ↵ · E X ,C  ij ⇥

Original: L X i !
Revised: L X i !

Original: X j
Revised: X j

Original: ( X j )  L X
Revised: ( X j )  L X

Original: i 6!
Revised: i 6!

Original: X j
Revised: X j

Original: ( X j ) +  sparse ⇤ C (1)
Revised: ( X j ) +  sparse ⇤ C (1)

Original: C (2)
Revised: C (2)

Original: [Table] [Table]
Revised: C (3)

Original: [Table]
Revised: [Table]

Original: @ @ ij L = ↵ · E X ,C  ij ⇥
Revised: @ @ ij L = ↵ · E X ,C  ij ⇥

Original: L X i !
Revised: [Equation]

Original: X j
Revised: j

Original: ( X j )  L X i 6!
Revised: [Equation]

Original: X j
Revised: sparse

Original: ( X j ) +  sparse ⇤ C (1) C (2)
Revised: [Equation] C [Equation]

Original: C (3)
Revised: C (3)

Original: L X 1 !
Revised: L X 1 !

Original: X 2 ( X 2 )
Revised: X 2 ( X 2 )

Original: L X 1 6!
Revised: L X 1 6!

Original: X 2 ( X 2 )
Revised: X 2 ( X 2 )

Original: L X 3 !
Revised: L X 3 !

Original: X 2 ( X 2 )
Revised: X 2 ( X 2 )

Original: L [Table] [Table]
Revised: L X 3 6!

Original:  ( X [Table]
Revised: X 2 ( X 2 )

Original: L X 1 !
Revised: L X 1 !

Original: [Table] ( X 
Revised: X 3 ( X 3 )

Original: [Table] L X [Table]
Revised: L X 1 6! X 3 ( X 3 )

Original: [Table]
Revised: L X 2 ! X 3 ( X 3 ) L X 2 6!

Original:  ( X 
Revised: X 3 ( X 3 )

Original: [Table]
Revised: [Table]

Original: L X 2 !

Original: X
Revised: X

Original: [Table]
Revised: [Table]

Original: ( X 3 )
Revised: X [Table]

Original: L
Revised: Samplegraphs X

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: X

Original: [Table]
Revised: [Table]

Original: ( X 
Revised: X [Table] [Equation]

Original: [Table]
Revised: [Table]

Revised: X

Original: [Table]
Revised: [Table]

Original: @ @ ij L = ↵ · E X ,C  ij ⇥ L X i !
Revised: @ @ ij L = ↵ · E X ,C  ij ⇥ L X i !

Original: X j
Revised: X j

Original: ( X j )  L X
Revised: ( X j )  L X

Original: i 6!
Revised: i 6!

Original: X j
Revised: X j

Original: ( X j ) +  sparse ⇤ C (1) C (2)
Revised: ( X j ) +  sparse ⇤ C (1) C (2)

Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: L X 3 !
Revised: L X 3 !

Original: X 2 ( X 2 )
Revised: X 2 ( X 2 )

Original: L X
Revised: L X

Original: [Table]
Revised: [Table]

Original: X 2 ( X 2 ) 
Revised: X 2 ( X 2 ) L X

Original: [Table]
Revised: [Table]

Revised: [Table]

Original:  [Equation]
Revised: [Table] [Table] [Table] [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: X
Revised: [Table] [Equation] [Table]

Original: [Equation]
Revised: [Equation]

Original: X
Revised: [Table] [Equation] [Table] [Equation]

Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Revised: X

Original: [Table]
Revised: [Equation]

Revised: X

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: X 2 ( X 2 )  L X 1 6!
Revised: X 2 ( X 2 )  L X 1 6!

Original: X 2 ( X 2 )] @
Revised: X 2 ( X 2 )] @

Original: @✓ 13 ˜ L =  @
Revised: @✓ 13 ˜ L =  @

Original: @✓ 31 ˜ L =  0 ( ✓ 13 )
Revised: @✓ 31 ˜ L =  0 ( ✓ 13 )

Original: ·  (  13 ) ·
Revised: ·  (  13 ) ·

Original: [ L X 1 !
Revised: [ L X 1 !

Original: X 3 ( X 3 )
Revised: X 3 ( X 3 )

Original:  L X 1 6!
Revised:  L X 1 6!

Original: X 3 ( X 3 )]
Revised: X 3 ( X 3 )]

Original: Estimate log-likelihoods and average … … … where p γ,θ ( C kl ) =
Revised: Estimate log-likelihoods and average … … … where p γ,θ ( C kl ) =

Original: σ ( γ kl ) · σ ( θ kl ) .
Revised: σ ( γ kl ) · σ ( θ kl ) .

Original: The ﬁrst expectation over p γ , θ ( C − kl ) is independent of γ kl as we have deﬁned the adjacency matrix distribution to be a product of independent edge probabilities.
Revised: The ﬁrst expectation over p γ , θ ( C − kl ) is independent of γ kl as we have deﬁned the adjacency matrix distribution to be a product of independent edge probabilities.

Original: The log-likelihood estimate of a variable, L C ( X i ) , depends on the adjacency matrix column C · ,i which represents the input connections to the node X i .
Revised: The log-likelihood estimate of a variable, L C ( X i ) , depends on the adjacency matrix column C · ,i which represents the input connections to the node X i .

Original: All other edges have no inﬂuence on the log-likelihood estimate of X i .
Revised: All other edges have no inﬂuence on the log-likelihood estimate of X i .

Original: Hence, the parameter γ kl only inﬂuences L C ( X l ) , and thus we can reduce the sum inside the expectation to:
Revised: Hence, the parameter γ kl only inﬂuences L C ( X l ) , and thus we can reduce the sum inside the expectation to:

Original: [Equation]
Revised: [Equation]

Original: The REINFORCE trick is a simple method to move the derivative of a discrete distribution inside the expectation.
Revised: The REINFORCE trick is a simple method to move the derivative of a discrete distribution inside the expectation.

Original: Applied to our situation, we obtain:
Revised: Applied to our situation, we obtain:

Original: [Equation]
Revised: [Equation]

Original: This leaves us with two cases in the expectation: C kl = 0 and C kl = 1 .
Revised: This leaves us with two cases in the expectation: C kl = 0 and C kl = 1 .

Original: In other words, we need to distinguish between samples of C where we have the edge X k
Revised: In other words, we need to distinguish between samples of C where we have the edge X k

Original: → X l , and where we do not have such an edge ( X k (cid:54)→ X l ).
Revised: → X l , and where we do not have such an edge ( X k (cid:54)→ X l ).

Original: Thus, we can also write the expectation as a weighted sum of those two cases:
Revised: Thus, we can also write the expectation as a weighted sum of those two cases:

Original: [Equation]
Revised: [Equation]

Original: We use L X k → X l ( X l ) to denote the (expected) negative log likelihood for X l under adjacency matrices where we have an edge from X k to X l :
Revised: We use L X k → X l ( X l ) to denote the (expected) negative log likelihood for X l under adjacency matrices where we have an edge from X k to X l :

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: The ﬁnal step is to solve the two derivative terms in Equation 14.
Revised: The ﬁnal step is to solve the two derivative terms in Equation 14.

Original: This is done as follows:
Revised: This is done as follows:

Original: [Equation]
Revised: [Equation]

Original: [Figure] 
Revised: [Figure] [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Revised: Parameter 0.

Original: G r a d i e n t es t i m a t e
Revised: G r a d i e n t es t i m a t e

Original: Gradient estimation variance of -parameters
Revised: Gradient estimation variance of -parameters

Original: Estimator W-REINFORCE
Revised: Estimator W-REINFORCE

Original: ENCO
Revised: ENCO

Original: The gradients have been scaled to match in terms of averages.
Revised: The gradients have been scaled to match in terms of averages.

Original: We can see a clear reduction in variance with the gradient estimator of ENCO allowing us to use lower sample sizes.
Revised: We can see a clear reduction in variance with the gradient estimator of ENCO allowing us to use lower sample sizes.

Original: Putting these results back in the original equation and adding the sparsity regularizer, we get:
Revised: Putting these results back in the original equation and adding the sparsity regularizer, we get:

Original: [Equation]
Revised: [Equation]

Original: To align the result with the gradient in Section 3.3, we switch the index notation from k, l to i, j again.
Revised: To align the result with the gradient in Section 3.3, we switch the index notation from k, l to i, j again.

Original: The expectation E X ,C − ij is a short form for the expectations
Revised: The expectation E X ,C − ij is a short form for the expectations

Original: E
Revised: E

Original: ˆ I ∼ p
Revised: ˆ I ∼ p

Original: I
Revised: I

Original: [Equation]
Revised: [Equation]

Original: From this expression, we can see that the gradients of γ ij are proportional to the difference of the expected negative log-likelihood of X j with having an edge between X i → X j , and the cases where
Revised: From this expression, we can see that the gradients of γ ij are proportional to the difference of the expected negative log-likelihood of X j with having an edge between X i → X j , and the cases where

Original: X i (cid:54)→ X j .
Revised: X i (cid:54)→ X j .

Original: The sparsity regularizer thereby biases the difference towards the no-edge case.
Revised: The sparsity regularizer thereby biases the difference towards the no-edge case.

Original: The value of γ ij and θ ij only scale the gradient, but do not inﬂuence its direction.
Revised: The value of γ ij and θ ij only scale the gradient, but do not inﬂuence its direction.

Original: In order to train this objective on a dataset of interventional data, we can use Monte-Carlo sampling to obtain an unbiased gradient estimator.
Revised: In order to train this objective on a dataset of interventional data, we can use Monte-Carlo sampling to obtain an unbiased gradient estimator.

Original: Note that the adjacency matrix samples to estimate
Revised: Note that the adjacency matrix samples to estimate

Original: L X i → X j ( X j ) and L X i (cid:54)→ X j ( X j ) are not required to be the same.
Revised: L X i → X j ( X j ) and L X i (cid:54)→ X j ( X j ) are not required to be the same.

Original: For efﬁciency, we instead sample K adjacency matrices from p γ , θ ( C ) , evaluate the likelihood of a batch X under all these graphs.
Revised: For efﬁciency, we instead sample K adjacency matrices from p γ , θ ( C ) , evaluate the likelihood of a batch X under all these graphs.

Original: Afterwards, we assign the evaluated samples to one of the two cases, depending on C ij being zero or one.
Revised: Afterwards, we assign the evaluated samples to one of the two cases, depending on C ij being zero or one.

Original: This way, we can reuse the same graph samples for all edge parameters γ .
Revised: This way, we can reuse the same graph samples for all edge parameters γ .

Original: We visualize the gradient calculation in Figure 5.
Revised: We visualize the gradient calculation in Figure 5.

Original: In the cases where we perform an intervention on X i , we do not optimize γ ij for this step and set the gradients to zero.
Revised: In the cases where we perform an intervention on X i , we do not optimize γ ij for this step and set the gradients to zero.

Original: The same holds for gradient steps where we do not have any samples for one of the two log-likelihood estimates.
Revised: The same holds for gradient steps where we do not have any samples for one of the two log-likelihood estimates.

Original: A.1. C OMPARISON TO PREVIOUS GRADIENT ESTIMATORS
Revised: A.1. C OMPARISON TO PREVIOUS GRADIENT ESTIMATORS

Original: As discussed in Section 3, previous work on similar structure learning methods (Bengio et al., 2020;
Revised: As discussed in Section 3, previous work on similar structure learning methods (Bengio et al., 2020;

Original: Ke et al., 2019) relied on a different estimator.
Revised: Ke et al., 2019) relied on a different estimator.

Original: In terms of derivation, the main difference is the continuation from Equation 14 on.
Revised: In terms of derivation, the main difference is the continuation from Equation 14 on.

Original: In our proposed method, we write the expectation as the sum of two terms that can independently be approximated via Monte-Carlo sampling.
Revised: In our proposed method, we write the expectation as the sum of two terms that can independently be approximated via Monte-Carlo sampling.

Original: In comparison,
Revised: In comparison,

Original: Bengio et al.
Revised: Bengio et al.

Original: (2020) proposed to directly apply a Monte-Carlo sampler to Equation 14, and apply an importance sampling weight to reduce the variance.
Revised: (2020) proposed to directly apply a Monte-Carlo sampler to Equation 14, and apply an importance sampling weight to reduce the variance.

Original: This estimator is also used in the method SDI (Ke et al., 2019) to which we have experimentally compared our method.
Revised: This estimator is also used in the method SDI (Ke et al., 2019) to which we have experimentally compared our method.

Original: Figure 2 compared the gradient estimator in terms of standard deviation.
Revised: Figure 2 compared the gradient estimator in terms of standard deviation.

Original: The gradient estimator of
Revised: The gradient estimator of

Original: ENCO achieves a 10 times lower standard deviation compared to Bengio et al.
Revised: ENCO achieves a 10 times lower standard deviation compared to Bengio et al.

Original: (2020) making it much more efﬁcient.
Revised: (2020) making it much more efﬁcient.

Original: Since the estimator by Bengio et al.
Revised: Since the estimator by Bengio et al.

Original: (2020) is biased and has a different mean, we have scaled both estimators to have the same mean.
Revised: (2020) is biased and has a different mean, we have scaled both estimators to have the same mean.

Original: Speciﬁcally, we have applied ENCO to random graphs from our experiments on synthetic graphs (see Section 4.2), and evaluated 64 , 000 sampled adjacency matrices in terms of log-likelihood estimates.
Revised: Speciﬁcally, we have applied ENCO to random graphs from our experiments on synthetic graphs (see Section 4.2), and evaluated 64 , 000 sampled adjacency matrices in terms of log-likelihood estimates.

Original: These 64 , 000 samples are grouped into sets of K samples which we have used to estimate the gradients of γ .
Revised: These 64 , 000 samples are grouped into sets of K samples which we have used to estimate the gradients of γ .

Original: We evaluated different values of K , from K = 20 to K = 4000 , and plotted the standard deviation of those estimates in Figure 2.
Revised: We evaluated different values of K , from K = 20 to K = 4000 , and plotted the standard deviation of those estimates in Figure 2.

Original: We have also visualized three examples as violin plots in Figure 6 that demonstrate that despite both estimators having a similar mean, the variance of gradient estimates is much higher for Bengio et al.
Revised: We have also visualized three examples as violin plots in Figure 6 that demonstrate that despite both estimators having a similar mean, the variance of gradient estimates is much higher for Bengio et al.

Original: To verify that the improvement of ENCO is not just because of the gradient estimators, we have performed an ablation study with ENCO deploying the gradient estimator of Bengio et al.
Revised: To verify that the improvement of ENCO is not just because of the gradient estimators, we have performed an ablation study with ENCO deploying the gradient estimator of Bengio et al.

Original: (2020) in
Revised: (2020) in

Original: Appendix D.3.
Revised: Appendix D.3.

Original: A. L OW - VARIANCE GRADIENT ESTIMATOR FOR ORIENTATION PARAMETERS
Revised: A. L OW - VARIANCE GRADIENT ESTIMATOR FOR ORIENTATION PARAMETERS

Original: To derive the gradients for the orientation parameters θ , we can mostly follow the same approach as for the edge existence parameters γ .
Revised: To derive the gradients for the orientation parameters θ , we can mostly follow the same approach as for the edge existence parameters γ .

Original: However, we have to keep in mind the constraint θ kl = − θ lk which ensures that the orientation probability sums to one: σ ( θ kl ) + σ ( θ lk ) = 1 .
Revised: However, we have to keep in mind the constraint θ kl = − θ lk which ensures that the orientation probability sums to one: σ ( θ kl ) + σ ( θ lk ) = 1 .

Original: To determine the gradient of the likelihood term, we can separate the two gradients of θ kl and θ lk .
Revised: To determine the gradient of the likelihood term, we can separate the two gradients of θ kl and θ lk .

Original: This is because θ kl only inﬂuences the expectation over L C ( X l ) , while θ lk concerns L C ( X k ) .
Revised: This is because θ kl only inﬂuences the expectation over L C ( X l ) , while θ lk concerns L C ( X k ) .

Original: We can follow Equation 11 to Equation 20 of Section A.1 by swapping θ kl and γ kl .
Revised: We can follow Equation 11 to Equation 20 of Section A.1 by swapping θ kl and γ kl .

Original: For the derivative through the expectation, we obtain the following gradient:
Revised: For the derivative through the expectation, we obtain the following gradient:

Original: [Equation]
Revised: [Equation]

Original: Since we have the condition that θ kl = − θ lk , the full gradient for θ kl would therefore consist of the gradient above minus the gradient of Equation 21 with respect to θ ji .
Revised: Since we have the condition that θ kl = − θ lk , the full gradient for θ kl would therefore consist of the gradient above minus the gradient of Equation 21 with respect to θ ji .

Original: However, as discussed in
Revised: However, as discussed in

Original: Section 3.3, the orientation of an edge cannot be learned from observational data in this framework.
Revised: Section 3.3, the orientation of an edge cannot be learned from observational data in this framework.

Original: Hence, we only want to use the gradients of θ kl if we intervene on node X k , which gives us the following gradient expression:
Revised: Hence, we only want to use the gradients of θ kl if we intervene on node X k , which gives us the following gradient expression:

Original: [Equation]
Revised: [Equation]

Original: To align the equation with the one in Section 3.3, we swap the indices k, l with i, j again.
Revised: To align the equation with the one in Section 3.3, we swap the indices k, l with i, j again.

Original: The ﬁrst line represents cases where we have an intervention on the variable X i , while we have it over interventions on the variable X j in the second line.
Revised: The ﬁrst line represents cases where we have an intervention on the variable X i , while we have it over interventions on the variable X j in the second line.

Original: The two terms are weighted based on the edge existence likelihood σ ( γ ij ) and σ ( γ ji ) respectively, and the likelihood of performing an intervention on X i or X j .
Revised: The two terms are weighted based on the edge existence likelihood σ ( γ ij ) and σ ( γ ji ) respectively, and the likelihood of performing an intervention on X i or X j .

Original: In our experiments, we use a uniform probability across interventions on variables, but emphasize that this is not strictly required.
Revised: In our experiments, we use a uniform probability across interventions on variables, but emphasize that this is not strictly required.

Original: Moreover, one could design heuristics that selects the intervention to update the parameters on with the aim of increasing computational efﬁciency.
Revised: Moreover, one could design heuristics that selects the intervention to update the parameters on with the aim of increasing computational efﬁciency.

Original: The gradient estimators presented in Equation 22 would still be valid in such a case.
Revised: The gradient estimators presented in Equation 22 would still be valid in such a case.

Original: We clarify that we do not consider the gradients of θ ij with respect to the edge regularizer.
Revised: We clarify that we do not consider the gradients of θ ij with respect to the edge regularizer.

Original: This is done for two reasons.
Revised: This is done for two reasons.

Original: Firstly, the orientation parameter models only the direction of the edge, not whether it exists or not.
Revised: Firstly, the orientation parameter models only the direction of the edge, not whether it exists or not.

Original: The regularizer would increase θ ij if the edge existence for the opposite direction would be greater than for the direction from X i to X j , i.e. γ ij < γ ji , and decrease θ ij if we have γ ij > γ ji .
Revised: The regularizer would increase θ ij if the edge existence for the opposite direction would be greater than for the direction from X i to X j , i.e. γ ij < γ ji , and decrease θ ij if we have γ ij > γ ji .

Original: However, the orientation should only model the causal direction of an edge.
Revised: However, the orientation should only model the causal direction of an edge.

Original: Hence, we do not gain any value from a causal perspective when adding the regularizer to the gradient.
Revised: Hence, we do not gain any value from a causal perspective when adding the regularizer to the gradient.

Original: Secondly, the regularizer would require us to take additional assumptions for guaranteeing the discovery of the true graph upon convergence.
Revised: Secondly, the regularizer would require us to take additional assumptions for guaranteeing the discovery of the true graph upon convergence.

Original: In experiments with using a regularizer in the θ -gradient, we did not observe any difference to the experiments without the regularizer.
Revised: In experiments with using a regularizer in the θ -gradient, we did not observe any difference to the experiments without the regularizer.

Original: We note that the orientation parameters are considered to be pairwise independent.
Revised: We note that the orientation parameters are considered to be pairwise independent.

Original: In other words, θ ij and θ kl are considered independent parameters if i (cid:54) =
Revised: In other words, θ ij and θ kl are considered independent parameters if i (cid:54) =

Original: k, l and j (cid:54) =
Revised: k, l and j (cid:54) =

Original: k, l .
Revised: k, l .

Original: Global order distributions such as Plackett-Luce (Luce, 1959; Plackett, 1975) can be used to also incorporate transitive relations.
Revised: Global order distributions such as Plackett-Luce (Plackett, 1975; Luce, 1959) can be used to also incorporate transitive relations.

Original: However, those require high variance gradient estimators and struggled with chains in early experiments.
Revised: However, those require high variance gradient estimators and struggled with chains in early experiments.

Original: The pairwise orientation parameters provide much easier optimization while still providing convergence guarantees for the full intervention setting.
Revised: The pairwise orientation parameters provide much easier optimization while still providing convergence guarantees for the full intervention setting.

Original: A. T RAINING LOOP
Revised: A. T RAINING LOOP

Original: Finally, we give an overview over the full training loop in Algorithm 1.
Revised: Finally, we give an overview over the full training loop in Algorithm 1.

Original: The distribution over interventions p ( I ) is set to a uniform distribution for all our experiments.
Revised: The distribution over interventions p ( I ) is set to a uniform distribution for all our experiments.

Original: However, the distribution can also be replaced by a heuristic which selects interventions to increase computational efﬁciency.
Revised: However, the distribution can also be replaced by a heuristic which selects interventions to increase computational efﬁciency.

Original: To keep the convergence guarantees, p ( I ) would have to guarantee a non-zero probability to pick any variable.
Revised: To keep the convergence guarantees, p ( I ) would have to guarantee a non-zero probability to pick any variable.

Original: In experiments, we experienced that the Adam optimizer (Kingma & Ba, 2015) speeds up the convergence of the parameters γ and θ while not interfering with the convergence guarantees in practice.
Revised: In experiments, we experienced that the Adam optimizer (Kingma & Ba, 2015) speeds up the convergence of the parameters γ and θ while not interfering with the convergence guarantees in practice.

Original: Algorithm 1: Learning algorithm of ENCO
Revised: Algorithm 1: Learning algorithm of ENCO

Original: Data: N variables { X 1 , ..., X N } ; observational dataset D obs ; interventional datasets D int (ˆ I ) for sparse, perfect interventions on all variables; distribution over interventions p ( I )
Revised: Data: N variables { X 1 , ..., X N } ; observational dataset D obs ; interventional datasets D int (ˆ I ) for sparse, perfect interventions on all variables; distribution over interventions p ( I )

Original: Result:
Revised: Result:

Original: A graph structure corresponding to the cuasal relations among variables
Revised: A graph structure corresponding to the cuasal relations among variables

Original: Initialize γ = 0 , θ =for number of epochs do /* Distribution fitting */ for F iterations do X ∼ D obs for i ← 1 to N do M l ∼ Ber ( σ ( γ li ) · σ ( θ li ))
Revised: Initialize γ = 0 , θ =for number of epochs do /* Distribution fitting */ for F iterations do X ∼ D obs for i ← 1 to N do M l ∼ Ber ( σ ( γ li ) · σ ( θ li ))

Original: L = − log f φ i
Revised: L = − log f φ i

Original: ( X i ; M − i (cid:12) X − i )
Revised: ( X i ; M − i (cid:12) X − i )

Original: φ i ← Adam ( φ i , ∇ φ i L ) end end /
Revised: φ i ← Adam ( φ i , ∇ φ i L ) end end /

Original: * Graph fitting */ for G iterations do /
Revised: * Graph fitting */ for G iterations do /

Original: * Sample an intervention */ ˆ I ∼ p
Revised: * Sample an intervention */ ˆ I ∼ p

Original: ( I ) with intervention target X t
Revised: ( I ) with intervention target X t

Original: X ∼ D int (ˆ I ) /
Revised: X ∼ D int (ˆ I ) /

Original: * Evaluate multiple graph samples for gradient estimator */ for k = 1 to K do
Revised: * Evaluate multiple graph samples for gradient estimator */ for k = 1 to K do

Original: C ( k ) ∼ Ber ( σ ( γ ) · σ ( θ )) for i = 1 to N do
Revised: C ( k ) ∼ Ber ( σ ( γ ) · σ ( θ )) for i = 1 to N do

Original: L C ( k ) ( X i ) ← − log f φ i ( X i ; C ( k ) : ,i (cid:12) X − i ) end end /
Revised: L C ( k ) ( X i ) ← − log f φ i ( X i ; C ( k ) : ,i (cid:12) X − i ) end end /

Original: * Update parameters */ for i = 1 to N do for j = 1 to N where j (cid:54)
Revised: * Update parameters */ for i = 1 to N do for j = 1 to N where j (cid:54)

Original: =
Revised: =

Original: i and j (cid:54)
Revised: i and j (cid:54)

Original: =
Revised: =

Original: t do /*
Revised: t do /*

Original: Considering edge X i → X j */
Revised: Considering edge X i → X j */

Original: L X i → X j ( X j ) ← (cid:80)
Revised: L X i → X j ( X j ) ← (cid:80)

Original: Kk =1 C ( k ) ij ·L C ( k ) ( X j ) (cid:80)
Revised: [Equation] (cid:80)

Original: Kk =1 C ( k ) ij
Revised: Kk =1 C ( k ) ij

Original: L X i (cid:54)→ X j ( X j ) ← (cid:80) K k =1 (1 − C ( k ) ij ) ·L C ( k ) ( X j ) (cid:80)
Revised: L X i (cid:54)→ X j ( X j ) ← (cid:80) K k =1 (1 − C ( k ) ij ) ·L C ( k ) ( X j ) (cid:80)

Original: Kk =1 (1 − C ( k ) ij )
Revised: Kk =1 (1 − C ( k ) ij )

Original: γ ij ← γ ij − σ (cid:48) ( γ ij ) · σ ( θ ij ) · (cid:2)
Revised: γ ij ← γ ij − σ (cid:48) ( γ ij ) · σ ( θ ij ) · (cid:2)

Original: L X i → X j ( X j ) − L X
Revised: L X i → X j ( X j ) − L X

Original: i (cid:54)→ X j ( X j ) + λ sparse (cid:3) if i ==
Revised: i (cid:54)→ X j ( X j ) + λ sparse (cid:3) if i ==

Original: t then θ ij ← θ ij − σ (cid:48)
Revised: t then θ ij ← θ ij − σ (cid:48)

Original: ( θ ij ) · σ ( γ ij ) · (cid:2)
Revised: ( θ ij ) · σ ( γ ij ) · (cid:2)

Original: L X i → X j ( X j ) − L X
Revised: L X i → X j ( X j ) − L X

Original: i (cid:54)→ X j ( X j ) (cid:3) θ ji ← − θ ij end end end end end return G =
Revised: i (cid:54)→ X j ( X j ) (cid:3) θ ji ← − θ ij end end end end end return G =

Original: ( V, E ) with V = X
Revised: ( V, E ) with V = X

Original: and
Revised: and

Original: E = { ( X i , X j ) | i, j ∈ [1 , N ] and i (cid:54) =
Revised: E = { ( X i , X j ) | i, j ∈ [1 , N ] and i (cid:54) =

Original: j and σ ( γ ij ) > 0 .
Revised: j and σ ( γ ij ) > 0 .

Original: 5 and σ ( θ ij ) > 0 . 5 }
Revised: 5 and σ ( θ ij ) > 0 . 5 }

Original: B C ONDITIONS FOR CONVERGING TO THE TRUE CAUSAL GRAPH
Revised: B C ONDITIONS FOR CONVERGING TO THE TRUE CAUSAL GRAPH

Original: The following section gives an overview and proves the conditions under which ENCO converges to the correct causal graph given sufﬁcient time and data.
Revised: The following section gives an overview and proves the conditions under which ENCO converges to the correct causal graph given sufﬁcient time and data.

Original: We emphasize that we provide conditions here for which no local optima exist, meaning that if ENCO converges, it returns the correct causal graph.
Revised: We emphasize that we provide conditions here for which no local optima exist, meaning that if ENCO converges, it returns the correct causal graph.

Original: This is a stronger statement than showing that the global optimum corresponds to the true graph, since a gradient-based algorithm can get stuck in a local optimum.
Revised: This is a stronger statement than showing that the global optimum corresponds to the true graph, since a gradient-based algorithm can get stuck in a local optimum.

Original: We will discuss the conditions for the global optimum in Appendix B.2.4.
Revised: We will discuss the conditions for the global optimum in Appendix B.2.5.

Original: To make the proof more accessible, we will ﬁrst discuss the assumptions that are needed for the guarantee, and then give a sketch of the proof.
Revised: To make the proof more accessible, we will ﬁrst discuss the assumptions that are needed for the guarantee, and then give a sketch of the proof.

Original: The proof will ﬁrst assume that we work in the data limit, i.e. have given sufﬁcient data, such that we can derive conditions that solely depend on the causal graphical model.
Revised: The proof will ﬁrst assume that we work in the data limit, i.e. have given sufﬁcient data, such that we can derive conditions that solely depend on the causal graphical model.

Original: In Appendix B.2.2, we extend the proof to the limited data setting.
Revised: In Appendix B.2.3, we extend the proof to the limited data setting.

Original: B. A SSUMPTIONS
Revised: B. A SSUMPTIONS

Original: Assumption 1 We are given a dataset of observational data from the joint distribution p ( X ) .
Revised: Assumption 1 We are given a dataset of observational data from the joint distribution p ( X ) .

Original: Additionally, we have N interventional datasets for N variables where in each intervention a different node is intervened on (the intervention size for each dataset is therefore 1).
Revised: Additionally, we have N interventional datasets for N variables where in each intervention a different node is intervened on (the intervention size for each dataset is therefore 1).

Original: Assumption 2 A common assumption in causal structure learning is that the data distribution over all variables p ( X ) is Markovian and faithful with respect to the causal graph we are trying to model.
Revised: Assumption 2 A common assumption in causal structure learning is that the data distribution over all variables p ( X ) is Markovian and faithful with respect to the causal graph we are trying to model.

Original: This means that the graph represents the (conditional) independence relations between variables in the data, and (conditional) independence relations in the data reﬂect the edges in the graph.
Revised: This means that the graph represents the (conditional) independence relations between variables in the data, and (conditional) independence relations in the data reﬂect the edges in the graph.

Original: For ENCO, faithfulness is not strictly required.
Revised: For ENCO, faithfulness is not strictly required.

Original: This is because we work with interventional data.
Revised: This is because we work with interventional data.

Original: Instead, we rely on the Markov property and assume that for all variables, the parent set pa ( X i ) reﬂects the inputs to the causal generation mechanism of X i .
Revised: Instead, we rely on the Markov property and assume that for all variables, the parent set pa ( X i ) reﬂects the inputs to the causal generation mechanism of X i .

Original: This allows us to also handle deterministic variables.
Revised: This allows us to also handle deterministic variables.

Original: Assumption 3 For this proof, we assume that all variables of the graph are known and observable, and no latent confounders exist.
Revised: Assumption 3 For this proof, we assume that all variables of the graph are known and observable, and no latent confounders exist.

Original: Latent confounders can introduce dependencies between variables which are not reﬂected by the ground truth graph solely on the observed variables.
Revised: Latent confounders can introduce dependencies between variables which are not reﬂected by the ground truth graph solely on the observed variables.

Original: We discuss the extension of latent confounders in Section 3.5 and Appendix B.3.
Revised: We discuss the extension of latent confounders in Section 3.5 and Appendix B.3.

Original: Assumption 4 ENCO relies on neural networks to determine the conditional data distributions p ( X i | ... ) .
Revised: Assumption 4 ENCO relies on neural networks to determine the conditional data distributions p ( X i | ... ) .

Original: Hence, for providing a guarantee, we assume that in the graph learning step the neural networks have been sufﬁciently trained such that they accurately model all possible conditional distribution p
Revised: Hence, for providing a guarantee, we assume that in the graph learning step the neural networks have been sufﬁciently trained such that they accurately model all possible conditional distribution p

Original: ( X i | ... ) .
Revised: ( X i | ... ) .

Original: In practice, the neural networks might have a slight error.
Revised: In practice, the neural networks might have a slight error.

Original: However, as long as enough data, network complexity, and training time is provided, it is fair to assume that the difference between the modeled distribution and the true conditional is smaller than an arbitrary constant (cid:15) , based on the universal approximation theorem (Hornik et al., 1989).
Revised: However, as long as enough data, network complexity, and training time is provided, it is fair to assume that the difference between the modeled distribution and the true conditional is smaller than an arbitrary constant (cid:15) , based on the universal approximation theorem (Hornik et al., 1989).

Original: For the limited data setting, see
Revised: For the limited data setting, see

Original: Appendix B.2.2.
Revised: Appendix B.2.3.

Original: Assumption 5 We are given a sufﬁciently large interventional dataset such that sampling data points from it models the exact interventional distribution under the true causal graph.
Revised: Assumption 5 We are given a sufﬁciently large interventional dataset such that sampling data points from it models the exact interventional distribution under the true causal graph.

Original: This can be achieved by, for example, sampling directly from the causal graph, or having an inﬁnitely large dataset.
Revised: This can be achieved by, for example, sampling directly from the causal graph, or having an inﬁnitely large dataset.

Original: For the limited data setting, see Appendix B.2.2.
Revised: For the limited data setting, see Appendix B.2.3.

Original: B. C ONVERGENCE CONDITIONS
Revised: B. C ONVERGENCE CONDITIONS

Original: The proof of the convergence conditions consists of the following three main steps:
Revised: The proof of the convergence conditions consists of the following three main steps:

Original: Step 1 We show under which conditions the orientation parameters θ ij will converge to + ∞ , i.e. σ ( θ ij ) → 1 , if X i is an ancestor of X j .
Revised: Step 1 We show under which conditions the orientation parameters θ ij will converge to + ∞ , i.e. σ ( θ ij ) → 1 , if X i is an ancestor of X j .

Original: Similarly, if X i is a descendant of X j , the parameter θ ij will converge to −∞ , i.e. σ ( θ ij ) → 0 .
Revised: Similarly, if X i is a descendant of X j , the parameter θ ij will converge to −∞ , i.e. σ ( θ ij ) → 0 .

Original: Step 2 Under the assumption that the orientation parameters have converged as in Step 1, we show that for edges in the true graph, γ ij will converge to 1.
Revised: Step 2 Under the assumption that the orientation parameters have converged as in Step 1, we show that for edges in the true graph, γ ij will converge to 1.

Original: Note that we need to take additional assumptions/conditions with respect to λ sparse here.
Revised: Note that we need to take additional assumptions/conditions with respect to λ sparse here.

Original: Step 3 Once the parameters γ ij and θ ij have converged for the edges in the ground truth graph, we show that all other edges will be removed by the sparsity regularizer.
Revised: Step 3 Once the parameters γ ij and θ ij have converged for the edges in the ground truth graph, we show that all other edges will be removed by the sparsity regularizer.

Original: The following paragraphs provide more details for each step.
Revised: The following paragraphs provide more details for each step.

Original: Note that causal graphs that do not fulﬁll all parts of the convergence guarantee can still eventually be recovered.
Revised: Note that causal graphs that do not fulﬁll all parts of the convergence guarantee can still eventually be recovered.

Original: The reason is that the conditions listed in the theorems below ensure that there exists no local minima for θ and γ to converge in.
Revised: The reason is that the conditions listed in the theorems below ensure that there exists no local minima for θ and γ to converge in.

Original: Although local minima exist, the optimization process might converge to the global minimum of the true causal graph.
Revised: Although local minima exist, the optimization process might converge to the global minimum of the true causal graph.

Original: Theorem B.1.
Revised: Theorem B.1.

Original: Consider the edge X i → X j in the true causal graph.
Revised: Consider the edge X i → X j in the true causal graph.

Original: The orientation parameter θ ij converges to σ ( θ ij ) = 1 if the following two conditions are fulﬁlled: (1) for all possible sets of parents of X j excluding X i , adding X i improves the log-likelihood estimate of X j under the intervention on X i , or leaves it unchanged:
Revised: The orientation parameter θ ij converges to σ ( θ ij ) = 1 if the following two conditions are fulﬁlled: (1) for all possible sets of parents of X j excluding X i , adding X i improves the log-likelihood estimate of X j under the intervention on X i , or leaves it unchanged:

Original: [Equation]
Revised: [Equation]

Original:  there exists a set of nodes
Revised: 2) there exists a set of nodes

Original: [Equation]
Revised: [Equation]

Original: pa
Revised: pa

Original: [Equation]
Revised: [Equation]

Original: X j ) , for which the probability to be sampled as parents of X j is greater than 0, and the following condition holds:
Revised: X j ) , for which the probability to be sampled as parents of X j is greater than 0, and the following condition holds:

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Based on the conditions in Equations 23 and 24, we need to show that the gradient of θ ij is negative in expectation, independent of other values of γ and θ .
Revised: Based on the conditions in Equations 23 and 24, we need to show that the gradient of θ ij is negative in expectation, independent of other values of γ and θ .

Original: For readability, we deﬁne the following function:
Revised: For readability, we deﬁne the following function:

Original: [Equation]
Revised: [Equation]

Original: Hence, the gradient of θ ij can be written as:
Revised: Hence, the gradient of θ ij can be written as:

Original: [Equation]
Revised: [Equation]

Original: Looking at the gradient of θ ij in Equation 26, the conditions correspond to T ( X i , X j ) being smaller or equals to zero.
Revised: Looking at the gradient of θ ij in Equation 26, the conditions correspond to T ( X i , X j ) being smaller or equals to zero.

Original: Note that the sign is ﬂipped because in T ( X i , X j ) , we have negative log-likelihoods represented by L X
Revised: Note that the sign is ﬂipped because in T ( X i , X j ) , we have negative log-likelihoods represented by L X

Original: k → X l ( X l ) , while in Equations 23 and 24, we have log-likelihoods.
Revised: k → X l ( X l ) , while in Equations 23 and 24, we have log-likelihoods.

Original: Further, the other factors of σ (cid:48) ( θ ij ) , σ ( γ ij ) and p ( I X ) are all limited in the range of (0 , 1) meaning that the sign of the gradient is solely determined by T ( X i , X j ) and T ( X j , X i ) .
Revised: Further, the other factors of σ (cid:48) ( θ ij ) , σ ( γ ij ) and p ( I X ) are all limited in the range of (0 , 1) meaning that the sign of the gradient is solely determined by T ( X i , X j ) and T ( X j , X i ) .

Original: If T ( X i , X j ) − T ( X j , X i ) is smaller than zero, then the gradient of θ ij is negative, i.e. increasing θ ij .
Revised: If T ( X i , X j ) − T ( X j , X i ) is smaller than zero, then the gradient of θ ij is negative, i.e. increasing θ ij .

Original: First, we look at when T ( X i , X j ) < 0 .
Revised: First, we look at when T ( X i , X j ) < 0 .

Original: The condition in Equation 23 ensures that conditioning X j on a true parent
Revised: The condition in Equation 23 ensures that conditioning X j on a true parent

Original: X i when intervening on X i does not lead to a worse log-likelihood estimate than without.
Revised: X i when intervening on X i does not lead to a worse log-likelihood estimate than without.

Original: While this condition might seem natural, there are special cases where this condition does not hold for all variables (see Section B.2.3).
Revised: While this condition might seem natural, there are special cases where this condition does not hold for all variables (see Section B.2.4).

Original: The second condition, Equation 24, guarantees that there is at least one parent set for which T
Revised: The second condition, Equation 24, guarantees that there is at least one parent set for which T

Original: ( X i , X j ) is negative.
Revised: ( X i , X j ) is negative.

Original: Therefore, in expectation over all possible adjacency matrices p γ , θ ( C ) , T ( X i , X j ) is smaller than zero if the two conditions hold.
Revised: Therefore, in expectation over all possible adjacency matrices p γ , θ ( C ) , T ( X i , X j ) is smaller than zero if the two conditions hold.

Original: To guarantee that the whole gradient of θ ij is negative, we also need to show that for interventions on X j , T ( X j , X i ) can only be positive.
Revised: To guarantee that the whole gradient of θ ij is negative, we also need to show that for interventions on X j , T ( X j , X i ) can only be positive.

Original: When intervening on X j , X i and X j become independent as the edge X i
Revised: When intervening on X j , X i and X j become independent as the edge X i

Original: → X j is removed in the intervened graph.
Revised: → X j is removed in the intervened graph.

Original: A distribution p ( X i |
Revised: A distribution p ( X i |

Original: X j , ... ) relying on correlations between X i and X j from observational data cannot achieve a better estimate than the same distribution when removing X j .
Revised: X j , ... ) relying on correlations between X i and X j from observational data cannot achieve a better estimate than the same distribution when removing X j .

Original: This is because the cross entropy is minimized when the sampled distribution, in this case p ( X i ) , is equal to the log-likelihood estimator (Cover & Thomas, 2005):
Revised: This is because the cross entropy is minimized when the sampled distribution, in this case p ( X i ) , is equal to the log-likelihood estimator (Cover & Thomas, 2005):

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: The only situation where X i and X j can become conditionally dependent under interventions on X j is if X i and X j share a collider X k , and X i is being conditioned on the collider X k and X j .
Revised: The only situation where X i and X j can become conditionally dependent under interventions on X j is if X i and X j share a collider X k , and X i is being conditioned on the collider X k and X j .

Original: However, this requires that θ ki has negative gradients, i.e. θ ki increasing, when intervening on X k .
Revised: However, this requires that θ ki has negative gradients, i.e. θ ki increasing, when intervening on X k .

Original: This cannot be the case since under interventions on X k , X i and X k become conditionally independent, and the correlations learned from observational data cannot be transferred to the interventional setting.
Revised: This cannot be the case since under interventions on X k , X i and X k become conditionally independent, and the correlations learned from observational data cannot be transferred to the interventional setting.

Original: If X k and X i again share a collider, we can apply this argumentation recursively until a node X n does not share a collider with X i .
Revised: If X k and X i again share a collider, we can apply this argumentation recursively until a node X n does not share a collider with X i .

Original: The recursion will always come to an end as we have a ﬁnite set of nodes, and the causal graph is assumed to be acyclic.
Revised: The recursion will always come to an end as we have a ﬁnite set of nodes, and the causal graph is assumed to be acyclic.

Original: Thus, if the conditions in Equations 23 and 24 holds for all edges X i → X j in the causal graph, we can guarantee that with sufﬁcient time and data, the corresponding orientation parameters θ ij will converge to σ ( θ ij )
Revised: Thus, if the conditions in Equations 23 and 24 hold for an edge X i → X j in the causal graph, we can guarantee that with sufﬁcient time and data, the corresponding orientation parameter θ ij will converge to σ ( θ ij )

Original: = 1 .
Revised: = 1 .

Original: Theorem B.2.
Revised: Theorem B.2.

Original: Consider a pair of variables
Revised: Consider a pair of variables

Original: X i , X j for which X i is an ancestor of X j without direct edge in the true causal graph.
Revised: X i , X j for which X i is an ancestor of X j without direct edge in the true causal graph.

Original: Then, the orientation parameter of the edge X i → X j converges to σ ( θ ij ) =
Revised: Then, the orientation parameter of the edge X i → X j converges to σ ( θ ij ) =

Original: 1 if the same conditions as in Theorem B.1 hold for the pair of X i , X j .
Revised: 1 if the same conditions as in Theorem B.1 hold for the pair of X i , X j .

Original: Proof.
Revised: Proof.

Original: To show this theorem, we need to consider two cases for a pair of variables
Revised: To show this theorem, we need to consider two cases for a pair of variables

Original: X i and X j : X i and X j are conditionally independent under a sampled adjacency matrix C , or X i and X j are not independent.
Revised: X i and X j : X i and X j are conditionally independent under a sampled adjacency matrix C , or X i and X j are not independent.

Original: Both cases need to be considered for an intervention on X i with the log-likelihood estimate of X j , and an intervention on X j with the log-likelihood estimate of X i .
Revised: Both cases need to be considered for an intervention on X i with the log-likelihood estimate of X j , and an intervention on X j with the log-likelihood estimate of X i .

Original: First, we discuss interventions on X i .
Revised: First, we discuss interventions on X i .

Original: If under the sampled adjacency matrix C , X j is conditionally independent of X i , the difference in the log-likelihood estimates T ( X i , X j ) is zero in expectation.
Revised: If under the sampled adjacency matrix C , X j is conditionally independent of X i , the difference in the log-likelihood estimates T ( X i , X j ) is zero in expectation.

Original: The variables can be independent if, for example, the parents of X j are all parents of the true causal graph.
Revised: The variables can be independent if, for example, the parents of X j are all parents of the true causal graph.

Original: If X j is not conditionally independent of X i , the conditions in Equations 23 and 24 from
Revised: If X j is not conditionally independent of X i , the conditions in Equations 23 and 24 from

Original: Theorem B.1 ensure that X i has, in expectation, a positive effect on the log-likelihood estimate of
Revised: Theorem B.1 ensure that X i has, in expectation, a positive effect on the log-likelihood estimate of

Original: X j .
Revised: X j .

Original: Thus, under interventions on X i , the gradient of θ ij is smaller or equals to zero, i.e. increases θ ij .
Revised: Thus, under interventions on X i , the gradient of θ ij is smaller or equals to zero, i.e. increases θ ij .

Original: Next, we consider interventions on X j .
Revised: Next, we consider interventions on X j .

Original: If under the sampled adjacency matrix X i is conditionally independent of X j , the difference in the log-likelihood estimates T ( X j , X i ) is zero.
Revised: If under the sampled adjacency matrix X i is conditionally independent of X j , the difference in the log-likelihood estimates T ( X j , X i ) is zero.

Original: The variables can be independent if X i is conditioned on variables that d-separate X i and X j in the true causal graph.
Revised: The variables can be independent if X i is conditioned on variables that d-separate X i and X j in the true causal graph.

Original: For instance, having the children of X i as parents of X i creates this scenario.
Revised: For instance, having the children of X i as parents of X i creates this scenario.

Original: However, for this scenario to take place, one or more orientation parameters of parent-child or ancestor-descendant pairs must be incorrectly converged.
Revised: However, for this scenario to take place, one or more orientation parameters of parent-child or ancestor-descendant pairs must be incorrectly converged.

Original: In case of a parent-child pair X i , X k , Theorem B.1 shows that σ ( θ ik ) will converge to one removing any possibility of a reversed edge to be sampled.
Revised: In case of a parent-child pair X i , X k , Theorem B.1 shows that σ ( θ ik ) will converge to one removing any possibility of a reversed edge to be sampled.

Original: In case of an ancestor-descendant pair X i , X l , we can apply a recursive argument: as X l d-separates X i and
Revised: In case of an ancestor-descendant pair X i , X l , we can apply a recursive argument: as X l d-separates X i and

Original: X j , X l must come before X j in the causal order.
Revised: X j , X l must come before X j in the causal order.

Original: If for the gradient θ il , we have a similar scenario with X i being conditionally independent of X j , the same argument applies.
Revised: If for the gradient θ il , we have a similar scenario with X i being conditionally independent of X j , the same argument applies.

Original: This can be recursively applied until no more variables except direct children of X i can d-separate X i and X j .
Revised: This can be recursively applied until no more variables except direct children of X i can d-separate X i and X j .

Original: In that case, σ ( θ ik ) will converge to one, which leads to all other orientation parameters to converge to one as well.
Revised: In that case, σ ( θ ik ) will converge to one, which leads to all other orientation parameters to converge to one as well.

Original: If X i is not conditionally independent of X j , we can rely back on the argumentation of Theorem B. when we have an edge X i → X j : as in the intervened causal graph, X i and X j are independent, any correlation learned from observational data can only lead to a worse log-likelihood estimate.
Revised: If X i is not conditionally independent of X j , we can rely back on the argumentation of Theorem B. when we have an edge X i → X j : as in the intervened causal graph, X i and X j are independent, any correlation learned from observational data can only lead to a worse log-likelihood estimate.

Original: In cases of colliders, we can rely on the recursive argument from before.
Revised: In cases of colliders, we can rely on the recursive argument from before.

Original: Thus, under interventions on X j , the gradient of θ ij must be smaller or equals to zero in expectation, i.e. increases θ ij .
Revised: Thus, under interventions on X j , the gradient of θ ij must be smaller or equals to zero in expectation, i.e. increases θ ij .

Original: Therefore, we can conclude that σ ( θ ij ) converges to one for any ancestor-descendant pairs
Revised: Therefore, we can conclude that σ ( θ ij ) converges to one for any ancestor-descendant pairs

Original: X i , X j under the conditions in Theorem B.1.
Revised: X i , X j under the conditions in Theorem B.1.

Original: Theorem B.3.
Revised: Theorem B.3.

Original: Consider an edge X i → X j in the true causal graph.
Revised: Consider an edge X i → X j in the true causal graph.

Original: The parameter γ ij converges to σ ( γ ij ) = 1 if the following condition holds:
Revised: The parameter γ ij converges to σ ( γ ij ) = 1 if the following condition holds:

Original: [Equation]
Revised: [Equation]

Original: where gpa i ( X j ) is the set of nodes excluding X i which, according to the ground truth graph, could have an edge to X j without introducing a cycle, and p I − j
Revised: where gpa i ( X j ) is the set of nodes excluding X i which, according to the ground truth graph, could have an edge to X j without introducing a cycle, and p I − j

Original: ( I ) refers to the distribution over interventions p
Revised: ( I ) refers to the distribution over interventions p

Original: I ( I ) excluding the intervention on variable X j .
Revised: I ( I ) excluding the intervention on variable X j .

Original: Proof.
Revised: Proof.

Original: To show this convergence, we assume that the orientation parameters have converged corresponding to Theorem B.1 and B.2.
Revised: To show this convergence, we assume that the orientation parameters have converged corresponding to Theorem B.1 and B.2.

Original: The parameter γ ij converges to σ ( γ ij ) = 1 if its gradient, ∂∂γ ij ˜ L , is negative independent of other values of γ and orientation parameters θ that are not included in Theorem B.1 and B.2.
Revised: The parameter γ ij converges to σ ( γ ij ) = 1 if its gradient, ∂∂γ ij ˜ L , is negative independent of other values of γ and orientation parameters θ that are not included in Theorem B.1 and B.2.

Original: The gradient of γ ij includes an expectation over adjacency matrices p γ , θ ( C ) .
Revised: The gradient of γ ij includes an expectation over adjacency matrices p γ , θ ( C ) .

Original: Based on the converged θ -values, we only need to consider sets of nodes as parents for
Revised: Based on the converged θ -values, we only need to consider sets of nodes as parents for

Original: X j that contain parents, ancestors, or (conditionally) independent nodes according to the ground truth graph.
Revised: X j that contain parents, ancestors, or (conditionally) independent nodes according to the ground truth graph.

Original: This sets of parents is represented by gpa i ( X j ) .
Revised: This sets of parents is represented by gpa i ( X j ) .

Original: Among those remaining parent sets, we need to ensure that for any such set, the gradient is negative.
Revised: Among those remaining parent sets, we need to ensure that for any such set, the gradient is negative.

Original: The condition in Equationcorresponds to the inequality ∂∂γ ij ˜ L < 0 since the term on the left represents the log-likelihood difference L X i →
Revised: The condition in Equationcorresponds to the inequality ∂∂γ ij ˜ L < 0 since the term on the left represents the log-likelihood difference L X i →

Original: X j ( X j ) − L X i (cid:54)→ X j ( X j ) in the gradients of γ ij in Equation 20 with a ﬂipped sign.
Revised: X j ( X j ) − L X i (cid:54)→ X j ( X j ) in the gradients of γ ij in Equation 20 with a ﬂipped sign.

Original: For readability and better interpretation, λ sparse has been moved on the right site of the inequality.
Revised: For readability and better interpretation, λ sparse has been moved on the right site of the inequality.

Original: This is possible as λ sparse is independent of the two expectations in Equation 28.
Revised: This is possible as λ sparse is independent of the two expectations in Equation 28.

Original: If the inequality holds for all parent sets ˆ pa ) , the gradient of γ ij can be guaranteed to be negative in expectation, independent of the other values of γ .
Revised: If the inequality holds for all parent sets ˆ pa ) , the gradient of γ ij can be guaranteed to be negative in expectation, independent of the other values of γ .

Original: Since the distribution over parent sets ˆ pa ) depends on other values of γ , the condition in Equation 28 ensures that even for the parent set with the lowest log-likelihood difference, it is still larger than λ sparse .
Revised: Since the distribution over parent sets ˆ pa ) depends on other values of γ , the condition in Equation 28 ensures that even for the parent set with the lowest log-likelihood difference, it is still larger than λ sparse .

Original: If this condition holds, then the gradient of ∂∂γ ij ˜ L will be smaller than zero independent of other values of γ .
Revised: If this condition holds, then the gradient of ∂∂γ ij ˜ L will be smaller than zero independent of other values of γ .

Original: The condition in Equation 28 introduces a dependency between convergence guarantees and the regularizer parameter λ sparse .
Revised: The condition in Equation 28 introduces a dependency between convergence guarantees and the regularizer parameter λ sparse .

Original: The lower we set the regularization weight λ sparse , the more edges we can guarantee to recover.
Revised: The lower we set the regularization weight λ sparse , the more edges we can guarantee to recover.

Original: If the regularization weight is set too high, we can eventually obtain false negative edge predictions.
Revised: If the regularization weight is set too high, we can eventually obtain false negative edge predictions.

Original: If the regularization weight is set very low, we take a longer time to converge as it requires lower gradient variance or more update steps, and is more sensitive in a limited data regime.
Revised: If the regularization weight is set very low, we take a longer time to converge as it requires lower gradient variance or more update steps, and is more sensitive in a limited data regime.

Original: Nonetheless, if sufﬁcient computational resources and data is provided, any value of λ sparse > 0 can be used.
Revised: Nonetheless, if sufﬁcient computational resources and data is provided, any value of λ sparse > 0 can be used.

Original: Theorem B.4.
Revised: Theorem B.4.

Original: Assume for all edges X i → X j in the true causal graph, σ ( θ ij ) and σ ( γ ij ) have converged to one.
Revised: Assume for all edges X i → X j in the true causal graph, σ ( θ ij ) and σ ( γ ij ) have converged to one.

Original: Then, the likelihood of all other edges, i.e. σ ( θ lk ) · σ ( θ lk ) , will converge to zero under the condition that λ sparse > 0 .
Revised: Then, the likelihood of all other edges, i.e. σ ( θ lk ) · σ ( θ lk ) , will converge to zero under the condition that λ sparse > 0 .

Original: Proof.
Revised: Proof.

Original: If all edges in the ground truth graph have converged, all other pairs of variables X l , X k are (conditionally) independent in the graph.
Revised: If all edges in the ground truth graph have converged, all other pairs of variables X l , X k are (conditionally) independent in the graph.

Original: This statement follows from the Markov property of the graph and excludes ancestor-descendant pairs
Revised: This statement follows from the Markov property of the graph and excludes ancestor-descendant pairs

Original: X i , X j .
Revised: X i , X j .

Original: The possibility of having edges from descendants to ancestors has been removed by the fact that the orientation parameters θ ij have converged according to Theorem B.2.
Revised: The possibility of having edges from descendants to ancestors has been removed by the fact that the orientation parameters θ ij have converged according to Theorem B.2.

Original: Thus, for those cases, we already have the guarantee that σ ( θ ij ) · σ ( θ ij ) converges to zero.
Revised: Thus, for those cases, we already have the guarantee that σ ( θ ij ) · σ ( θ ij ) converges to zero.

Original: For a conditionally independent pair X l , X k , the difference of the log-likelihood estimate in the gradient of γ lk , i.e. L X l → X k ( X k ) − L X l (cid:54)→ X k
Revised: For a conditionally independent pair X l , X k , the difference of the log-likelihood estimate in the gradient of γ lk , i.e. L X l → X k ( X k ) − L X l (cid:54)→ X k

Original: ( X k ) , is zero in expectation since independent nodes do not share any information.
Revised: ( X k ) , is zero in expectation since independent nodes do not share any information.

Original: Thus, the gradient remaining is:
Revised: Thus, the gradient remaining is:

Original: [Equation]
Revised: [Equation]

Original: Since the gradient is positive independent of the values of γ lk and θ lk , γ lk will decrease until it converges to σ ( γ lk ) = 0 .
Revised: Since the gradient is positive independent of the values of γ lk and θ lk , γ lk will decrease until it converges to σ ( γ lk ) = 0 .

Original: Hence, if γ lk decreases for all pairs of (conditionally) independent variables X l , X k in the ground truth graph, and σ ( θ lk ) converged to zero for children and descendants, the product σ ( γ lk ) · σ ( θ lk ) will converge to zero for all edges not existing in the ground truth graph.
Revised: Hence, if γ lk decreases for all pairs of (conditionally) independent variables X l , X k in the ground truth graph, and σ ( θ lk ) converged to zero for children and descendants, the product σ ( γ lk ) · σ ( θ lk ) will converge to zero for all edges not existing in the ground truth graph.

Original: For graphs that fulﬁll all conditions in the Theorems B.1 and B.4, ENCO is guaranteed to converge given sufﬁcient data and time.
Revised: For graphs that fulﬁll all conditions in the Theorems B.1 and B.4, ENCO is guaranteed to converge given sufﬁcient data and time.

Original: The conditions in the theorems ensure that there exist no local minima or saddle points in the loss surface of the objective in Equation 2 with respect to γ and θ .
Revised: The conditions in the theorems ensure that there exist no local minima or saddle points in the loss surface of the objective in Equation 2 with respect to γ and θ .

Original: Summary We can summarize the conditions discussed above as follows.
Revised: Summary We can summarize the conditions discussed above as follows.

Original: Given a causal graph G with variables X 1 , ..., X N and sparse interventions on all variables, the proposed method ENCO will converge to the true, causal graph G , if the following three conditions hold for all edges X i → X j in the true causal graph G :
Revised: Given a causal graph G with variables X 1 , ..., X N and sparse interventions on all variables, the proposed method ENCO will converge to the true, causal graph G , if the following three conditions hold for all edges X i → X j in the true causal graph G :

Original: For all possible sets of parents of X j excluding X i , adding X i improves the log-likelihood estimate of X j under the intervention on X i , or leaves it unchanged:
Revised: For all possible sets of parents of X j excluding X i , adding X i improves the log-likelihood estimate of X j under the intervention on X i , or leaves it unchanged:

Original: [Equation]
Revised: [Equation]

Original: There exists a set of nodes
Revised: There exists a set of nodes

Original: [Equation]
Revised: [Equation]

Original: pa
Revised: pa

Original: [Equation]
Revised: [Equation]

Original: X j ) , for which the probability to be sampled as parents of X j is greater than
Revised: X j ) , for which the probability to be sampled as parents of X j is greater than

Original: [Equation]
Revised: [Equation]

Original: and the following condition holds:
Revised: and the following condition holds:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Revised: The conditional distributions are given in Equation 33 to 35.

Original: The effect of X i on X j cannot be described by other variables up to λ sparse :
Revised: The effect of X i on X j cannot be described by other variables up to λ sparse :

Original: [Equation]
Revised: [Equation]

Original: where gpa i ( X j ) is the set of nodes excluding X i which, according to the ground truth graph, could have an edge to X j without introducing a cycle.
Revised: where gpa i ( X j ) is the set of nodes excluding X i which, according to the ground truth graph, could have an edge to X j without introducing a cycle.

Original: Further, for all other pairs X i , X j for which X j is a descendant of X i , conditions (1) and (2) need to hold as well.
Revised: Further, for all other pairs X i , X j for which X j is a descendant of X i , conditions (1) and (2) need to hold as well.

Original: B.2. E XAMPLE FOR CHECKING CONVERGENCE CONDITIONS
Revised: B.2. E XAMPLE FOR CHECKING CONVERGENCE CONDITIONS

Original: In the following, we will provide a walkthrough for how the conditions above can be checked on a simple example graph.
Revised: In the following, we will provide a walkthrough for how the conditions above can be checked on a simple example graph.

Original: For further details on the precise calculations, we provide a Jupyter Notebook that contains all calculations in this example 1 .
Revised: For further details on the precise calculations, we provide a Jupyter Notebook that contains all calculations in this example 1 .

Original: Suppose we have a graph with 3 binary variables, X 1 , X 2 , X 3 , with the causal graph being X 1 →
Revised: Suppose we have a graph with 3 binary variables, X 1 , X 2 , X 3 , with the causal graph being X 1 →

Original: X 2 → X 3 , i.e., a small chain.
Revised: X 2 → X 3 , i.e., a small chain.

Original: For simplicity, let us assume that the true, conditional distributions are the following:
Revised: For simplicity, let us assume that the true, conditional distributions are the following:

Original: [Equation]
Revised: [Equation]

Original: In other words, X 2 is equals to the value of X 1 with a probability of 0 . 6 , and the opposite binary value otherwise.
Revised: In other words, X 2 is equals to the value of X 1 with a probability of 0 . 6 , and the opposite binary value otherwise.

Original: Similarly, X 3 is equals to the value of X 2 with a probability of 0 . 2 , and the opposite binary value with a probability of 0 . 8 .
Revised: Similarly, X 3 is equals to the value of X 2 with a probability of 0 . 2 , and the opposite binary value with a probability of 0 . 8 .

Original: Therefore, the sample with the highest probability in this joint distribution would be X 1 = 1 , X 2 = 1 , X 3 = 0 .
Revised: Therefore, the sample with the highest probability in this joint distribution would be X 1 = 1 , X 2 = 1 , X 3 = 0 .

Original: Further, we assume that all interventions replace the respective conditional distribution by a uniform distribution, i.e., p I Xi
Revised: Further, we assume that all interventions replace the respective conditional distribution by a uniform distribution, i.e., p I Xi

Original: ( X i ) =
Revised: ( X i ) =

Original: Bern (0 . 5) .
Revised: Bern (0 . 5) .

Original: Next, we will check the conditions for the edges in G , i.e., X 1 → X 2 and X 2 → X 3 , and the remaining ancestor-descendant pair X 1 , X 3 .
Revised: Next, we will check the conditions for the edges in G , i.e., X 1 → X 2 and X 2 → X 3 , and the remaining ancestor-descendant pair X 1 , X 3 .

Original: Edge X 1 → X 2 :
Revised: Edge X 1 → X 2 :

Original: • Condition 1: the possible parent sets that exclude X 1 and X 2 are X 3 and the empty set.
Revised: • Condition 1: the possible parent sets that exclude X 1 and X 2 are X 3 and the empty set.

Original: For the empty set, we get:
Revised: For the empty set, we get:

Original: [Equation]
Revised: [Equation]

Original: For conditioning on X  , we obtain:
Revised: For conditioning on X 3 , we obtain:

Original: [Equation]
Revised: [Equation]

Original: Since both values are greater than zero condition 1 is fulﬁlled for X 1 → X 2 .
Revised: Since both values are greater than zero, condition 1 is fulﬁlled for X 1 → X 2 .

Original: • Condition 2: is already fulﬁlled by the equations in condition 1 since all parent sets have a difference greater than zero.
Revised: • Condition 2: is already fulﬁlled by the equations in condition 1 since all parent sets have a difference greater than zero.

Original: • Condition 3: the set gpa 1 ( X 2 ) is the empty set since X 3 is a descendant of X 2 , and no other nodes exist in the graph. 
Revised: • Condition 3: the set gpa 1 ( X 2 ) is the empty set since X 3 is a descendant of X 2 , and no other nodes exist in the graph. Thus, the parent set minimizing

Revised: the

Original: Thus, the parent set minimizing the expression on the left can only be the empty set, and we can calculate it as follows:
Revised: expression on the left can only be the empty set , and we can calculate it as follows:

Original: [Equation]
Revised: [Equation]

Original: with assuming p
Revised: with assuming p

Original: I ( I ) being the uniform distribution, and excluding I X 2 since we do not update γ12 in this case.
Revised: I ( I ) being the uniform distribution and excluding I X 2 since we do not update γ in this case Hence

Original: Hence, as long as λ sparse is smaller than 0 .
Revised: , as long as λ sparse is smaller than 0 .

Original: 02 , the condition is fulﬁlled.
Revised: 02 , the condition is fulﬁlled

Original: Edge X 2 → X 
Revised: Edge X 2 → X 3 :

Original: [Equation]

Original: • Condition 1: the possible parent sets that exclude X 2 and X 3 are X 1 and the empty set.
Revised: • Condition 1: the possible parent sets that exclude X 2 and X 3 are X 1 and the empty set.

Original: For the empty set, we get:
Revised: For the empty set, we get:

Original: [Equation]
Revised: [Equation]

Original: For conditioning on X1 , we obtain:
Revised: For conditioning on X we obtain:

Original: [Equation]
Revised: [Equation]

Original: Since both values are greater than zero, condition 1 is fulﬁlled for X 2 → X 3 .
Revised: Since both values are greater than zero, condition 1 is fulﬁlled for X 2 → X 3 .

Original: • Condition 2: is already fulﬁlled by the equations in condition 1 since all parent sets have a difference greater than zero.
Revised: • Condition 2: is already fulﬁlled by the equations in condition 1 since all parent sets have a difference greater than zero.

Original: • Condition 3: the set gpa 2 ( X 3 ) contains the variable X 1 since we can introduce an edge X 1 → X 3 without introducing acyclicity in the true, causal graph.
Revised: • Condition 3: the set gpa 2 ( X 3 ) contains the variable X 1 since we can introduce an edge X 1 → X 3 without introducing acyclicity in the true, causal graph.

Original: Thus, we need to compare two parent sets for ﬁnding the minimum of the left-side term: X 1 and the empty set.
Revised: Thus, we need to compare two parent sets for ﬁnding the minimum of the left-side term: X 1 and the empty set.

Original: First
Revised: First,

Original: , we consider the empty set:
Revised: we consider the empty set:

Original: [Equation]
Revised: [Equation]

Original: 194 >
Revised: 194 >

Original: λ sparse
Revised: λ sparse

Original: Again, we exclude I X 3 since we do not update γ 23 in this case.
Revised: Again, we exclude I X 3 since we do not update γ 23 in this case.

Original: The second case considers
Revised: The second case considers

Original: X 1 as additional parent set ˆ pa:
Revised: X 1 as additional parent set ˆ pa:

Original: [Equation]
Revised: [Equation]

Original:  > 
Revised: = [Equation] 193 > λ

Original: [Equation]
Revised: [Equation]

Original: The minimum of both values is 0 .
Revised: The minimum of both values is 0 .

Original: Hence, the edge X 2 → X 3 can be recovered if 0 .
Revised: Hence, the edge X 2 → X 3 can be recovered if 0 .

Original: 193 >
Revised: 193 >

Original: λ sparse .
Revised: λ sparse .

Original: Ancestor-descendant pair X 1 , X 3 :
Revised: Ancestor-descendant pair X 1 , X 3 :

Original: • Condition 1: the possible parent sets that exclude X 1 and X 3 are X 2 and the empty set.
Revised: • Condition 1: the possible parent sets that exclude X 1 and X 3 are X 2 and the empty set.

Original: For the empty set, we get:
Revised: For the empty set, we get:

Original: [Equation]
Revised: [Equation]

Original: For conditioning on X 2 , we obtain:
Revised: For conditioning on X 2 , we obtain:

Original: [Equation]
Revised: [Equation]

Original: The difference is zero because X 3 is independent of X 1 when conditioned on X 2 : p ( X 3 | X 1 , X 2 )
Revised: The difference is zero because X 3 is independent of X 1 when conditioned on X 2 : p ( X 3 | X 1 , X 2 )

Original: = p ( X 3 | X 2 )
Revised: = p ( X 3 | X 2 )

Original: Since both values are greater or equals to zero, condition 1 is 
Revised: Since both values are greater or equals to zero, condition 1 is fulﬁlled

Original: fulﬁlled for the pair X 1 , X 3 .
Revised: for the pair X 1 , X 3 .

Original: • Condition 2: from condition 1, we can see that the parent set of
Revised: • Condition 2: from condition 1, we can see that the parent set of

Original: [Equation]
Revised: [Equation]

Original: pa
Revised: pa

Original: [Equation]
Revised: [Equation]

Original: X 3 ) being the empty set is the only option that fulﬁlls the condition being greater than zero.
Revised: X 3 ) being the empty set is the only option that fulﬁlls the condition being greater than zero.

Original: Since we start the optimization process with an initialization that assigns a non-zero probability to all possible parent sets, it follows that
Revised: Since we start the optimization process with an initialization that assigns a non-zero probability to all possible parent sets, it follows that

Original: [Equation]
Revised: [Equation]

Original: ) being the empty set has a probability greater than zero throughout the optimization process.
Revised: ) being the empty set has a probability greater than zero throughout the optimization process.

Original: Hence, condition 2 is fulﬁlled as well.
Revised: Hence, condition 2 is fulﬁlled as well.

Original: Summary : in conclusion, for the discussed example, we can guarantee that ENCO converges to the correct causal graph if λ sparse < 0 .
Revised: Summary : in conclusion, for the discussed example, we can guarantee that ENCO converges to the correct causal graph if λ sparse < 0 .

Original: To experimentally verify this results, we applied ENCO on this graph with two hyperparameter settings for the sparsity regularizer: λ sparse = 0 .
Revised: To experimentally verify this results, we applied ENCO on this graph with two hyperparameter settings for the sparsity regularizer: λ sparse = 0 .

Original: 019 and λ sparse
Revised: 019 and λ sparse

Original: = 0 .
Revised: = 0 .

Original: We considered a very large sample size, more speciﬁcally 10k per intervention and 100k observational samples, to simulate the data limit regime.
Revised: We considered a very large sample size, more speciﬁcally 10k per intervention and 100k observational samples, to simulate the data limit regime.

Original: For λ sparse = 0 .
Revised: For λ sparse = 0 .

Original: 019 , ENCO was able to recover the graph without errors while for λ sparse = 0 .
Revised: 019 , ENCO was able to recover the graph without errors while for λ sparse = 0 .

Original: 021 , the edge X 1 → X 2 was, as expected, missed.
Revised: 021 , the edge X 1 → X 2 was, as expected, missed.

Original: This veriﬁes the theoretical result above with respect to λ sparse .
Revised: This veriﬁes the theoretical result above with respect to λ sparse .

Original: Note that if the condition is not fulﬁlled by selecting a too large sparsity regularizer, this does not necessarily mean that ENCO will not be able to recover the graph.
Revised: Note that if the condition is not fulﬁlled by selecting a too large sparsity regularizer, this does not necessarily mean that ENCO will not be able to recover the graph.

Original: This is because we consider the ’worst-case’ parent set in condition 3, while this case might not be in the true causal graph to which the other edges converge.
Revised: This is because we consider the ’worst-case’ parent set in condition 3, while this case might not be in the true causal graph to which the other edges converge.

Revised: B.2. I NTUITION BEHIND C ONDITION 1 AND

Revised: As mentioned in the text, condition 1 and 2 of Theorem 3.1 ensure that the orientation probabilities cannot converge to any local optima.

Revised: Since the conditions explicitly involve the data distributions and implicitly the gradient estimators, we provide below an assumption from a data generation mechanism perspective as an alternative, that ensures condition 1 and 2 to be satisﬁed.

Revised: Firstly, we assume that ancestors and descendants are not independent under interventions on the ancestors.

Original: B
Revised: Note that there can exist graphs where the ancestors are independent of descendants, for instance in a linear Gaussian setting when the ancestor has a weight of zero on the descendant. However, those graphs, violating faithfulness, are impossible to ﬁnd for any causal discovery method since the variables are independent under any setting. In terms of condition 1 and 2, it would imply that the inequality is always zero. Next, we show that under the previous assumption, local optima of the orientation probabilities can only occur in the following structure: for an edge X i → X j , there exist one or more parent(s) of X j sharing a common confounder X k with X i , where X k is not a direct parent of X j .

Revised: An example of this structure is the following: X 1 → X 2 , X 3 ; X 2 , X 3 → X 4 where the orientations of the edges

Revised: X 2 → X 4 and X 3 → X 4 could have a local optimum.

Revised: This statement can be proven as follows by using the do-calculus (Pearl, 2009).

Revised: Suppose a graph that includes the three variables X 1 , X 2 , Xwith X 1 → X 2 , X 3 → X 2 , and X 2 having no parents besides X 1 and X 3 .

Revised: If X 1 and X 3 do not share a confounder, then, from do-calculus, we know that p ( X| do ( X 1 = x 1 ))

Revised: = p ( X 2 | X 1 = x 1 ) and p ( X 2 | do ( X 3 = x 3 ))

Revised: = p ( X 2 | X 3 = x 3 ) .

Revised: Furthermore, since the conditional entropy of a variable can only be smaller or equals to the marginal, i.e. H ( X ) ≥ H ( X | Y ) , estimating X 2 under interventions on X 1 can only be improved by conditioning on X 1 , and similarly for X 3 .

Revised: Thus, condition 1 is strictly fulﬁlled when parents do not share a confounder under the previous assumption of no independence in all possible settings.

Revised: Now, consider the situation where X 1 and X 3 share a common confounder.

Revised: Then, from do-calculus, we can state that there can exist a parameterization of the conditional distributions for which p ( X 2 | do ( X 1 = x 1 ))

Revised: (cid:54) =

Revised: p ( X 2 | X 1 = x 1 ) .

Revised: Under this setting, we cannot guarantee that condition 1 is always fulﬁlled.

Revised: However, whether this parent-confounder structure above actually leads to a local optimum or not depends on the distributions, which condition 1 models.

Revised: Intuitively, this requires the mutual information between the two or more parents to be very high, and the initial edge probabilities of those edges to be very low.

Revised: Further, as the results show, this combination of events is not very common in practice, meaning that as long as the ancestor and descendant are not independent under the interventions, we usually converge to a graph with the correct orientation.

Revised: Besides, if the confounder X j of X 1 and X 3 is a parent of X 2 , then the local optimum would disappear with learning that edge since p ( X 2 | do ( X 1 = x 1 ) , X j ) =

Revised: p ( X 2 | X 1 = x 1 , X j ) .

Revised: In conclusion, for many of the graph structures like chain, bidiag, collider, full and jungle, this shows that there does not exist any local optima for the orientation probabilities.

Revised: Only for the certain structures of confounded parents, there may exist local optima that depend on the speciﬁc distribution parameterization.

Original:  L IMITED DATA REGIME
Revised: B.2. L IMITED DATA REGIME

Original: Assumption (3) and (4) are taken with respect to the data limit such that the conditions derived in the next section solely depend on the given causal graphical model.
Revised: Assumption (3) and (4) are taken with respect to the data limit such that the conditions derived in the next section solely depend on the given causal graphical model.

Original: However, in practice, we often have a limited data set.
Revised: However, in practice, we often have a limited data set.

Original: The proof presented for the data limit is straightforward to extend to this setting with the following modiﬁcation:
Revised: The proof presented for the data limit is straightforward to extend to this setting with the following modiﬁcation:

Original:  • The conditional distributions p ( X | ... ) are replaced by the conditional distributions that follow from the given, observational data.
Revised: [Figure] • The conditional distributions p ( X | ... ) are replaced by the conditional distributions that follow from the given, observational data.

Original: • The expectations over the interventional data ˜
Revised: • The expectations over the interventional data ˜

Original: p
Revised: p

Original: ˆ I ( X ) is replaced by the joint distribution over samples given for the intervention ˆ I .
Revised: ˆ I ( X ) is replaced by the joint distribution over samples given for the intervention ˆ I .

Original: • Theorem B.1 and B.2 for the edge
Revised: • Theorem B.1 and B.2 for the edge

Original: X i → X j are extended as follows: (3) For all possible sets of parents of X i excluding X j , adding X j does not improve the log-likelihood estimate of X i under the intervention on X j , or leaves it unchanged:∀ (cid:98) pa ( X i ) ⊆ X − i,j : E I Xj , X [log p ( X i | (cid:98)
Revised: X i → X j are extended as follows: (3) For all possible sets of parents of X i excluding X j , adding X j does not improve the log-likelihood estimate of X i under the intervention on X j , or leaves it unchanged:

Original: [Equation]
Revised: [Equation]

Original: This condition is the inverse statement of Equation 23, in the sense that we consider interventions on the child/descendant X j .
Revised: This condition is the inverse statement of Equation 23, in the sense that we consider interventions on the child/descendant X j .

Original: In the data limit, this naturally follows from Equation 23 and Equation 24, but in the limited data regime, we might have violations of Equation 36 due to biases in our samples.
Revised: In the data limit, this naturally follows from Equation 23 and Equation 24, but in the limited data regime, we might have violations of Equation 36 due to biases in our samples.

Original: Violations of Equation 36 are the cause of ENCO predicting cyclic graphs as seen in Section 4.2.
Revised: Violations of Equation 36 are the cause of ENCO predicting cyclic graphs as seen in Section 4.2.

Original: • Finally, Theorem B.4 does not necessarily hold anymore since noise in our data can lead to an overestimation of edges.
Revised: • Finally, Theorem B.4 does not necessarily hold anymore since noise in our data can lead to an overestimation of edges.

Original: Thus, we add the following condition:(1) For all pairs of variables
Revised: Thus, we add the following condition:

Original:  X i , X j for which there exists no direct causal relation in the true causal graph, and X j not being the ancestor of X i , the following condition has to hold:
Revised: (1) For all pairs of variables X i , X j for which there exists no direct causal relation in the true causal graph, and X j not being the ancestor of X i , the following condition has to hold:

Original: [Equation]
Revised: [Equation]

Original: where gpa i ( X j ) is the set of nodes excluding X i which, according to the ground truth graph, could have an edge to X j without introducing a cycle.
Revised: where gpa i ( X j ) is the set of nodes excluding X i which, according to the ground truth graph, could have an edge to X j without introducing a cycle.

Original: This condition ensures that no correlations due to sample biases introduce additional edges in the causal graphs.
Revised: This condition ensures that no correlations due to sample biases introduce additional edges in the causal graphs.

Original: If the conditions discussed above hold with respect to the given observational and interventional dataset, we can guarantee that ENCO will converge to the true causal graph given sufﬁcient time.
Revised: If the conditions discussed above hold with respect to the given observational and interventional dataset, we can guarantee that ENCO will converge to the true causal graph given sufﬁcient time.

Original: B.2. G RAPHS WITHOUT GUARANTEES
Revised: B.2. G RAPHS WITH LIMITED GUARANTEES

Original: Most common causal graphs fulﬁll the conditions mentioned above as long as a small enough value for λ sparse is chosen.
Revised: Most common causal graphs fulﬁll the conditions mentioned above, as long as a small enough value for λ sparse is chosen.

Original: Still, there are situations where we cannot guarantee that ENCO convergences to the correct causal graph independent of the chosen value of λ sparse .
Revised: Still, there are situations where we cannot guarantee that ENCO convergences to the correct causal graph independent of the chosen value of λ sparse .

Original: Here, we want to discuss two scenarios visualized in Figure 8 under which the guarantees fail.
Revised: Here, we want to discuss two scenarios visualized in Figure 8 under which the guarantees fail.

Original: Still, we want to emphasize that despite graphs not fulﬁlling the conditions, ENCO might still converge to the correct DAG for those as the guarantee conditions assume the worst-case scenarios for θ and γ in all situations.
Revised: Still, we want to emphasize that despite graphs not fulﬁlling the conditions, ENCO might still converge to the correct DAG for those as the guarantee conditions assume the worst-case scenarios for θ and γ in all situations.

Original: [Figure] The ﬁrst example we discuss is based on a fork structure where we have three binary variables, { X 1 , X 2 , X 3 } , and the edges X 1 → X 3 and X 2 → X 3 (see Figure 8a).
Revised: The ﬁrst example we discuss is based on a fork structure where we have three binary variables, { X 1 , X 2 , X 3 } , and the edges X 1 → X 3 and X 2 → X 3 (see Figure 8a).

Original: The parameterization we look at is a (noisy) XOR-gate for X 3 with its two input variables
Revised: The parameterization we look at is a (noisy) XOR-gate for X 3 with its two input variables

Original: X 1 , X 2 being independent of each other and uniformly distributed.
Revised: X 1 , X 2 being independent of each other and uniformly distributed.

Original: The conditional probability distribution p ( X 3 | X 1 , X 2 ) can be summarized in the following probability function:
Revised: The conditional probability distribution p ( X 3 | X 1 , X 2 ) can be [Table]

Original: [Equation]
Revised: [Equation]

Original: In other words, if X 1 (cid:54) =
Revised: In other words, if X 1 (cid:54) =

Original: X 2 , X 3 is equals 1 with a likelihood of 1 − (cid:15) .
Revised: X 2 , X 3 is equals 1 with a likelihood of 1 − (cid:15) .

Original: If X 

Revised: If X

Original: = X 2 , X 3 is equalswith a likelihood of (cid:15) .
Revised: = X 2 , X 3 is equalswith a likelihood of (cid:15) .

Original: The issue that this probability table creates is the following.
Revised: The issue that this probability table creates is the following.

Original: Knowing only one out of the two variables does not improve the log likelihood estimate for the output.
Revised: Knowing only one out of the two variables does not improve the log likelihood estimate for the output.

Original: This is because
Revised: This is because

Original: X 1 and X 2 are independent of each other, and p ( X 3 | X 1 ) =
Revised: X 1 and X 2 are independent of each other, and p ( X 3 | X 1 ) =

Original: p ( X 3 ) is a uniform distribution.
Revised: p ( X 3 ) is a uniform distribution.

Original: Hence, the worst-case parent set in Equation 6 would be the empty set, and leads to an expected difference log-likelihood difference of zero.
Revised: Hence, the worst-case parent set in Equation 6 would be the empty set, and leads to an expected difference log-likelihood difference of zero.

Original: As λ sparse is required to be greater than zero for Theorem B.4, we cannot fulﬁll the condition for that graph.
Revised: As λ sparse is required to be greater than zero for Theorem B.4, we cannot fulﬁll the condition for that graph.

Original: This means that an empty graph without any edges is a local minimum to which ENCO could converge.
Revised: This means that an empty graph without any edges is a local minimum to which ENCO could converge.

Original: Yet, when the edge probabilities are non-zero, we will sample adjacency matrices with both input variables being a parent of X 3 with a non-zero probability.
Revised: Yet, when the edge probabilities are non-zero, we will sample adjacency matrices with both input variables being a parent of X 3 with a non-zero probability.

Original: Hence, the log-likelihood difference for X 1 and X 2 to X 3 is unequal zero.
Revised: Hence, the log-likelihood difference for X 1 and X 2 to X 3 is unequal zero.

Original: Further, this graph is still often correctly discovered despite ENCO not having a convergence guarantee for it.
Revised: Further, this graph is still often correctly discovered despite ENCO not having a convergence guarantee for it.

Original: We have conducted experiments on this graph with (cid:15) = { 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 45 } using a sparsity regularizer of λ sparse = 1 e- 4 , and in all cases, ENCO converged to the correct, acyclic graph.
Revised: We have conducted experiments on this graph with (cid:15) = { 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 45 } using a sparsity regularizer of λ sparse = 1 e- 4 , and in all cases, ENCO converged to the correct, acyclic graph.

Original: Note that values close to 0.5 for (cid:15) are most challenging, because the difference between the true conditional and marginal distribution goes against zero.
Revised: Note that values close to 0.5 for (cid:15) are most challenging, because the difference between the true conditional and marginal distribution goes against zero.

Original: The second example we want to discuss aims at graphs that violate the condition in Theorem B.1, more speciﬁcally Equation 23.
Revised: The second example we want to discuss aims at graphs that violate the condition in Theorem B.1, more speciﬁcally Equation 23.

Original: The graph we consider is a fully connected graph with three variables
Revised: The graph we consider is a fully connected graph with three variables

Original: X 1 , X 2 , X 3 (see Figure 8b).
Revised: X 1 , X 2 , X 3 (see Figure 8b).

Original: The scenario can be described as follows: if knowing X 2 informs the log-likelihood estimate of X 3 more about X 1 than about X 2 itself, an intervention on X 2 and a sampled graph with the edge X 2 → X 3 could lead to a worse likelihood estimate of X 3 than without the edge.
Revised: The scenario can be described as follows: if knowing X 2 informs the log-likelihood estimate of X 3 more about X 1 than about X 2 itself, an intervention on X 2 and a sampled graph with the edge X 2 → X 3 could lead to a worse likelihood estimate of X 3 than without the edge.

Original: For this scenario to happen, p ( X 2 | X 1 ) must be close to deterministic.
Revised: For this scenario to happen, p ( X 2 | X 1 ) must be close to deterministic.

Original: Additionally, p ( X 3 | X 1 , X 2 ) must be much less reliant on X 2 than on X 1 , such as in the following probability density:
Revised: Additionally, p ( X 3 | X 1 , X 2 ) must be much less reliant on X 2 than on X 1 , such as in the following probability density:

Original: [Equation]
Revised: [Equation]

Original: The two variables (cid:15) 1 , (cid:15) 2 represent small constants close to zero.
Revised: The two variables (cid:15) 1 , (cid:15) 2 represent small constants close to zero.

Original: In this case, the graph can violate the condition in Equation 23 since intervening on X 2 breaks the dependency between X 1 and X 2 .
Revised: In this case, the graph can violate the condition in Equation 23 since intervening on X 2 breaks the dependency between X 1 and X 2 .

Original: The conditional distribution p ( X 3 | X 2 ) learned from observational data relies on the dependency between
Revised: The conditional distribution p ( X 3 | X 2 ) learned from observational data relies on the dependency between

Original: X 1 and X 2 which can make it to a worse estimate than p ( X 3 ) .
Revised: X 1 and X 2 which can make it to a worse estimate than p ( X 3 ) .

Original: Note that if the edge X 1 → X 3 is learned by ENCO though, this will not constitute a problem anymore since with conditioning on X 1 , i.e. p ( X 3 | X 2 , X 1 ) , the edge X 2 → X 3 will gain a gradient towards the correct graph.
Revised: Note that if the edge X 1 → X 3 is learned by ENCO though, this will not constitute a problem anymore since with conditioning on X 1 , i.e. p ( X 3 | X 2 , X 1 ) , the edge X 2 → X 3 will gain a gradient towards the correct graph.

Original: Thus, when γ and θ are not initialized with the worst-case values, the graph with both X 1 and X 2 as parents of Xcan be sampled and provides gradients in the correct direction.
Revised: Thus, when γ and θ are not initialized with the worst-case values, the graph with both X 1 and X 2 as parents of Xcan be sampled and provides gradients in the correct direction.

Original: Further, we did not observe any of these situations in the synthetic and real-world graphs we experimented on.
Revised: Further, we did not observe any of these situations in the synthetic and real-world graphs we experimented on.

Original: B.2. C ONDITIONS FOR THE GLOBAL OPTIMUM
Revised: B.2. C ONDITIONS FOR THE GLOBAL OPTIMUM

Original: So far, the discussion focused on proving that the optimization space does not contain any local optima with respect to the graph parameters γ and θ besides the global optimum.
Revised: So far, the discussion focused on proving that the optimization space does not contain any local optima with respect to the graph parameters γ and θ besides the global optimum.

Original: If these conditions are violated, ENCO might still converge to the correct solution, since we are not guaranteed to ﬁnd and get stuck in one of these local optima.
Revised: If these conditions are violated, ENCO might still converge to the correct solution, since we are not guaranteed to ﬁnd and get stuck in one of these local optima.

Original: Thus, in this section, we provide conditions under which the ground truth graph is the global optimum of the objective in Equation 2.
Revised: Thus, in this section, we provide conditions under which the ground truth graph is the global optimum of the objective in Equation 2.

Original: Graphs that fulﬁll these conditions are very likely to be correctly identiﬁed by ENCO, but with a suboptimal choice of hyperparameters, initial starting conditions etc.
Revised: Graphs that fulﬁll these conditions are very likely to be correctly identiﬁed by ENCO, but with a suboptimal choice of hyperparameters, initial starting conditions etc.

Original: , we could return an incorrect graph.
Revised: , we could return an incorrect graph.

Original: The conditions and proof follow a similar structure to those as before for the local optima.
Revised: The conditions and proof follow a similar structure to those as before for the local optima.

Original: We ﬁrst discuss when we can guarantee that the global optima has the same orientation of edges, and then when we also ﬁnd the correct parent set of the remaining variables.
Revised: We ﬁrst discuss when we can guarantee that the global optima has the same orientation of edges, and then when we also ﬁnd the correct parent set of the remaining variables.

Original: Theorem B.5.
Revised: Theorem B.5.

Original: For every pair of variables X i , X j where X i is a parent of X j , the graph ˆ G that optimizes objective in Equation 2 models the orientation X i → X j , if there exists an edge between
Revised: For every pair of variables X i , X j where X i is a parent of X j , the graph ˆ G that optimizes objective in Equation 2 models the orientation X i → X j , if there exists an edge between

Original: X i and X j in ˆ G , under the following conditions:
Revised: X i and X j in ˆ G , under the following conditions:

Original: • X i and X j are not independent under observational data.
Revised: • X i and X j are not independent under observational data.

Original: • Under interventions on X i , X i and X j are not independent given the true parent set of X j .
Revised: • Under interventions on X i , X i and X j are not independent given the true parent set of X j .

Original: Proof.
Revised: Proof.

Original: If X i and X j are independent under observational data, the observational distributions would not identify any correlation among those two variables.
Revised: If X i and X j are independent under observational data, the observational distributions would not identify any correlation among those two variables.

Original: Hence, transferring them for any graph to interventional data would have p ( X i | ... )
Revised: Hence, transferring them for any graph to interventional data would have p ( X i | ... )

Original: =
Revised: =

Original: p ( X i |
Revised: p ( X i |

Original: X j , ... ) , thus making the objective invariant to the orientation of the edge, and removing any edge between X i and X j for sparsity.
Revised: X j , ... ) , thus making the objective invariant to the orientation of the edge, and removing any edge between X i and X j for sparsity.

Original: If X i and X j are dependent, we can prove the statement by showing that modeling the orientation
Revised: If X i and X j are dependent, we can prove the statement by showing that modeling the orientation