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

Original: X j → X i will strictly lead to a worse estimate under the intervention on X j since the orientation parameters are optimized by comparing the interventions of the two adjacent variables.
Revised: X j → X i will strictly lead to a worse estimate under the intervention on X j since the orientation parameters are optimized by comparing the interventions of the two adjacent variables.

Original: Under interventions on X i , the causal mechanism p ( X j | pa ( X j )) , with pa ( X j ) being the parent set of the ground truth graph including X i , remains invariant under interventions on X i , and is strictly better than p ( X j | pa ( X j ) \ X i ) for estimating X j due to the direct causal relation.
Revised: Under interventions on X i , the causal mechanism p ( X j | pa ( X j )) , with pa ( X j ) being the parent set of the ground truth graph including X i , remains invariant under interventions on X i , and is strictly better than p ( X j | pa ( X j ) \ X i ) for estimating X j due to the direct causal relation.

Original: Under interventions on X j , the causal mechanism p
Revised: Under interventions on X j , the causal mechanism p

Original: ( X i |
Revised: ( X i |

Original: X j , ... ) leads to a strictly worse estimate as discussed in Theorem B.1, since the dependency between X i and X j does not exist in the interventional regime.
Revised: X j , ... ) leads to a strictly worse estimate as discussed in Theorem B.1, since the dependency between X i and X j does not exist in the interventional regime.

Original: Hence, the inverse orientation of the edge X i → X j , i.e. X j → X i cannot be part of the global optimum.
Revised: Hence, the inverse orientation of the edge X i → X j , i.e. X j → X i cannot be part of the global optimum.

Original: Theorem B.6.
Revised: Theorem B.6.

Original: For every pair of variables X i , X j where X i is an ancestor but not direct parents of
Revised: For every pair of variables X i , X j where X i is an ancestor but not direct parents of

Original: X j , the graph ˆ G that optimizes objective in Equation 2 does not include the edge X j → X i if the conditions in Theorem B.5 hold.
Revised: X j , the graph ˆ G that optimizes objective in Equation 2 does not include the edge X j → X i if the conditions in Theorem B.5 hold.

Original: Proof.
Revised: Proof.

Original: To show this statement, we need to consider different independence relations between X i and X j .
Revised: To show this statement, we need to consider different independence relations between X i and X j .

Original: First, if X i and X j are independent in the observational dataset given any conditional set, the edge will be removed since any edge between two independent variables is removed for any λ sparse > 0 .
Revised: First, if X i and X j are independent in the observational dataset given any conditional set, the edge will be removed since any edge between two independent variables is removed for any λ sparse > 0 .

Original: The same holds if X i and X j are independent for interventions on X i and X j .
Revised: The same holds if X i and X j are independent for interventions on X i and X j .

Original: If they are dependent, we can follow a similar argument as in Theorem B.2.
Revised: If they are dependent, we can follow a similar argument as in Theorem B.2.

Original: The causal mechanism p ( X j | X i , ... ) transfers from observational to interventional data on X i since on interventions on
Revised: The causal mechanism p ( X j | X i , ... ) transfers from observational to interventional data on X i since on interventions on

Original: X i , the causal mechanism of X j is not changed.
Revised: X i , the causal mechanism of X j is not changed.

Original: Further, when intervening on X j , X i and X j become independent such that any mechanism p ( X i | X j , ... ) cannot transfer except if X i and X j are independent under interventions on X j .
Revised: Further, when intervening on X j , X i and X j become independent such that any mechanism p ( X i | X j , ... ) cannot transfer except if X i and X j are independent under interventions on X j .

Original: In this case, the edge will be again removed by the sparsity regularizer.
Revised: In this case, the edge will be again removed by the sparsity regularizer.

Original: This shows that for any setting, the orientation of the edge X j → X i cannot lead to a better estimate than X i → X j , and in case of independence, the edge X j → X i will be removed as well.
Revised: This shows that for any setting, the orientation of the edge X j → X i cannot lead to a better estimate than X i → X j , and in case of independence, the edge X j → X i will be removed as well.

Original: Theorem B.7.
Revised: Theorem B.7.

Original: The graph ˆ G that optimizes objective in Equation 2 models the same parent sets for each variable under the following conditions:
Revised: The graph ˆ G that optimizes objective in Equation 2 models the same parent sets for each variable under the following conditions:

Original: • The conditions in Theorem B.5 hold.
Revised: • The conditions in Theorem B.5 hold.

Original: • For any variable X i with its true parent set pa ( X i ) , there does not exist a smaller parent set ˆ pa ⊂ X \ X i , descendants ( X i ) which approximates the log-likelihood of X i up to λ sparse · ( | pa ( X i ) | − | ˆ pa | ) on average.
Revised: • For any variable X i with its true parent set pa ( X i ) , there does not exist a smaller parent set ˆ pa ⊂ X \ X i , descendants ( X i ) which approximates the log-likelihood of X i up to λ sparse · ( | pa ( X i ) | − | ˆ pa | ) on average.

Original: • The regularization parameter λ sparse is greater than zero.
Revised: • The regularization parameter λ sparse is greater than zero.

Original: Proof.
Revised: Proof.

Original: If the orientations for all edges are according to the ground truth graph in the global optimum following Theorem B.5 and B.6, the parent set for a variable X i is limited to those variables which are not descendants of X i .
Revised: If the orientations for all edges are according to the ground truth graph in the global optimum following Theorem B.5 and B.6, the parent set for a variable X i is limited to those variables which are not descendants of X i .

Original: From the ground truth graph, we know that, conditioned on the true parent set pa ( X i ) ,
Revised: From the ground truth graph, we know that, conditioned on the true parent set pa ( X i ) ,

Original: X i is independent of all other non-descendant variables.
Revised: X i is independent of all other non-descendant variables.

Original: Thus, the log-likelihood estimate of X i , i.e. the left part of the objective in Equation 2, is optimized by p ( X i | pa ( X i )) .
Revised: Thus, the log-likelihood estimate of X i , i.e. the left part of the objective in Equation 2, is optimized by p ( X i | pa ( X i )) .

Original: To show that this is also the global optimum when combining with the regularizer, we need to consider those parent sets of X i which obtain a lower penalty, i.e. smaller parent sets.
Revised: To show that this is also the global optimum when combining with the regularizer, we need to consider those parent sets of X i which obtain a lower penalty, i.e. smaller parent sets.

Original: The difference between two parent sets pa ( X i ) and ˆ pa in terms of the regularizer corresponds to λ sparse · ( | pa ( X i ) | − | ˆ pa | ) .
Revised: The difference between two parent sets pa ( X i ) and ˆ pa in terms of the regularizer corresponds to λ sparse · ( | pa ( X i ) | − | ˆ pa | ) .

Original: [Figure]
Revised: [Figure]

Original: Thus, if there exists no parent set for which this difference is greater than the penalty for the worse log-likelihood estimate, the true parent set pa ( X i ) constitutes the global optimum.
Revised: Thus, if there exists no parent set for which this difference is greater than the penalty for the worse log-likelihood estimate, the true parent set pa ( X i ) constitutes the global optimum.

Original: B. C ONVERGENCE CONDITIONS FOR LATENT CONFOUNDER DETECTION
Revised: B. C ONVERGENCE CONDITIONS FOR LATENT CONFOUNDER DETECTION

Original: In Section 3.5, we have discussed that ENCO can be extended to graph with latent confounders.
Revised: In Section 3.5, we have discussed that ENCO can be extended to graph with latent confounders.

Original: For this, we have to record the gradients of γ ij for the interventional data on X i and all other interventional data separately.
Revised: For this, we have to record the gradients of γ ij for the interventional data on X i and all other interventional data separately.

Original: 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: 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: The score to detect latent confounders is:
Revised: The score to detect latent confounders is:

Original: [Equation]
Revised: [Equation]

Original: In this section, we show under which conditions the score
Revised: In this section, we show under which conditions the score

Original: lc
Revised: lc

Original: ( X i , X j ) converges to one if X i and X j share a latent confounder.
Revised: ( X i , X j ) converges to one if X i and X j share a latent confounder.

Original: We restrict our discussion to 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 restrict our discussion to 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: We assume that the causal graph based on the observed variable fulﬁlls all conditions of Theorem B.1 to B.4 in Section B.2, meaning that without the latent confounders, the graph could have been recovered without errors.
Revised: We assume that the causal graph based on the observed variable fulﬁlls all conditions of Theorem B.1 to B.4 in Section B.2, meaning that without the latent confounders, the graph could have been recovered without errors.

Original: Under those conditions, we can also show that the graph among observed variables with latent confounders is also correctly recovered.
Revised: Under those conditions, we can also show that the graph among observed variables with latent confounders is also correctly recovered.

Original: This is since the latent confounders only affect Theorem B.4: if X i and X j share a latent confounder, they are not conditionally independent given their observed parents.
Revised: This is since the latent confounders only affect Theorem B.4: if X i and X j share a latent confounder, they are not conditionally independent given their observed parents.

Original: Thus, we can rely on the fact that all edges in the true causal graph will be found according to Theorem B.1 to B.4, and the edges with latent confounders do not fulﬁll Theorem B.4.
Revised: Thus, we can rely on the fact that all edges in the true causal graph will be found according to Theorem B.1 to B.4, and the edges with latent confounders do not fulﬁll Theorem B.4.

Original: For all pairs of variables that do not share a latent confounder, lc
Revised: For all pairs of variables that do not share a latent confounder, lc

Original: ( X i , X j ) converges to zero.
Revised: ( X i , X j ) converges to zero.

Original: The edges that are removed in Theorem B.4 converge to σ ( γ ( O ) ij ) = 0 which sets lc ( X i , X j ) to zero.
Revised: The edges that are removed in Theorem B.4 converge to σ ( γ ( O ) ij ) = 0 which sets lc ( X i , X j ) to zero.

Original: For edges that have been recovered, we state in Equation 24 that the gradient for interventional data must be negative for interventions on the parent.
Revised: For edges that have been recovered, we state in Equation 24 that the gradient for interventional data must be negative for interventions on the parent.

Original: Hence, σ ( γ ( I ) ji ) converges to one which brings lc ( X i , X j ) to zero again.
Revised: Hence, σ ( γ ( I ) ji ) converges to one which brings lc ( X i , X j ) to zero again.

Original: For variables that share a latent confounder, we distinguish between two cases that are visualized in Figure 9.
Revised: For variables that share a latent confounder, we distinguish between two cases that are visualized in Figure 9.

Original: In the ﬁrst case, we assume that X i and X j are independent in the true causal graph excluding the latent confounder.
Revised: In the ﬁrst case, we assume that X i and X j are independent in the true causal graph excluding the latent confounder.

Original: This means that an intervention on X i does not cause any change in
Revised: This means that an intervention on X i does not cause any change in

Original: X j , and vice versa.
Revised: X j , and vice versa.

Original: The second case describes the situation where X i is an ancestor of X j .
Revised: The second case describes the situation where X i is an ancestor of X j .

Original: The case of X i being a parent of X j has been excluded in earlier assumptions as in those cases, we cannot separate the causal effect of X i on X j based on its causal relation and the latent confounder.
Revised: The case of X i being a parent of X j has been excluded in earlier assumptions as in those cases, we cannot separate the causal effect of X i on X j based on its causal relation and the latent confounder.

Original: In case that the two children of the latent confounder are not an ancestor-descendant pair, we can provide a guarantee under the following conditions.
Revised: In case that the two children of the latent confounder are not an ancestor-descendant pair, we can provide a guarantee under the following conditions.

Original: Theorem B.8.
Revised: Theorem B.8.

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

Original: X i , X j that share a latent confounder X l .
Revised: X i , X j that share a latent confounder X l .

Original: Assume that
Revised: Assume that

Original: X i and X j are conditionally independent given the latent confounder and their observed parents.
Revised: X i and X j are conditionally independent given the latent confounder and their observed parents.

Original: Further, all other edges in the causal graph have been recovered under the conditions of Theorem B. to B.4.
Revised: Further, all other edges in the causal graph have been recovered under the conditions of Theorem B. to B.4.

Original: The confounder score lc
Revised: The confounder score lc

Original: ( X i , X j ) converges to one if the following two conditions hold:
Revised: ( X i , X j ) converges to one if the following two conditions hold:

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: We need to show that under the two conditions above, σ ( γ ( O ) ij ) and σ ( γ ( O ) ji ) are guaranteed to converge to one while σ ( γ ( I ) ij ) and σ ( γ ( I ) ji ) converge to zero.
Revised: We need to show that under the two conditions above, σ ( γ ( O ) ij ) and σ ( γ ( O ) ji ) are guaranteed to converge to one while σ ( γ ( I ) ij ) and σ ( γ ( I ) ji ) converge to zero.

Original: The distribution p I - Xk ( I ) represents the distribution over interventions excluding the ones performed on the variable X k .
Revised: The distribution p I - Xk ( I ) represents the distribution over interventions excluding the ones performed on the variable X k .

Original: The two conditions resemble Equation 28 with the difference that the intervention on the potential parent variable is excluded, and the parent set is the true parent set of the correct causal graph.
Revised: The two conditions resemble Equation 28 with the difference that the intervention on the potential parent variable is excluded, and the parent set is the true parent set of the correct causal graph.

Original: This is because all other edges have been correctly recovered, and the two conditions are concerning σ ( γ ( O ) ij ) .
Revised: This is because all other edges have been correctly recovered, and the two conditions are concerning σ ( γ ( O ) ij ) .

Original: If the condition in Equation 41 holds, it corresponds to a negative gradient in γ ( O ) ij following the argumentation in
Revised: If the condition in Equation 41 holds, it corresponds to a negative gradient in γ ( O ) ij following the argumentation in

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

Original: The same holds for γ ( O ) ji .
Revised: The same holds for γ ( O ) ji .

Original: Therefore, σ ( γ ( O ) ij ) and σ ( γ ( O ) ji ) are guaranteed to converge to one under the conditions given in Theorem B.8.
Revised: Therefore, σ ( γ ( O ) ij ) and σ ( γ ( O ) ji ) are guaranteed to converge to one under the conditions given in Theorem B.8.

Original: For the interventional parameters γ ( I ) ij and γ ( I ) ji , we show that the gradient can only be positive, i.e. decreasing γ ( I ) ij and γ ( I ) ji .
Revised: For the interventional parameters γ ( I ) ij and γ ( I ) ji , we show that the gradient can only be positive, i.e. decreasing γ ( I ) ij and γ ( I ) ji .

Original: Under the intervention on X i , X i and X j become independent since we assume perfect interventions.
Revised: Under the intervention on X i , X i and X j become independent since we assume perfect interventions.

Original: In this case, the log-likelihood estimate of X j cannot be improved by conditioning on X i .
Revised: In this case, the log-likelihood estimate of X j cannot be improved by conditioning on X i .

Original: Hence, the difference L X i → X j ( X j ) − L X i (cid:54)→ X j ( X j ) is greater or equal to zero.
Revised: Hence, the difference L X i → X j ( X j ) − L X i (cid:54)→ X j ( X j ) is greater or equal to zero.

Original: When further considering the sparsity regularizer λ sparse , the gradient of γ ij under interventions on X i can only be positive, i.e. decreasing γ ( I ) ij .
Revised: When further considering the sparsity regularizer λ sparse , the gradient of γ ij under interventions on X i can only be positive, i.e. decreasing γ ( I ) ij .

Original: The same argument holds for γ ( I ) ji .
Revised: The same argument holds for γ ( I ) ji .

Original: Thus, we can conclude that σ ( γ ( I ) ij ) and σ ( γ ( I ) ji ) converge to zero.
Revised: Thus, we can conclude that σ ( γ ( I ) ij ) and σ ( γ ( I ) ji ) converge to zero.

Original: If the conditions of Theorem B.8 are not fulﬁlled, σ ( γ ( O ) ij ) and σ ( γ ( O ) ji ) might converge to zero.
Revised: If the conditions of Theorem B.8 are not fulﬁlled, σ ( γ ( O ) ij ) and σ ( γ ( O ) ji ) might converge to zero.

Original: This results in the score lc
Revised: This results in the score lc

Original: ( X i , X j ) being zero, but also σ ( γ ij ) converging to zero.
Revised: ( X i , X j ) being zero, but also σ ( γ ij ) converging to zero.

Original: Hence, we do not get any false positive edge predictions as we have seen in the experiments of Section 4.5.
Revised: Hence, we do not get any false positive edge predictions as we have seen in the experiments of Section 4.5.

Original: For the second case where X i is an ancestor of X j , we cannot give such a guarantee because of Theorem B.2.
Revised: For the second case where X i is an ancestor of X j , we cannot give such a guarantee because of Theorem B.2.

Original: Theorem B.2 states that σ ( θ ij ) converges to one for ancestor-descendant pairs.
Revised: Theorem B.2 states that σ ( θ ij ) converges to one for ancestor-descendant pairs.

Original: However, σ ( θ ji ) is a factor in the gradients of γ ji .
Revised: However, σ ( θ ji ) is a factor in the gradients of γ ji .

Original: This means that if σ ( θ ji ) converges to zero according to Theorem B.2, we cannot guarantee that γ ji converges to the desired value since its gradient becomes zero.
Revised: This means that if σ ( θ ji ) converges to zero according to Theorem B.2, we cannot guarantee that γ ji converges to the desired value since its gradient becomes zero.

Original: Nevertheless, 59 . 2% of the latent confounders in our experiments of
Revised: Nevertheless, 59 . 2% of the latent confounders in our experiments of

Original: Section 4.5 were on ancestor-descendant pairs.
Revised: Section 4.5 were on ancestor-descendant pairs.

Original: ENCO detects a majority of those confounders, showing that ENCO still works on such confounders despite not having guarantees.
Revised: ENCO detects a majority of those confounders, showing that ENCO still works on such confounders despite not having guarantees.

Original: Further, we show in Section C.3 that the confounder scores
Revised: Further, we show in Section C.3 that the confounder scores

Original: lc
Revised: lc

Original: ( X i , X j ) indeed converge to one for detected confounders, and zero for all other edges.
Revised: ( X i , X j ) indeed converge to one for detected confounders, and zero for all other edges.

Original: B. C ONVERGENCE FOR PARTIAL INTERVENTION SETS
Revised: B. C ONVERGENCE FOR PARTIAL INTERVENTION SETS

Original: In Section 4.4, we have experimentally shown that ENCO works on partial intervention sets as well.
Revised: In Section 4.4, we have experimentally shown that ENCO works on partial intervention sets as well.

Original: Here, we will discuss convergence guarantees in the situation when interventions are not provided on all variables.
Revised: Here, we will discuss convergence guarantees in the situation when interventions are not provided on all variables.

Original: We start with discussing the case where, for a graph with N variables, we are given samples of interventions on N − 1 variables.
Revised: We start with discussing the case where, for a graph with N variables, we are given samples of interventions on N − 1 variables.

Original: In this case, we can rely on the previous convergence guarantees discussed in Appendix B.2 with minor modiﬁcations.
Revised: In this case, we can rely on the previous convergence guarantees discussed in Appendix B.2 with minor modiﬁcations.

Original: Speciﬁcally, for the variable X i on which we do not have interventions, the orientation parameters θ i · are only updated by interventions on other variables.
Revised: Speciﬁcally, for the variable X i on which we do not have interventions, the orientation parameters θ i · are only updated by interventions on other variables.

Original: Hence, for this variable X i , the following conditions need to hold instead of Theorem B.1:
Revised: Hence, for this variable X i , the following conditions need to hold instead of Theorem B.1:

Original: • 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:
Revised: • 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: For at least one parent set
Revised: For at least one parent set

Original: [Equation]
Revised: [Equation]

Original: , which has a probability greater than zero to be sampled, this inequality is strictly smaller than zero.
Revised: , which has a probability greater than zero to be sampled, this inequality is strictly smaller than zero.

Original: This condition ensures that θ ij converges to the correct values, where X i is the parent or ancestor of
Revised: This condition ensures that θ ij converges to the correct values, where X i is the parent or ancestor of

Original: X j .
Revised: X j .

Original: Thus, in conclusion, we can provide convergence guarantees if N − 1 interventions are provided.
Revised: Thus, in conclusion, we can provide convergence guarantees if N − 1 interventions are provided.

Original: Next, we can consider the case of having N − 2 interventions.
Revised: Next, we can consider the case of having N − 2 interventions.

Original: With the conditions above, we can ensure that the all orientation parameters are learned, excluding θ ij where X i and X j are the variables for which we have not obtained interventions.
Revised: With the conditions above, we can ensure that the all orientation parameters are learned, excluding θ ij where X i and X j are the variables for which we have not obtained interventions.

Original: In this case, we cannot give strict convergence guarantees for the edge
Revised: In this case, we cannot give strict convergence guarantees for the edge

Original: X i ↔ X j , especially when X i and X j have a direct causal relationship.
Revised: X i ↔ X j , especially when X i and X j have a direct causal relationship.

Original: If X j is the child of X i , we might obtain the edge X j → X i which violates the assumptions in the second and third step of the proof.
Revised: If X j is the child of X i , we might obtain the edge X j → X i which violates the assumptions in the second and third step of the proof.

Original: Therefore, we cannot give guarantees of correctness for incoming/outgoing edges of X i and X j , and might make incorrect predictions of edges between these two variables.
Revised: Therefore, we cannot give guarantees of correctness for incoming/outgoing edges of X i and X j , and might make incorrect predictions of edges between these two variables.

Original: When taking the next step to having M interventions provided, ENCO can create more incorrect predictions.
Revised: When taking the next step to having M interventions provided, ENCO can create more incorrect predictions.

Original: For the variables for which interventions are provided, we can use the same convergence guarantees (Theorem B.1-B.4) since all conditions are independent across variables.
Revised: For the variables for which interventions are provided, we can use the same convergence guarantees (Theorem B.1-B.4) since all conditions are independent across variables.

Original: For variables without interventions, we cannot rely on those.
Revised: For variables without interventions, we cannot rely on those.

Original: While we have observed that learning the missing θ ’s from other interventions give reasonable results, we see a degradation of performance the further the distance is between a node and the closest intervened variable.
Revised: While we have observed that learning the missing θ ’s from other interventions give reasonable results, we see a degradation of performance the further the distance is between a node and the closest intervened variable.

Original: As an example, suppose we have a chain with 5 variables, i.e. X 1 → X 2 → X 3 → X 4 → X 5 , and we are provided with an intervention on X 1 only.
Revised: As an example, suppose we have a chain with 5 variables, i.e. X 1 → X 2 → X 3 → X 4 → X 5 , and we are provided with an intervention on X 1 only.

Original: This allows us to learn the orientation between X 1 and X 2 .
Revised: This allows us to learn the orientation between X 1 and X 2 .

Original: The orientation between Xand X 3 is often learned correctly as well because adding the edge X 2 → X 3 instead of X 3 → X 2 gives a greater decrease in overall log-likelihood, since part of the information from X 3 to predict Xis already included in X 1 .
Revised: The orientation between Xand X 3 is often learned correctly as well because adding the edge X 2 → X 3 instead of X 3 → X 2 gives a greater decrease in overall log-likelihood, since part of the information from X 3 to predict Xis already included in X 1 .

Original: However, the further we go away from X 1 , the less information is shared between the child and the intervened variable.
Revised: However, the further we go away from X 1 , the less information is shared between the child and the intervened variable.

Original: Moreover, the likelihood of a mistake occurring due to limited data further increases.
Revised: Moreover, the likelihood of a mistake occurring due to limited data further increases.

Original: This is why the orientation of the edge X 4 → X 5 is not always learned correctly, which can also cause false positive edges.
Revised: This is why the orientation of the edge X 4 → X 5 is not always learned correctly, which can also cause false positive edges.

Original: Many scenarios of predicting false positive edges can, in theory, be solved by providing an undirected skeleton of the graph, for example, obtained from observational data.
Revised: Many scenarios of predicting false positive edges can, in theory, be solved by providing an undirected skeleton of the graph, for example, obtained from observational data.

Original: Still, one of the cornerstones of ENCO is that it does not assume faithfulness.
Revised: Still, one of the cornerstones of ENCO is that it does not assume faithfulness.

Original: Without faithfulness or any other assumption on the functional form of the causal mechanisms, the correct undirected graph cannot be recovered by any method.
Revised: Without faithfulness or any other assumption on the functional form of the causal mechanisms, the correct undirected graph cannot be recovered by any method.

Original: One of the future directions will be to include faithfulness in ENCO to solve the scenarios mentioned above, although this would imply that we might not be able to recover edges of deterministic variables anymore.
Revised: One of the future directions will be to include faithfulness in ENCO to solve the scenarios mentioned above, although this would imply that we might not be able to recover edges of deterministic variables anymore.

Original: B. E XAMPLE FOR NON - FAITHFUL GRAPHS
Revised: B. E XAMPLE FOR NON - FAITHFUL GRAPHS

Original: Below we give an example of a distribution which is not faithful with respect to its graph structure, but can yet be found by ENCO.
Revised: Below we give an example of a distribution which is not faithful with respect to its graph structure, but can yet be found by ENCO.

Original: Suppose we have a chain of three variables, X 1 → X 2 → X 3 .
Revised: Suppose we have a chain of three variables, X 1 → X 2 → X 3 .

Original: For simplicity, we assume here that all the variables are binary, but the argument can similarly hold for any categorical data.
Revised: For simplicity, we assume here that all the variables are binary, but the argument can similarly hold for any categorical data.

Original: The distribution p ( X 1 ) is an arbitrary function with 0 < p ( X 1 = 1) < 1 , and the other two conditionals are deterministic functions:
Revised: The distribution p ( X 1 ) is an arbitrary function with 0 < p ( X 1 = 1) < 1 , and the other two conditionals are deterministic functions:

Original: p ( X 2 | X 1 ) =
Revised: p ( X 2 | X 1 ) =

Original: δ [ X 2 = X 1 ] , p ( X 3 | X 2 ) =
Revised: δ [ X 2 = X 1 ] , p ( X 3 | X 2 ) =

Original: δ [ X 3 =
Revised: δ [ X 3 =

Original: X 2 ] .
Revised: X 2 ] .

Original: This joint distribution is not faithful to the graph, since X 3 is independent of X 2 given
Revised: This joint distribution is not faithful to the graph, since X 3 is independent of X 2 given

Original: X 1 , which is not implied by the graph.
Revised: X 1 , which is not implied by the graph.

Original: We will now focus our discussion on the edge X 1 → Xto show that, despite the independence, the proposed method ENCO can identify the true parent set of X 3 .
Revised: We will now focus our discussion on the edge X 1 → Xto show that, despite the independence, the proposed method ENCO can identify the true parent set of X 3 .

Original: The ﬁrst step of ENCO is to ﬁt the neural networks to the observational distributions, which include p ( X 3 ) , p ( X 3 | X 1 ) , p ( X 3 | X 2 ) , p ( X 3 | X 1 , X 2 ) .
Revised: The ﬁrst step of ENCO is to ﬁt the neural networks to the observational distributions, which include p ( X 3 ) , p ( X 3 | X 1 ) , p ( X 3 | X 2 ) , p ( X 3 | X 1 , X 2 ) .

Original: Now, the update of the edge parameter γ 13 under interventions on X 2 can be summarized as follows, where we marginalize out over graph samples:
Revised: Now, the update of the edge parameter γ 13 under interventions on X 2 can be summarized as follows, where we marginalize out over graph samples:

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: where the samples X are sampled from the graph under interventions on X 2 .
Revised: where the samples X are sampled from the graph under interventions on X 2 .

Original: Intuitively, the gradient points towards increasing the probability of the edge X 1 → X 3 if adding X 1 to the conditionals improves the log-likelihood estimate of X 3 in both graph samples, i.e. where we have the edge
Revised: Intuitively, the gradient points towards increasing the probability of the edge X 1 → X 3 if adding X 1 to the conditionals improves the log-likelihood estimate of X 3 in both graph samples, i.e. where we have the edge

Original: X 2 → X 3 or not.
Revised: X 2 → X 3 or not.

Original: Thus, to show that the gradient points towards decreasing the probability of the edge X 1 → X 3 , we need to show that both of the above log-likelihood differences are greater than zero (note that positive gradients lead to a decrease since we minimize the objective).
Revised: Thus, to show that the gradient points towards decreasing the probability of the edge X 1 → X 3 , we need to show that both of the above log-likelihood differences are greater than zero (note that positive gradients lead to a decrease since we minimize the objective).

Original: In the ﬁrst difference, E X [ − log p ( X 3 | X 1 , X 2 ) + log p ( X 3 | X 2 )] , we know that p ( X 3 | X 2 ) is the optimal estimator, since the ground truth data is generated via this conditional.
Revised: In the ﬁrst difference, E X [ − log p ( X 3 | X 1 , X 2 ) + log p ( X 3 | X 2 )] , we know that p ( X 3 | X 2 ) is the optimal estimator, since the ground truth data is generated via this conditional.

Original: Thus, independent of what function the network has learned for p ( X 3 | X 1 , X 2 ) , the difference above can be only greater or equals to zero.
Revised: Thus, independent of what function the network has learned for p ( X 3 | X 1 , X 2 ) , the difference above can be only greater or equals to zero.

Original: Note that for p ( X 3 | X 1 , X 2 ) , there are value combinations that have never been observed, i.e. X 1 (cid:54) =
Revised: Note that for p ( X 3 | X 1 , X 2 ) , there are value combinations that have never been observed, i.e. X 1 (cid:54) =

Original: X 2 , such that in practice we can’t guarantee a speciﬁc distribution to be learned for such conditionals.
Revised: X 2 , such that in practice we can’t guarantee a speciﬁc distribution to be learned for such conditionals.

Original: Still, this does not constitute a problem for ﬁnding the graph as shown above.
Revised: Still, this does not constitute a problem for ﬁnding the graph as shown above.

Original: The second difference, E X [ − log p ( X 3 | X 1 ) + log p ( X 3 )] , can be similarly reasoned.
Revised: The second difference, E X [ − log p ( X 3 | X 1 ) + log p ( X 3 )] , can be similarly reasoned.

Original: Since X 1 is independent of X 3 under interventions on X 2 , p ( X 3 | X 1 ) cannot lead to a better estimator than p ( X 3 ) , even when trained on the interventional data.
Revised: Since X 1 is independent of X 3 under interventions on X 2 , p ( X 3 | X 1 ) cannot lead to a better estimator than p ( X 3 ) , even when trained on the interventional data.

Original: However, since p ( X 3 | X 1 ) was trained on observational data, where there exist a strong correlation between X 1 and X 3 , this estimator must be strictly worse than p ( X 3 ) .
Revised: However, since p ( X 3 | X 1 ) was trained on observational data, where there exist a strong correlation between X 1 and X 3 , this estimator must be strictly worse than p ( X 3 ) .

Original: Hence, this difference will be strictly positive.
Revised: Hence, this difference will be strictly positive.

Original: To summarize, under interventions on X 2 , the edge X 1 → X 3 will be trained towards decreasing its probability.
Revised: To summarize, under interventions on X 2 , the edge X 1 → X 3 will be trained towards decreasing its probability.

Original: Further, under interventions on X 1 , the effect of X 1 on X 3 can be fully expressed by conditioning on X 2 , making this gradient going to zero when the edge probability X 2 → X 3 goes towards one.
Revised: Further, under interventions on X 1 , the effect of X 1 on X 3 can be fully expressed by conditioning on X 2 , making this gradient going to zero when the edge probability X 2 → X 3 goes towards one.

Original: For the edge X 2 → X 3 itself, the same reasoning as here can be followed such that independent of whether the edge X 1 → X 3 is included in the graph or not, conditioning X 3 on Xcan only lead to an improvement in its estimator.
Revised: For the edge X 2 → X 3 itself, the same reasoning as here can be followed such that independent of whether the edge X 1 → X 3 is included in the graph or not, conditioning X 3 on Xcan only lead to an improvement in its estimator.

Original: Therefore, ENCO is able to ﬁnd the correct graph despite it not being faithful.
Revised: Therefore, ENCO is able to ﬁnd the correct graph despite it not being faithful.

Original: bidiag chain collider
Revised: bidiag chain collider

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X
Revised: X

Original: Xfull jungle random
Revised: Xfull jungle random

Original: X
Revised: X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X X
Revised: X X

Original: X
Revised: X

Original: C E XPERIMENTAL DETAILS
Revised: C E XPERIMENTAL DETAILS

Original: The following section gives an overview of the hyperparameters used across experiments.
Revised: The following section gives an overview of the hyperparameters used across experiments.

Original: Additionally, we discuss details of the graph generation and the learning process of different algorithms.
Revised: Additionally, we discuss details of the graph generation and the learning process of different algorithms.

Original: C. C OMMON GRAPH STRUCTURE EXPERIMENTS C.1. D ATASETS
Revised: C. C OMMON GRAPH STRUCTURE EXPERIMENTS C.1. D ATASETS

Original: Graph generation The six common graph structures we have used for the experiments in Tableare visualized in Figure 10.
Revised: Graph generation The six common graph structures we have used for the experiments in Tableare visualized in Figure 10.

Original: In the graph bidiag , a variable X i has X i − 

Revised: In the graph bidiag , a variable X i has X i −

Original: and X i − 1 as parents, and consequently X i +

Revised: and X i − 1 as parents, and consequently X i +

Original: and X i +2 as children.
Revised: and X i +2 as children.

Original: Hence, this graph represents a chain with 2-hop connections.
Revised: Hence, this graph represents a chain with 2-hop connections.

Original: The graph chain is a bidiag with a single hop, meaning that X i is the parent of X i − 1 but not
Revised: The graph chain is a bidiag with a single hop, meaning that X i is the parent of X i − 1 but not

Original: X i − 2 .
Revised: X i − 2 .

Original: In the graph collider , the variable X N has all other variables, X − i , as parents.
Revised: In the graph collider , the variable X N has all other variables, X − i , as parents.

Original: In the graph full , the parents of a variable X i are all previous variables: pa ( X i ) =
Revised: In the graph full , the parents of a variable X i are all previous variables: pa ( X i ) =

Original: { X 1 , X 2 , ...,
Revised: { X 1 , X 2 , ...,

Original: X i − 1 } .
Revised: X i − 1 } .

Original: Hence, it is the densest connected graph possible.
Revised: Hence, it is the densest connected graph possible.

Original: The graph jungle represents a binary tree where a node is also connected to its parent’s parent.
Revised: The graph jungle represents a binary tree where a node is also connected to its parent’s parent.

Original: Finally, the graph random follows a randomly sampled adjacency matrix.
Revised: Finally, the graph random follows a randomly sampled adjacency matrix.

Original: For every possible pair of variables X i , X j , we sample an edge with a likelihood of 0 . 3 .
Revised: For every possible pair of variables X i , X j , we sample an edge with a likelihood of 0 . 3 .

Original: To determine the orientation of the edges, we assume the causal ordering of X i (cid:31)
Revised: To determine the orientation of the edges, we assume the causal ordering of X i (cid:31)

Original: X i +1 .
Revised: X i +1 .

Original: In other words, if we have an edge between X i and X j , it is oriented X
Revised: In other words, if we have an edge between X i and X j , it is oriented X

Original: i
Revised: i

Original: →
Revised: →

Original: X j is i <
Revised: X j is i <

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

Original: Conditional distributions In the generated graphs, we use categorical variables that each havecategories.
Revised: Conditional distributions In the generated graphs, we use categorical variables that each havecategories.

Original: To model the ground-truth conditional distributions, we use randomly initialized neural networks.
Revised: To model the ground-truth conditional distributions, we use randomly initialized neural networks.

Original: Speciﬁcally, we use MLPs of two layers where the categorical inputs are represented by embedding vectors.
Revised: Speciﬁcally, we use MLPs of two layers where the categorical inputs are represented by embedding vectors.

Original: For a variable X i with M parents, we stack the M embedding vectors to form the input to the following MLPs.
Revised: For a variable X i with M parents, we stack the M embedding vectors to form the input to the following MLPs.

Original: Each embedding has a dimensionality of 4, hence the input size to the ﬁrst linear layer is 4 M .
Revised: Each embedding has a dimensionality of 4, hence the input size to the ﬁrst linear layer is 4 M .

Original: The hidden size of the layers is 48, and we use a LeakyReLU activation function in between the two linear layers.
Revised: The hidden size of the layers is 48, and we use a LeakyReLU activation function in between the two linear layers.

Original: Finally, a softmax activation function is used on
Revised: Finally, a softmax activation function is used on

Original: [Table]
Revised: [Table]

Original: the output to obtain a distribution over the 10 categories.
Revised: the output to obtain a distribution over the 10 categories.

Original: The MLP and hyperparameters have been chosen based on the design of networks used in ENCO, SDI (Ke et al., 2019) and DCDI (Brouillard et al., 2020).
Revised: The MLP and hyperparameters have been chosen based on the design of networks used in ENCO, SDI (Ke et al., 2019) and DCDI (Brouillard et al., 2020).

Original: For the initialization of the networks
Revised: For the initialization of the networks

Original: [Table]
Revised: [Table]

Original: we follow Ke et al.
Revised: we follow Ke et al.

Original: and use the orthogonal initialize with a gain of 2 . 5 .
Revised: and use the orthogonal initialize with a gain of 2 . 5 .

Original: The biases are thereby initialized uniformly between − 0 .
Revised: The biases are thereby initialized uniformly between − 0 .

Original: 5 and 0 .
Revised: 5 and 0 .

Original: This way, we obtain non-trivial, random distributions.
Revised: This way, we obtain non-trivial, random distributions.

Original: Experiments with different synthetic distributions are provided in Appendix D.7.
Revised: Experiments with different synthetic distributions are provided in Appendix D.7.

Original: C.1. M ETHODS AND HYPERPARAMETERS
Revised: C.1. M ETHODS AND HYPERPARAMETERS

Original: Baseline implementation We used existing implementations to run the baselines GIES (Hauser &
Revised: Baseline implementation We used existing implementations to run the baselines GIES (Hauser &

Original: Bühlmann, 2012), IGSP (Wang et al., 2017), GES (Chickering, 2002) and DCDI (Brouillard et al., 2020).
Revised: Bühlmann, 2012), IGSP (Wang et al., 2017), GES (Chickering, 2002) and DCDI (Brouillard et al., 2020).

Original: For GIES, we used the implementation from the R package pcalg 2 .
Revised: For GIES, we used the implementation from the R package pcalg 2 .

Original: To run categorical data, we used the GaussL0penIntScore score function.
Revised: To run categorical data, we used the GaussL0penIntScore score function.

Original: For IGSP, we used the implementation of the python package causaldag 3 .
Revised: For IGSP, we used the implementation of the python package causaldag 3 .

Original: As IGSP uses conditional independence tests in its score function, we cast the categorical data into continuous space ﬁrst and experiment with different kernel-based independence tests.
Revised: As IGSP uses conditional independence tests in its score function, we cast the categorical data into continuous space ﬁrst and experiment with different kernel-based independence tests.

Original: Due to its long runtime for large dataset sizes, we limit the interventional and observational data set size to 25k.
Revised: Due to its long runtime for large dataset sizes, we limit the interventional and observational data set size to 25k.

Original: Larger dataset sizes did not show any signiﬁcant improvements.
Revised: Larger dataset sizes did not show any signiﬁcant improvements.

Original: For details on the observational GES experiments, see Section D.6.
Revised: For details on the observational GES experiments, see Section D.6.

Original: Finally, we have used the original python code for DCDI published by the authors 4 .
Revised: Finally, we have used the original python code for DCDI published by the authors 4 .

Original: We have added the same neural networks used by
Revised: We have added the same neural networks used by

Original: ENCO into the framework to perform structure learning on categorical data.
Revised: ENCO into the framework to perform structure learning on categorical data.

Original: Bugs in the original code were corrected to the best of our knowledge.
Revised: Bugs in the original code were corrected to the best of our knowledge.

Original: Since SDI (Ke et al., 2019) has a similar learning structure as ENCO, we have implemented it in the same code base as ENCO.
Revised: Since SDI (Ke et al., 2019) has a similar learning structure as ENCO, we have implemented it in the same code base as ENCO.

Original: This allows us to compare the learning algorithms under exact same perquisites.
Revised: This allows us to compare the learning algorithms under exact same perquisites.

Original: Further, all methods with neural networks used the deep learning framework PyTorch (Paszke et al., 2019) which ensures a fair run time comparison across methods.
Revised: Further, all methods with neural networks used the deep learning framework PyTorch (Paszke et al., 2019) which ensures a fair run time comparison across methods.

Original: Hyperparameters To ensure a fair comparison, we performed a hyperparameter search for all methods.
Revised: Hyperparameters To ensure a fair comparison, we performed a hyperparameter search for all methods.

Original: The hyperparameter search was performed on a hold-out set of graphs containing two of each graph structure.
Revised: The hyperparameter search was performed on a hold-out set of graphs containing two of each graph structure.

Original: GIES We performed a hyperparameter search over the regularizer values λ ∈ { 0 . 01 , 0 . 02 , 0 .
Revised: GIES We performed a hyperparameter search over the regularizer values λ ∈ { 0 . 01 , 0 . 02 , 0 .

Original: The value obtaining the best results in terms of structural hamming distance (SHD) was λ = 20 .
Revised: The value obtaining the best results in terms of structural hamming distance (SHD) was λ = 20 .

Original: The average run time of GIES was 2mins per graph.
Revised: The average run time of GIES was 2mins per graph.

Original: [Figure]
Revised: [Figure]

Original: IGSP We experimented with two different conditional independence tests, kci and hsic , and different cutoff values α
Revised: IGSP We experimented with two different conditional independence tests, kci and hsic , and different cutoff values α

Original: =
Revised: =

Original: 1 e- 5 , 1 e- 4 , 1 e-
Revised: 1 e- 5 , 1 e- 4 , 1 e-

Original: 3 , 1 e- 2 } .
Revised: 3 , 1 e- 2 } .

Original: The best hyperparameter setting was kci with α = 1 e- 3 .
Revised: The best hyperparameter setting was kci with α = 1 e- 3 .

Original: The average run time of IGSP was 13mins.
Revised: The average run time of IGSP was 13mins.

Original: SDI We focused the hyperparameter search for SDI on its two regularizers, λ sparse and λ DAG , as well as its learning rate for γ .
Revised: SDI We focused the hyperparameter search for SDI on its two regularizers, λ sparse and λ DAG , as well as its learning rate for γ .

Original: The other hyperparameters with respect to the neural networks were kept the same as ENCO for a fair comparison.
Revised: The other hyperparameters with respect to the neural networks were kept the same as ENCO for a fair comparison.

Original: We show all details of the hyperparameter search in Table 4.
Revised: We show all details of the hyperparameter search in Table 4.

Original: The best combination of regularizers found was λ sparse = 0 .
Revised: The best combination of regularizers found was λ sparse = 0 .

Original: 02 and λ DAG = 0 .
Revised: 02 and λ DAG = 0 .

Original: Lower values of λ sparse lead to more false positives, especially in sparse graphs, while a lower value of λ DAG caused many two-variable loops.
Revised: Lower values of λ sparse lead to more false positives, especially in sparse graphs, while a lower value of λ DAG caused many two-variable loops.

Original: Compared to the reported hyperparameter by Ke et al.
Revised: Compared to the reported hyperparameter by Ke et al.

Original: ( λ DAG = 0 .
Revised: ( λ DAG = 0 .

Original: 5 , λ sparse = 0 . 1 ), we found a lower sparsity regularizer to work better.
Revised: 5 , λ sparse = 0 . 1 ), we found a lower sparsity regularizer to work better.

Original: This is likely because of testing SDI on larger graphs.
Revised: This is likely because of testing SDI on larger graphs.

Original: In contrast to ENCO, SDI needed a lower learning rate for γ due to its higher variance gradient estimators.
Revised: In contrast to ENCO, SDI needed a lower learning rate for γ due to its higher variance gradient estimators.

Original: To compensate for it, we ran it for 50 instead of 30 epochs.
Revised: To compensate for it, we ran it for 50 instead of 30 epochs.

Original: In general,
Revised: In general,

Original: SDI achieved lower scores than in the original experiments by Ke et al.
Revised: SDI achieved lower scores than in the original experiments by Ke et al.

Original: (2019) which was because of the larger graph size and smaller dataset size.
Revised: (2019) which was because of the larger graph size and smaller dataset size.

Original: The average run time of SDI was 4mins per graph.
Revised: The average run time of SDI was 4mins per graph.

Original: DCDI
Revised: DCDI

Original: The most crucial hyperparameter in DCDI was the initialization of the constraint factor µ 0 .
Revised: The most crucial hyperparameter in DCDI was the initialization of the constraint factor µ 0 .

Original: We experimented with a range of µ 0 ∈ { 1 e- 10 , 1 e- 9 , 1 e- 8 , 1 e- 7 , 1 e-
Revised: We experimented with a range of µ 0 ∈ { 1 e- 10 , 1 e- 9 , 1 e- 8 , 1 e- 7 , 1 e-

Original: 6 , 1 e- 5 }
Revised: 6 , 1 e- 5 }

Original: and found µ 

Revised: and found µ

Original: = 

Revised: =

Original: e-
Revised: e-

Original: 9 to work best.
Revised: 9 to work best.

Original: This is close to the reported value of 1 e- 8 by Brouillard et al.
Revised: This is close to the reported value of 1 e- 8 by Brouillard et al.

Original: Higher values lead to empty graphs, while lower values slowed down the optimization.
Revised: Higher values lead to empty graphs, while lower values slowed down the optimization.

Original: Additionally, we search over the regularizer hyperparameter λ ∈ { 1 e- 3 , 1 e- 2 , 1 e- 1 , 1 . 0 , 10 . 0 } where we found λ = 0 .
Revised: Additionally, we search over the regularizer hyperparameter λ ∈ { 1 e- 3 , 1 e- 2 , 1 e- 1 , 1 . 0 , 10 . 0 } where we found λ = 0 .

Original: 1 , which is the same value used by Brouillard et al.
Revised: 1 , which is the same value used by Brouillard et al.

Original: We stop the search after the Lagrangian constraint is below 1 e- 7 , following Brouillard et al.
Revised: We stop the search after the Lagrangian constraint is below 1 e- 7 , following Brouillard et al.

Original: (2020), or 50k iterations have been used which was sufﬁcient to converge on all graphs.
Revised: (2020), or 50k iterations have been used which was sufﬁcient to converge on all graphs.

Original: We have experimented with using weight decay to prevent overﬁtting, but did not ﬁnd it to provide any gain in terms of structure learning performance.
Revised: We have experimented with using weight decay to prevent overﬁtting, but did not ﬁnd it to provide any gain in terms of structure learning performance.

Original: The average run time of DCDI wasminutes.
Revised: The average run time of DCDI wasminutes.

Original: ENCO We outline the hyperparameters of ENCO in Table 4.
Revised: ENCO We outline the hyperparameters of ENCO in Table 4.

Original: As discussed in Section 3.4, the most crucial hyperparameter in ENCO is the sparsity regularizer λ sparse .
Revised: As discussed in Section 3.4, the most crucial hyperparameter in ENCO is the sparsity regularizer λ sparse .

Original: The larger it is, the faster ENCO converges, but at the same time might miss edges in the ground-truth graph.
Revised: The larger it is, the faster ENCO converges, but at the same time might miss edges in the ground-truth graph.

Original: [Table]
Revised: [Table]

Original: Lower values allow the detection of more edges for the price of longer training times.
Revised: Lower values allow the detection of more edges for the price of longer training times.

Original: We have found that for the graph structures given, only the graph full was sensitive to the value of λ sparse where λ sparse = 0 .
Revised: We have found that for the graph structures given, only the graph full was sensitive to the value of λ sparse where λ sparse = 0 .

Original: 002 and λ sparse = 0 .
Revised: 002 and λ sparse = 0 .

Original: 004 performed almost equally well.
Revised: 004 performed almost equally well.

Original: In comparison to SDI, ENCO can make use of larger learning rates due to lower variance gradient estimators.
Revised: In comparison to SDI, ENCO can make use of larger learning rates due to lower variance gradient estimators.

Original: Especially for θ , we have noticed that high learning rates are beneﬁcial.
Revised: Especially for θ , we have noticed that high learning rates are beneﬁcial.

Original: This is in line with our theoretical guarantees which require the orientation parameters to converge ﬁrst.
Revised: This is in line with our theoretical guarantees which require the orientation parameters to converge ﬁrst.

Original: We use the Adam optimizer for γ and θ with the β -hyperparameters (0 . 9 , 0 . 9) and (0 . 9 , 0 . 999) respectively.
Revised: We use the Adam optimizer for γ and θ with the β -hyperparameters (0 . 9 , 0 . 9) and (0 . 9 , 0 . 999) respectively.

Original: A lower β 2 hyperparameter for γ allows the second momentum to adapt faster to a change of gradient scale which happens for initial false positive predictions.
Revised: A lower β 2 hyperparameter for γ allows the second momentum to adapt faster to a change of gradient scale which happens for initial false positive predictions.

Original: The average run time of ENCO was 2mins per graph.
Revised: The average run time of ENCO was 2mins per graph.

Original: The algorithm could be sped up even more by reducing the number of graph samples K and model ﬁtting iterations.
Revised: The algorithm could be sped up even more by reducing the number of graph samples K and model ﬁtting iterations.

Original: However, for graphs of larger than 100 nodes, K = 100 and longer model ﬁtting times showed to be beneﬁcial.
Revised: However, for graphs of larger than 100 nodes, K = 100 and longer model ﬁtting times showed to be beneﬁcial.

Original: The learning curves in terms of recall and precision are shown in Figure 10.
Revised: The learning curves in terms of recall and precision are shown in Figure 10.

Original: Enforcing acyclicity ENCO is guaranteed to converge to acyclic graphs in the data limit; arguably, an assumption that does not always hold.
Revised: Enforcing acyclicity ENCO is guaranteed to converge to acyclic graphs in the data limit; arguably, an assumption that does not always hold.

Original: In the presence of cycles, which can occur especially when low data is available, a simple heuristic is to keep the graph, which maximizes the orientation probabilities.
Revised: In the presence of cycles, which can occur especially when low data is available, a simple heuristic is to keep the graph, which maximizes the orientation probabilities.

Original: Speciﬁcally, we aim to ﬁnd the order O ∈ S N , where S N represents the set of all permutations from 1 to the number of variables N , for which we maximize the following objective:
Revised: Speciﬁcally, we aim to ﬁnd the order O ∈ S N , where S N represents the set of all permutations from 1 to the number of variables N , for which we maximize the following objective:

Original: [Equation]
Revised: [Equation]

Original: For small cycles it is easy to do this exhaustively by checking all permutations.
Revised: For small cycles it is easy to do this exhaustively by checking all permutations.

Original: For larger cycles, we apply a simple greedy search that works just as well.
Revised: For larger cycles, we apply a simple greedy search that works just as well.

Original: Once the order ˆ O has been found, we remove all edges X i → X j where i comes after j in ˆ O .
Revised: Once the order ˆ O has been found, we remove all edges X i → X j where i comes after j in ˆ O .

Original: This guarantees to result in an acyclic graph.
Revised: This guarantees to result in an acyclic graph.

Original: The intuition behind this heuristic is the following.
Revised: The intuition behind this heuristic is the following.

Original: Cycles are often caused by a single orientation pair being incorrect due to noise in the interventional data.
Revised: Cycles are often caused by a single orientation pair being incorrect due to noise in the interventional data.

Original: For example, in a chain X 1 → X 2 →
Revised: For example, in a chain X 1 → X 2 →

Original: X 3 → X 4 → X 5 , it can happen that the orientation parameter θ 14 is incorrectly learned as orientation of the edge between X 1 and X 4 as X 4 → X 1 if the interventional data on X 4 does not show the independence of X 1 and X 4 .
Revised: X 3 → X 4 → X 5 , it can happen that the orientation parameter θ 14 is incorrectly learned as orientation of the edge between X 1 and X 4 as X 4 → X 1 if the interventional data on X 4 does not show the independence of X 1 and X 4 .

Original: However, most other orientation parameters, e.g. θ 12 , θ 13 , θ 24 , θ 34 , etc., have been likely learned correctly.
Revised: However, most other orientation parameters, e.g. θ 12 , θ 13 , θ 24 , θ 34 , etc., have been likely learned correctly.

Original: Thus, it is easy to spot that θ 14 is an outlier, and this is what the simple heuristic above implements.
Revised: Thus, it is easy to spot that θ 14 is an outlier, and this is what the simple heuristic above implements.

Original: Results Besides the structural hamming distance, a common alternative metric is structural intervention distance (SID) (Peters & Bühlmann, 2015).
Revised: Results Besides the structural hamming distance, a common alternative metric is structural intervention distance (SID) (Peters & Bühlmann, 2015).

Original: In contrast to SHD, SID quantiﬁes the closeness between two DAGs in terms of their corresponding causal inference statements.
Revised: In contrast to SHD, SID quantiﬁes the closeness between two DAGs in terms of their corresponding causal inference statements.

Original: Hence, it is suited for comparing causal discovery methods.
Revised: Hence, it is suited for comparing causal discovery methods.

Original: The results of the experiments on the synthetic graphs in terms of SID are shown in Table 5, and show a similar trend as before, namely that ENCO is outperforming all baselines.
Revised: The results of the experiments on the synthetic graphs in terms of SID are shown in Table 5, and show a similar trend as before, namely that ENCO is outperforming all baselines.

Original: C. S CALABILITY EXPERIMENTS
Revised: C. S CALABILITY EXPERIMENTS

Original: Graph generation For generating the graphs, we use the same strategy as for the graph random in the previous experiments.
Revised: Graph generation For generating the graphs, we use the same strategy as for the graph random in the previous experiments.

Original: The probability of sampling an edge is set to 8 /N , meaning that on
Revised: The probability of sampling an edge is set to 8 /N , meaning that on

Original: [Figure]
Revised: [Figure]

Original: average, every node has 8 in- and outgoing edges.
Revised: average, every node has 8 in- and outgoing edges.

Original: We limit the number of parents to 10 per node since, otherwise, we cannot guarantee that the randomly sampled distributions take all parents faithfully into account.
Revised: We limit the number of parents to 10 per node since, otherwise, we cannot guarantee that the randomly sampled distributions take all parents faithfully into account.

Original: This is also in line with the real-world inspired graphs of the BnLearn repository, which have a maximum of 6 parents.
Revised: This is also in line with the real-world inspired graphs of the BnLearn repository, which have a maximum of 6 parents.

Original: To give an intuition on the complexity of such graphs, we show an example graph of 100 nodes in Figure 12a.
Revised: To give an intuition on the complexity of such graphs, we show an example graph of 100 nodes in Figure 12a.

Original: Accordingly to the number of variables, we have increased the data set size to 4096 samples per intervention, and 100k observational samples.
Revised: Accordingly to the number of variables, we have increased the data set size to 4096 samples per intervention, and 100k observational samples.

Original: We did not apply the order heuristic on the predictions, since ENCO was able to recover acyclic graphs by itself with the given data.
Revised: We did not apply the order heuristic on the predictions, since ENCO was able to recover acyclic graphs by itself with the given data.

Original: γ -freezing stage For ENCO, one challenge of large graphs is that the orientation parameters θ are updated very sparsely.
Revised: γ -freezing stage For ENCO, one challenge of large graphs is that the orientation parameters θ are updated very sparsely.

Original: The gradients for θ ij require data from an intervention on one of its adjacent nodes X i or X j , which we evaluate less frequently with increasing N as we iterate over interventions on all N nodes.
Revised: The gradients for θ ij require data from an intervention on one of its adjacent nodes X i or X j , which we evaluate less frequently with increasing N as we iterate over interventions on all N nodes.

Original: Hence, we require more iterations/epochs just for training the orientation parameters while wasting a lot of computational resources.
Revised: Hence, we require more iterations/epochs just for training the orientation parameters while wasting a lot of computational resources.

Original: To accelerate training of large graphs, we freeze γ in every second graph ﬁtting stage.
Revised: To accelerate training of large graphs, we freeze γ in every second graph ﬁtting stage.

Original: Updating only θ allows us to use the same graph sample C − ij for both L X i → X j ( X j ) and L X i (cid:54)→ X j ( X j ) since the log-likelihood estimate of X j only needs to be evaluated for θ ij .
Revised: Updating only θ allows us to use the same graph sample C − ij for both L X i → X j ( X j ) and L X i (cid:54)→ X j ( X j ) since the log-likelihood estimate of X j only needs to be evaluated for θ ij .

Original: With this gradient estimator, we experience that as little as 4 graph samples are sufﬁcient to obtain a reasonable gradient variance.
Revised: With this gradient estimator, we experience that as little as 4 graph samples are sufﬁcient to obtain a reasonable gradient variance.

Original: Hence, it is possible to perform more gradient updates of θ in the same computation time.
Revised: Hence, it is possible to perform more gradient updates of θ in the same computation time.

Original: Note that this is estimator not efﬁcient when training γ as we require different C − ij samples for every i .
Revised: Note that this is estimator not efﬁcient when training γ as we require different C − ij samples for every i .

Original: In experiments, we alternate the standard graph ﬁtting step with this pure θ -training stage.
Revised: In experiments, we alternate the standard graph ﬁtting step with this pure θ -training stage.

Original: We want to emphasize that this approach can also be used for small graphs obtaining similar results as in Table 1.
Revised: We want to emphasize that this approach can also be used for small graphs obtaining similar results as in Table 1.

Original: However, it is less needed because the orientation parameters are more frequently updated in the ﬁrst place.
Revised: However, it is less needed because the orientation parameters are more frequently updated in the ﬁrst place.

Original: Such an approach is not possible for the baselines, SDI and DCDI, because they do not model the orientation as a separate variable.
Revised: Such an approach is not possible for the baselines, SDI and DCDI, because they do not model the orientation as a separate variable.

Original: Hyperparameters To experiment with large graphs, we mostly keep to the same hyperparameters as reported in Section C.1.
Revised: Hyperparameters To experiment with large graphs, we mostly keep to the same hyperparameters as reported in Section C.1.

Original: However, all methods showed to gain by a small hyperparameter search.
Revised: However, all methods showed to gain by a small hyperparameter search.

Original: For SDI and ENCO, we increase the number of distribution ﬁtting iterations as the neural networks need to model a larger set of possible parents.
Revised: For SDI and ENCO, we increase the number of distribution ﬁtting iterations as the neural networks need to model a larger set of possible parents.

Original: We also increase the learning rate of γ to 2 e- 2 .
Revised: We also increase the learning rate of γ to 2 e- 2 .

Original: However, while SDI reaches better performance with the increased learning rate at epoch 30, it showed to perform worse when training for longer.
Revised: However, while SDI reaches better performance with the increased learning rate at epoch 30, it showed to perform worse when training for longer.

Original: This indicates that high learning rates can lead to local minima in SDI.
Revised: This indicates that high learning rates can lead to local minima in SDI.

Original: Additionally, we noticed that a slightly higher sparsity regularizer improved convergence speed for ENCO while SDI did not improve with a higher sparsity regularizer.
Revised: Additionally, we noticed that a slightly higher sparsity regularizer improved convergence speed for ENCO while SDI did not improve with a higher sparsity regularizer.

Original: Tableshows a hyperparameter overview of ENCO on large-scale graphs, and Figure 12b the learning curve on graphs of 1 , 000 nodes.
Revised: Tableshows a hyperparameter overview of ENCO on large-scale graphs, and Figure 12b the learning curve on graphs of 1 , 000 nodes.

Original: For DCDI, we noticed that the hyperparameters around the Lagrangian constraint needed to be carefully ﬁne-tuned.
Revised: For DCDI, we noticed that the hyperparameters around the Lagrangian constraint needed to be carefully ﬁne-tuned.

Original: The Lagrangian constraint can reach values larger than possible to represent with double, and starts with 8 e 216 for graphs of 1 , 000 nodes.
Revised: The Lagrangian constraint can reach values larger than possible to represent with double, and starts with 8 e 216 for graphs of 1 , 000 nodes.

Original: Following Brouillard et al.
Revised: Following Brouillard et al.

Original: (2020), we normalize the constraint by the value after initialization, which gives us a more reasonable value to start learning.
Revised: (2020), we normalize the constraint by the value after initialization, which gives us a more reasonable value to start learning.

Original: We performed another hyperparameter search on µ 0 , noticed however that it did not have a major impact.
Revised: We performed another hyperparameter search on µ 0 , noticed however that it did not have a major impact.

Original: In the run time of ENCO, DCDI just starts to increase the weighting factor of the augmented Lagrangian while the DAG constraint is lower than 1 e- 10 for the smallest graph.
Revised: In the run time of ENCO, DCDI just starts to increase the weighting factor of the augmented Lagrangian while the DAG constraint is lower than 1 e- 10 for the smallest graph.

Original: The best value found was µ 0 = 1 e- 7 .
Revised: The best value found was µ 0 = 1 e- 7 .

Original: [Table]
Revised: [Table]

Original: Results For clarity, we report the results of all methods below.
Revised: Results For clarity, we report the results of all methods below.

Original: The exact values might not be easily readable in Figure 3 due to large differences in performance.
Revised: The exact values might not be easily readable in Figure 3 due to large differences in performance.

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

Original: C. L ATENT CONFOUNDER EXPERIMENTS
Revised: C. L ATENT CONFOUNDER EXPERIMENTS

Original: Graph generation
Revised: Graph generation

Original: The graphs used for testing the latent confounding strategy are based on the random graphs from Section 4.2.
Revised: The graphs used for testing the latent confounding strategy are based on the random graphs from Section 4.2.

Original: We use graphs of 25 nodes, and add 5 extra nodes that represent latent confounders.
Revised: We use graphs of 25 nodes, and add 5 extra nodes that represent latent confounders.

Original: Each latent confounder X l is connected to two randomly sampled nodes X i ,
Revised: Each latent confounder X l is connected to two randomly sampled nodes X i ,

Original: X j that do not have a direct connection.
Revised: X j that do not have a direct connection.

Original: However, X i and X j can be an ancestor-descendant pair and have any other (shared) parent set (see Figure 13a).
Revised: However, X i and X j can be an ancestor-descendant pair and have any other (shared) parent set (see Figure 13a).

Original: In the adjacency matrix, we add the edges
Revised: In the adjacency matrix, we add the edges

Original: X l → X i and X l → X j , and perform the data generation as for the previous graphs.
Revised: X l → X i and X l → X j , and perform the data generation as for the previous graphs.

Original: After data generation, we remove the 5 latent confounders from both observational and interventional data.
Revised: After data generation, we remove the 5 latent confounders from both observational and interventional data.

Original: The task is to learn the graph structure of the remaining 25 observable variables, as well as detecting whether there exists a latent confounder between any pair of variables.
Revised: The task is to learn the graph structure of the remaining 25 observable variables, as well as detecting whether there exists a latent confounder between any pair of variables.

Original: We use the same setup in terms of dataset size as before for the observational samples, namely 5k, but increased the samples per intervention to 512.
Revised: We use the same setup in terms of dataset size as before for the observational samples, namely 5k, but increased the samples per intervention to 512.

Original: Little interventional data showed to cause a high variance in the interventional gradients, γ ( I ) ij , which is why more false positives occured.
Revised: Little interventional data showed to cause a high variance in the interventional gradients, γ ( I ) ij , which is why more false positives occured.

Original: The results in for the limited data withinterventions, and results in the data limit, i.e. for 10k interventional samples and 100k observational samples, are shown in Table 8.
Revised: The results in for the limited data withinterventions, and results in the data limit, i.e. for 10k interventional samples and 100k observational samples, are shown in Table 8.

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: the detection of latent confounders in ENCO is not sensitive to the speciﬁc value of τ .
Revised: the detection of latent confounders in ENCO is not sensitive to the speciﬁc value of τ .

Original: Baselines None of our previous continuous optimization baselines, i.e. , SDI and DCDI, are able to deal with latent confounders.
Revised: Baselines None of our previous continuous optimization baselines, i.e. , SDI and DCDI, are able to deal with latent confounders.

Original: To the best of our knowledge, other methods that are able to handle latent confounders commonly take assumptions that do not hold in our experimental setup.
Revised: To the best of our knowledge, other methods that are able to handle latent confounders commonly take assumptions that do not hold in our experimental setup.

Original: Further, most methods are able to deal with latent confounders in the sense that they obtain the correct results despite latent confounding being present.
Revised: Further, most methods are able to deal with latent confounders in the sense that they obtain the correct results despite latent confounding being present.

Original: However, in our case, we explicitly predict latent confounders which is a different task by itself.
Revised: However, in our case, we explicitly predict latent confounders which is a different task by itself.

Original: Hyperparameters To show that we can perform latent confounder detection without speciﬁc hyperparameters, we use the same hyperparameters as for the experiment on the previous graph structures (see Appendix C.3).
Revised: Hyperparameters To show that we can perform latent confounder detection without speciﬁc hyperparameters, we use the same hyperparameters as for the experiment on the previous graph structures (see Appendix C.3).

Original: To record γ ( I ) ij and γ ( O ) ij separately, we use separate ﬁrst and second order momentum parameters in the Adam optimizer.
Revised: To record γ ( I ) ij and γ ( O ) ij separately, we use separate ﬁrst and second order momentum parameters in the Adam optimizer.

Original: We plot in Figure 13b the latent confounder scores
Revised: We plot in Figure 13b the latent confounder scores

Original: lc ( X i , X j ) calculated based on Equation 8.
Revised: lc ( X i , X j ) calculated based on Equation 8.

Original: We see that the score converges close to 1 for pairs with a latent confounder, and for all other, it converges to 0.
Revised: We see that the score converges close to 1 for pairs with a latent confounder, and for all other, it converges to 0.

Original: This veriﬁes our motivation of the score function discussed in Section 3.5, and also shows that the method is not sensitive to the threshold hyperparameter τ .
Revised: This veriﬁes our motivation of the score function discussed in Section 3.5, and also shows that the method is not sensitive to the threshold hyperparameter τ .

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

Original: 4 which was slightly higher than the highest value recorded for any other pair at early stages of training.
Revised: 4 which was slightly higher than the highest value recorded for any other pair at early stages of training.

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

Original: Datasets We perform the experiments of interventions on fewer variables on the same graphs and datasets as used for the initial synthetic graphs (see Section C.1).
Revised: Datasets We perform the experiments of interventions on fewer variables on the same graphs and datasets as used for the initial synthetic graphs (see Section C.1).

Original: To simulate having interventions on fewer variables, we randomly sample a subset of variables for which we include the interventional data, and remove for all others.
Revised: To simulate having interventions on fewer variables, we randomly sample a subset of variables for which we include the interventional data, and remove for all others.

Original: The sampled variables are the same for both ENCO and DCDI, and differ across graphs.
Revised: The sampled variables are the same for both ENCO and DCDI, and differ across graphs.

Original: The dataset size is the same as before, namely 200 samples per intervention and 5k observational datasets.
Revised: The dataset size is the same as before, namely 200 samples per intervention and 5k observational datasets.

Original: ENCO for partial intervention sets While the theoretical guarantees for convergence to an acyclic graph apply when interventions on all variables are possible, it is straightforward to extend the ENCO algorithm to support partial interventions as well.
Revised: ENCO for partial intervention sets While the theoretical guarantees for convergence to an acyclic graph apply when interventions on all variables are possible, it is straightforward to extend the ENCO algorithm to support partial interventions as well.

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: All other orientation parameters would remain unchanged throughout the training, since their gradients rely on interventions missing from the dataset.
Revised: All other orientation parameters would remain unchanged throughout the training, since their gradients rely on interventions missing from the dataset.

Original: Instead, we extend the gradient estimator in Equation 4 to not be exclusive to adjacent interventions, but include interventions on all variables.
Revised: Instead, we extend the gradient estimator in Equation 4 to not be exclusive to adjacent interventions, but include interventions on all variables.

Original: Speciﬁcally, for the orientation parameter θ ij without any interventions on X i or X j , we use the following gradient estimator:
Revised: Speciﬁcally, for the orientation parameter θ ij without any interventions on X i or X j , we use the following gradient estimator:

Original: [Equation]
Revised: [Equation]

Original: where we have an intervention on an arbitrary variable X k with k (cid:54)
Revised: where we have an intervention on an arbitrary variable X k with k (cid:54)

Original: =
Revised: =

Original: i, k (cid:54) =
Revised: i, k (cid:54) =

Original: j .
Revised: j .

Original: This still represents an unbiased gradient estimator since in the derivation of the estimator, we excluded interventions on other variables only to reduce noise.
Revised: This still represents an unbiased gradient estimator since in the derivation of the estimator, we excluded interventions on other variables only to reduce noise.

Original: ENCO has been designed under the assumption that interventional data is provided.
Revised: ENCO has been designed under the assumption that interventional data is provided.

Original: When we have interventional data on only a very small subset of variables, we might not optimally use the information that is provided by the observational data.
Revised: When we have interventional data on only a very small subset of variables, we might not optimally use the information that is provided by the observational data.

Original: To overcome this issue, we can run a causal discovery method that solely work on observational data and return an undirected graph.
Revised: To overcome this issue, we can run a causal discovery method that solely work on observational data and return an undirected graph.

Original: This skeleton can be used as a prior, and prevents false positive edges between conditionally independent variables.
Revised: This skeleton can be used as a prior, and prevents false positive edges between conditionally independent variables.

Original: Hyperparameters We reuse the hyperparameters of the experiments on the synthetic graph except that we use a slighly smaller sparsity regularizer, λ sparse = 0 .
Revised: Hyperparameters We reuse the hyperparameters of the experiments on the synthetic graph except that we use a slighly smaller sparsity regularizer, λ sparse = 0 .

Original: 002 , and a weight decay of 4 e - 5 .
Revised: 002 , and a weight decay of 4 e - 5 .

Original: For the orientation parameters without adjacent intervention, we use a learning rate of 0 .
Revised: For the orientation parameters without adjacent intervention, we use a learning rate of 0 .

Original: 1 · lr θ which is 1 efor this experiment.
Revised: 1 · lr θ which is 1 efor this experiment.

Original: For DCDI, we observed that a higher regularization term of λ = 1 . 0 obtained best performance.
Revised: For DCDI, we observed that a higher regularization term of λ = 1 . 0 obtained best performance.

Original: All other hyperparameters are the same as in Section C.1.
Revised: All other hyperparameters are the same as in Section C.1.

Original: Results For additional experimental results, see Section D.2.
Revised: Results For additional experimental results, see Section D.2.

Original: C. R EAL - WORLD INSPIRED EXPERIMENTS
Revised: C. R EAL - WORLD INSPIRED EXPERIMENTS

Original: Datasets We perform experiments on a collection of causal graphs from the Bayesian Network
Revised: Datasets We perform experiments on a collection of causal graphs from the Bayesian Network

Original: Repository (BnLearn) (Scutari, 2010).
Revised: 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: We chose the graphs to reﬂect a variety of sizes and different challenges (rare events, deterministic variables, etc.).
Revised: We chose the graphs to reﬂect a variety of sizes and different challenges (rare events, deterministic variables, etc.).

Original: The chosen graphs are cancer (Korb & Nicholson, 2010), earthquake (Korb & Nicholson, 2010), asia (Lauritzen & Spiegelhalter, 1988), sachs (Sachs et al., 2005), child (Spiegelhalter & Cowell, 1992), alarm (Beinlich et al., 1989), diabetes (Andreassen et al., 1991), and pigs (Scutari, 2010).
Revised: The chosen graphs are cancer (Korb & Nicholson, 2010), earthquake (Korb & Nicholson, 2010), asia (Lauritzen & Spiegelhalter, 1988), sachs (Sachs et al., 2005), child (Spiegelhalter & Cowell, 1992), alarm (Beinlich et al., 1989), diabetes (Andreassen et al., 1991), and pigs (Scutari, 2010).

Original: The graphs have been downloaded from the BnLearn website 5 .
Revised: The graphs have been downloaded from the BnLearn website 5 .

Original: For the small graphs, we have used a dataset size of 50k observational samples and 512 samples per intervention.
Revised: For the small graphs, we have used a dataset size of 50k observational samples and 512 samples per intervention.

Original: This is a larger dataset size than for the synthetic graph because many edges in the real-world graphs have very small causal effects that cannot be recovered from limited data, and the goal of the experiment was to show that the convergence conditions also hold on real-world graphs.
Revised: This is a larger dataset size than for the synthetic graph because many edges in the real-world graphs have very small causal effects that cannot be recovered from limited data, and the goal of the experiment was to show that the convergence conditions also hold on real-world graphs.

Original: Hence, we need more observational and interventional samples.
Revised: Hence, we need more observational and interventional samples.

Original: The results with a smaller dataset size, i.e. 5k observational and 200 interventional samples as before, are shown in Table 9.
Revised: The results with a smaller dataset size, i.e. 5k observational and 200 interventional samples as before, are shown in Table 9.

Original: For the large graphs, we follow the dataset size for the scalability experiments (see
Revised: For the large graphs, we follow the dataset size for the scalability experiments (see

Original: Section C.2).
Revised: Section C.2).

Original: Hyperparameters We reuse most of the hyperparameters of the previous experiments.
Revised: Hyperparameters We reuse most of the hyperparameters of the previous experiments.

Original: For all graphs less than 100 nodes, we use the hyperparameters of Appendix C.1, i.e. the synthetic graphs of
Revised: For all graphs less than 100 nodes, we use the hyperparameters of Appendix C.1, i.e. the synthetic graphs of

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

Original: 25 nodes.
Revised: 25 nodes.

Original: For all graphs larger than 100 nodes, we use the hyperparameters of Appendix C.2, i.e. the large-scale graphs.
Revised: For all graphs larger than 100 nodes, we use the hyperparameters of Appendix C.2, i.e. the large-scale graphs.

Original: One exception is that we allow the ﬁne-tuning of the regularizer parameter for both sets.
Revised: One exception is that we allow the ﬁne-tuning of the regularizer parameter for both sets.

Original: For ENCO, we used a slightly smaller regularizer, λ sparse = 0 .
Revised: For ENCO, we used a slightly smaller regularizer, λ sparse = 0 .

Original: 002 , for the small graphs, and a larger one, λ sparse = 0 .
Revised: 002 , for the small graphs, and a larger one, λ sparse = 0 .

Original: 02 , for the large graphs.
Revised: 02 , for the large graphs.

Original: Due to the large amount of deterministic variables, ENCO tends to predict more false positives in the beginning before removing them one by one.
Revised: Due to the large amount of deterministic variables, ENCO tends to predict more false positives in the beginning before removing them one by one.

Original: For SDI, we also found a smaller regularizer, λ sparse = 0 .
Revised: For SDI, we also found a smaller regularizer, λ sparse = 0 .

Original: 01 , to work best for the small graphs.
Revised: 01 , to work best for the small graphs.

Original: However, in line with the results of Ke et al.
Revised: However, in line with the results of Ke et al.

Original: (2019), SDI was not able to detect all edges.
Revised: (2019), SDI was not able to detect all edges.

Original: Even lower regularizers showed to perform considerably worse on the child dataset, while minor improvements were made on the small graphs.
Revised: Even lower regularizers showed to perform considerably worse on the child dataset, while minor improvements were made on the small graphs.

Original: Hence, we settled for λ sparse = 0 .
Revised: Hence, we settled for λ sparse = 0 .

Original: In terms of run time, both methods used 100 epochs for the small graphs and 50 for the large graphs.
Revised: In terms of run time, both methods used 100 epochs for the small graphs and 50 for the large graphs.

Original: Results
Revised: Results

Original: The results including standard deviations can be found in Table 9.
Revised: The results including standard deviations can be found in Table 9.

Original: The low standard deviation for ENCO shows that the approach is stable across seeds, even for large graphs.
Revised: The low standard deviation for ENCO shows that the approach is stable across seeds, even for large graphs.

Original: SDI has a zero standard deviation for a few graphs.
Revised: SDI has a zero standard deviation for a few graphs.

Original: In those cases, SDI converged to the same graph across seeds, but not necessarily the correct graph.
Revised: In those cases, SDI converged to the same graph across seeds, but not necessarily the correct graph.

Original: We have also applied DCDI (Brouillard et al., 2020) to the real-world datasets and report the results in Table 9 and 10.
Revised: We have also applied DCDI (Brouillard et al., 2020) to the real-world datasets and report the results in Table 9 and 10.

Original: DCDI performs relatively similar to SDI, making a few more mistakes on the very small graphs ( < 10 nodes) while being slightly better on sachs and child .
Revised: DCDI performs relatively similar to SDI, making a few more mistakes on the very small graphs ( < 10 nodes) while being slightly better on sachs and child .

Original: Nonetheless, ENCO outperforms DCDI on all graphs.
Revised: Nonetheless, ENCO outperforms DCDI on all graphs.

Original: We do not report results of DCDI on the largest graphs, diabetes and pigs , because it ran out of memory for diabetes (larger number of max.
Revised: We do not report results of DCDI on the largest graphs, diabetes and pigs , because it ran out of memory for diabetes (larger number of max.

Original: categories per variable) and did not converge within the same time limitations as SDI and ENCO (see Section 4.3 for a comparison on scalability).
Revised: categories per variable) and did not converge within the same time limitations as SDI and ENCO (see Section 4.3 for a comparison on scalability).

Original: D A DDITIONAL EXPERIMENTS
Revised: D A DDITIONAL EXPERIMENTS

Original: In this section, we show additional experiments performed as ablation studies of ENCO.
Revised: In this section, we show additional experiments performed as ablation studies of ENCO.

Original: First, we discuss further experiments We then discuss the effect of using our gradient estimators proposed in
Revised: First, we discuss further experiments We then discuss the effect of using our gradient estimators proposed in

Original: Section 3.4 compared to Bengio et al.
Revised: Section 3.4 compared to Bengio et al.

Original: Next, we show experiments on synthetic graphs with deterministic variables violating faithfulness, and experiments on continuous data with Normalizing
Revised: Next, we show experiments on synthetic graphs with deterministic variables violating faithfulness, and experiments on continuous data with Normalizing

Original: Flows.
Revised: Flows.

Original: Finally, we discuss experiments with different causal mechanism functions for generating synthetic, conditional categorical distributions besides neural networks.
Revised: Finally, we discuss experiments with different causal mechanism functions for generating synthetic, conditional categorical distributions besides neural networks.

Original: D. E FFECT OF THE SAMPLE SIZE
Revised: D. E FFECT OF THE SAMPLE SIZE

Original: The number of samples provided as observational and interventional data is crucial for causal structure learning methods since the more data we have, the better we can estimate the underlying causal mechanisms.
Revised: The number of samples provided as observational and interventional data is crucial for causal structure learning methods since the more data we have, the better we can estimate the underlying causal mechanisms.

Original: To gain further insights in the effect of the sample size on ENCO and the compared baselines, we repeat the experiments of Section 4.2 with different sample sizes.
Revised: To gain further insights in the effect of the sample size on ENCO and the compared baselines, we repeat the experiments of Section 4.2 with different sample sizes.

Original: Large sample size First, we use very large sample sizes to ﬁnd the upper bound performance level that we can expect from each method.
Revised: Large sample size First, we use very large sample sizes to ﬁnd the upper bound performance level that we can expect from each method.

Original: For this, we sample 100k observational samples per graph, and 10k samples per intervention.
Revised: For this, we sample 100k observational samples per graph, and 10k samples per intervention.

Original: We observed that this is sufﬁcient to model most conditional probabilities up to a negligible error.
Revised: We observed that this is sufﬁcient to model most conditional probabilities up to a negligible error.

Original: The results are shown in Table 11.
Revised: The results are shown in Table 11.

Original: We ﬁnd that, in line with the theoretical guarantees, ENCO can reliably recover most graphs, only making 0 .
Revised: We ﬁnd that, in line with the theoretical guarantees, ENCO can reliably recover most graphs, only making 0 .

Original: 3 mistakes on average on the full graph.
Revised: 3 mistakes on average on the full graph.

Original: Of the baselines, only DCDI is able to recover the collider graph without errors since its edges can be independently orientated.
Revised: Of the baselines, only DCDI is able to recover the collider graph without errors since its edges can be independently orientated.

Original: For all other graphs, DCDI converges to acyclic graphs, but incorrectly orients some edges and predicts false positive edges, while being 8 times slower than ENCO on the same hardware.
Revised: For all other graphs, DCDI converges to acyclic graphs, but incorrectly orients some edges and predicts false positive edges, while being 8 times slower than ENCO on the same hardware.

Original: All other baselines show improved SHD scores than in
Revised: All other baselines show improved SHD scores than in

Original: Table 1 as well, but are not able to match ENCO’s performance.
Revised: Table 1 as well, but are not able to match ENCO’s performance.

Original: This shows that, even in the data limit, ENCO achieves notably better results than concurrent methods.
Revised: This shows that, even in the data limit, ENCO achieves notably better results than concurrent methods.

Original: Next, we consider situations where data is very limited.
Revised: Next, we consider situations where data is very limited.

Original: Thereby, we consider two data sample axes: observational and interventional data.
Revised: Thereby, we consider two data sample axes: observational and interventional data.

Original: Limited interventional data sample sizes We repeat the experiments of Table 1 for ENCO while limiting the sample size per intervention to 20, 50, and 100 (200 before).
Revised: Limited interventional data sample sizes We repeat the experiments of Table 1 for ENCO while limiting the sample size per intervention to 20, 50, and 100 (200 before).

Original: The observational dataset size of 5000 samples is thereby kept constant.
Revised: The observational dataset size of 5000 samples is thereby kept constant.

Original: We plot the performance for all graph structures in
Revised: We plot the performance for all graph structures in

Original: Figure 14.
Revised: Figure 14.

Original: Overall, the decrease of performance with lower interventional sample size is consistent across graph structures.
Revised: Overall, the decrease of performance with lower interventional sample size is consistent across graph structures.

Original: With only 20 samples per intervention, it becomes especially hard to reason about variables with many parents, since the variable’s distribution is determined by many other parents as well.
Revised: With only 20 samples per intervention, it becomes especially hard to reason about variables with many parents, since the variable’s distribution is determined by many other parents as well.

Original: Yet, for four out of the six graphs, we obtain an SHD of less than 1 withinterventional samples, and less than 6 when only 20 samples are available.
Revised: Yet, for four out of the six graphs, we obtain an SHD of less than 1 withinterventional samples, and less than 6 when only 20 samples are available.

Original: In conclusion, ENCO works well with little interventional data if most variables have a small parent set.
Revised: In conclusion, ENCO works well with little interventional data if most variables have a small parent set.

Original: Limited observational data sample sizes Similarly as above
Revised: Limited observational data sample sizes Similarly as above

Original: , we repeat the experiments of Tablefor ENCO but limit the observational sample size to 1000 and 2000 (5000 before) while keeping 200 samples per interventions.
Revised: , we repeat the experiments of Tablefor ENCO but limit the observational sample size to 1000 and 2000 (5000 before) while keeping 200 samples per interventions.

Original: Observational data is important in ENCO for learning the conditional distributions.
Revised: Observational data is important in ENCO for learning the conditional distributions.

Original: For variables with many parents, this becomes more difﬁcult when fewer samples are available, because the input space grows exponentially with the number of parents.
Revised: For variables with many parents, this becomes more difﬁcult when fewer samples are available, because the input space grows exponentially with the number of parents.

Original: Thus, we
Revised: Thus, we

Original: [Table]
Revised: [Table]

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

Original: [Figure]
Revised: [Figure]

Original: would expect the collider and full graph suffer the most from having less observational data, and this is indeed the case as shown by the results in Figure 15.
Revised: would expect the collider and full graph suffer the most from having less observational data, and this is indeed the case as shown by the results in Figure 15.

Original: The results of all other graphs are less affected, although interestingly, some become even better with less observational data.
Revised: The results of all other graphs are less affected, although interestingly, some become even better with less observational data.

Original: For the chain
Revised: For the chain

Original: X 1 → X 2 → ...X N , for instance, we observed that the learned conditional distributions picked up spurious correlations among variables, e.g., between X 1 and X 3 when modeling p ( X| X 1 , X 2 ) which are, in the data limit, independent given X 2 .
Revised: X 1 → X 2 → ...X N , for instance, we observed that the learned conditional distributions picked up spurious correlations among variables, e.g., between X 1 and X 3 when modeling p ( X| X 1 , X 2 ) which are, in the data limit, independent given X 2 .

Original: Since those correlations do not necessarily transfer to the interventional setting, it is easier to spot false positive edges, and we can obtain even better results than for the larger sample sizes.
Revised: Since those correlations do not necessarily transfer to the interventional setting, it is easier to spot false positive edges, and we can obtain even better results than for the larger sample sizes.

Original: In conclusion, having sufﬁcient observational data is crucial in ENCO for graphs with variables that have larger parent sets, while being less important for sparser graphs.
Revised: In conclusion, having sufﬁcient observational data is crucial in ENCO for graphs with variables that have larger parent sets, while being less important for sparser graphs.

Original: Limited interventional and observational data sample sizes
Revised: Limited interventional and observational data sample sizes

Original: Finally, we combine the smallest interventional and observational data sample sizes, and also include the results of the previously best baselines, SDI and DCDI, in Table 12.
Revised: Finally, we combine the smallest interventional and observational data sample sizes, and also include the results of the previously best baselines, SDI and DCDI, in Table 12.

Original: The results of ENCO show the combination of the previous two effects: graphs consisting of variables with small parent sets can still be recovered well by ENCO, while errors increase for the collider and full graph.
Revised: The results of ENCO show the combination of the previous two effects: graphs consisting of variables with small parent sets can still be recovered well by ENCO, while errors increase for the collider and full graph.

Original: Similar trends are observed for SDI, while DCDI showed a considerable decrease in performance for all graphs.
Revised: Similar trends are observed for SDI, while DCDI showed a considerable decrease in performance for all graphs.

Original: In conclusion, ENCO still works well for graphs with smaller parent sets under a small observational and interventional data regime, and outperforms related baselines in this setting.
Revised: In conclusion, ENCO still works well for graphs with smaller parent sets under a small observational and interventional data regime, and outperforms related baselines in this setting.

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: different scale of the y-axis for the six graphs.
Revised: different scale of the y-axis for the six graphs.

Original: For four out of six graphs, ENCO outperforms DCDI even for as few as 4 interventions, especially when enforcing acyclicity.
Revised: For four out of six graphs, ENCO outperforms DCDI even for as few as 4 interventions, especially when enforcing acyclicity.

Original: The detailed numbers of the results are listed in Table 13.
Revised: The detailed numbers of the results are listed in Table 13.

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

Original: We have performed the experiments in Section 4.4 using fewer interventions for all six synthetic graph structures.
Revised: We have performed the experiments in Section 4.4 using fewer interventions for all six synthetic graph structures.

Original: The results are visualized in Figure 16, and presented in table-form in Table 13.
Revised: The results are visualized in Figure 16, and presented in table-form in Table 13.

Original: From the experiments, we can see that ENCO with interventions on only 4 variables matches or outperforms DCDI with 10 interventions for 4 out of the 6 graph structures (bidiag, chain, full, jungle) when enforcing acyclicity.
Revised: From the experiments, we can see that ENCO with interventions on only 4 variables matches or outperforms DCDI with 10 interventions for 4 out of the 6 graph structures (bidiag, chain, full, jungle) when enforcing acyclicity.

Original: Especially on chain-like graphs such as jungle, ENCO achieves lower
Revised: Especially on chain-like graphs such as jungle, ENCO achieves lower

Original: SHD scores for the same number of interventions on variables, while DCDI incorrectly orientates many edges and predicts false positive edges between children.
Revised: SHD scores for the same number of interventions on variables, while DCDI incorrectly orientates many edges and predicts false positive edges between children.

Original: On the graph collider, we observed a high variance for settings with very few interventions.
Revised: On the graph collider, we observed a high variance for settings with very few interventions.

Original: This is because when we intervene on the collider node itself, ENCO can deduce the orientation for all edges.
Revised: This is because when we intervene on the collider node itself, ENCO can deduce the orientation for all edges.

Original: Finally, on the graph random, we observe that enforcing acyclicity for ENCO reduces the error a lot.
Revised: Finally, on the graph random, we observe that enforcing acyclicity for ENCO reduces the error a lot.

Original: This is because incorrectly orientated edges cause more false positives in this densely connected graph, which are removed with the cycles.
Revised: This is because incorrectly orientated edges cause more false positives in this densely connected graph, which are removed with the cycles.

Original: We include a longer discussion of the limitations of ENCO on fewer interventions in
Revised: We include a longer discussion of the limitations of ENCO on fewer interventions in

Original: Appendix B.4.
Revised: Appendix B.4.

Original: Still, we conclude that, in practice, ENCO performs competitively to DCDI, even when very few interventions are provided, and scales better to more interventions.
Revised: Still, we conclude that, in practice, ENCO performs competitively to DCDI, even when very few interventions are provided, and scales better to more interventions.

Original: D. A BLATION STUDY ON GRADIENT ESTIMATORS
Revised: D. A BLATION STUDY ON GRADIENT ESTIMATORS

Original: To analyze the importance of the low-variance gradient estimators in ENCO, we repeat the experiments on synthetic graph structure from Section 4.2 where the gradient estimators of ENCO have been replaced by those from Bengio et al.
Revised: To analyze the importance of the low-variance gradient estimators in ENCO, we repeat the experiments on synthetic graph structure from Section 4.2 where the gradient estimators of ENCO have been replaced by those from Bengio et al.

Original: The results are shown in Table 14.
Revised: The results are shown in Table 14.

Original: Overall, the scores are very similar with minor differences for the graphs full and bidiag .
Revised: Overall, the scores are very similar with minor differences for the graphs full and bidiag .

Original: In comparisons to the learning curves in Figure 11, the curves with the gradient estimator of Bengio et al.
Revised: In comparisons to the learning curves in Figure 11, the curves with the gradient estimator of Bengio et al.

Original: (2020) are more noisy, with recall and precision jumping up and down.
Revised: (2020) are more noisy, with recall and precision jumping up and down.

Original: While ENCO easily converged early to the correct graphs for all graph types, this model often required the full 30 iterations to reach the optimal recovery.
Revised: While ENCO easily converged early to the correct graphs for all graph types, this model often required the full 30 iterations to reach the optimal recovery.

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

Original: The difference between the two gradient estimators becomes more apparent on large graphs.
Revised: The difference between the two gradient estimators becomes more apparent on large graphs.

Original: We repeated the experiments of Section 4.3 on the graphs with 100 nodes using the gradient estimator of
Revised: We repeated the experiments of Section 4.3 on the graphs with 100 nodes using the gradient estimator of

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

Original: Within the 30 epochs, the model obtained an SHD of 15 . 4 on average overexperiments, which is considerably higher than ENCO with the proposed gradient estimators ( 0 . 0 ).
Revised: Within the 30 epochs, the model obtained an SHD of 15 . 4 on average overexperiments, which is considerably higher than ENCO with the proposed gradient estimators ( 0 . 0 ).

Original: Still, this is only half of the errors that SDI (Ke et al., 2019) with the same gradient estimator achieved.
Revised: Still, this is only half of the errors that SDI (Ke et al., 2019) with the same gradient estimator achieved.

Original: Hence, we can conclude that the proposed gradient estimators are beneﬁcial for ENCO but not strictly necessary for small graphs.
Revised: Hence, we can conclude that the proposed gradient estimators are beneﬁcial for ENCO but not strictly necessary for small graphs.

Original: For large graphs, the low variance of the estimator becomes much more important.
Revised: For large graphs, the low variance of the estimator becomes much more important.

Original: D. D ETERMINISTIC VARIABLES
Revised: D. D ETERMINISTIC VARIABLES

Original: In contrast to algorithms working on observational data, ENCO does not strictly require the faithfulness assumption.
Revised: In contrast to algorithms working on observational data, ENCO does not strictly require the faithfulness assumption.

Original: Hence, we can apply ENCO to graphs with deterministic variables.
Revised: Hence, we can apply ENCO to graphs with deterministic variables.

Original: Deterministic variables have a distribution that is deﬁned by a one-to-one mapping of its parents’ inputs to an output value.
Revised: Deterministic variables have a distribution that is deﬁned by a one-to-one mapping of its parents’ inputs to an output value.

Original: In other words, we have the following distribution:
Revised: In other words, we have the following distribution:

Original: [Equation]
Revised: [Equation]

Original: (47) where f is an arbitrary function.
Revised: (47) where f is an arbitrary function.

Original: The difﬁculty of deterministic variables is that a variable X i can be fully replaced by its parents pa ( X i ) in any conditional distribution.
Revised: The difﬁculty of deterministic variables is that a variable X i can be fully replaced by its parents pa ( X i ) in any conditional distribution.

Original: The only way we can identify deterministic variables is from interventional data, where an intervention on X i breaks the dependency to its parents.
Revised: The only way we can identify deterministic variables is from interventional data, where an intervention on X i breaks the dependency to its parents.

Original: We have already tested ENCO on deterministic variables in the context of the real-world inspired graphs of Section 4.6.
Revised: We have already tested ENCO on deterministic variables in the context of the real-world inspired graphs of Section 4.6.

Original: To have a more detailed analysis, we created synthetic graphs following the random graph setup with an edge probability of 0 .
Revised: To have a more detailed analysis, we created synthetic graphs following the random graph setup with an edge probability of 0 .

Original: 1 and an average of two parents, and maximum of three parents.
Revised: 1 and an average of two parents, and maximum of three parents.

Original: An example graph is shown in Figure 17.
Revised: An example graph is shown in Figure 17.

Original: All variables except the leaf nodes have deterministic distributions, where the function f ( pa ( X i )) is randomly created by picking a
Revised: All variables except the leaf nodes have deterministic distributions, where the function f ( pa ( X i )) is randomly created by picking a

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: random output category for any pair of input values.
Revised: random output category for any pair of input values.

Original: We create 10 such graphs and use the same hyperparameter setup as for the synthetic graphs, except that we increase the sparsity regularizer to λ sparse = 0 .
Revised: We create 10 such graphs and use the same hyperparameter setup as for the synthetic graphs, except that we increase the sparsity regularizer to λ sparse = 0 .

Original: We report the results in Table 15.
Revised: We report the results in Table 15.

Original: In line with the results on the real-world graphs,
Revised: In line with the results on the real-world graphs,

Original: ENCO is able to recover most graphs with less than two errors.
Revised: ENCO is able to recover most graphs with less than two errors.

Original: As a baseline, we apply SDI (Ke et al., 2019) and see a signiﬁcant higher error rate.
Revised: As a baseline, we apply SDI (Ke et al., 2019) and see a signiﬁcant higher error rate.

Original: The method predicts many false positives, including two-variable loops, but was also missing out some true positive edges.
Revised: The method predicts many false positives, including two-variable loops, but was also missing out some true positive edges.

Original: We conclude that ENCO also works well on deterministic graphs.
Revised: We conclude that ENCO also works well on deterministic graphs.

Original: Cancellation of paths Besides deterministic nodes, a common example for faithfulness violation is the cancellation of two paths.
Revised: Cancellation of paths Besides deterministic nodes, a common example for faithfulness violation is the cancellation of two paths.

Original: For instance, consider the causal graph with the three variables
Revised: For instance, consider the causal graph with the three variables

Original: X 1 , X 2 , X 3 shown in Figure 18, and the conditional distribution p ( X 2 | X 1 ) =
Revised: X 1 , X 2 , X 3 shown in Figure 18, and the conditional distribution p ( X 2 | X 1 ) =

Original: δ [ X 2 = X 1 ] .
Revised: δ [ X 2 = X 1 ] .

Original: In this case, the two paths X 1 → X 2 → X 3 and X 1 → X 3 cancel each other, i.e. X 3 is independent of X 2 when conditioned on X 1 , and independent of X 1 when conditioned on X 2 .
Revised: In this case, the two paths X 1 → X 2 → X 3 and X 1 → X 3 cancel each other, i.e. X 3 is independent of X 2 when conditioned on X 1 , and independent of X 1 when conditioned on X 2 .

Original: This implies that only one of the two graphs is necessary for describing the relations.
Revised: This implies that only one of the two graphs is necessary for describing the relations.

Original: Yet, ENCO can ﬁnd the edge X 1 → X 3 by observing interventions on X 2 , since in this case, X 1 ⊥⊥ X 2 and X 1 (cid:54)⊥⊥ X 3 | X 2 .
Revised: Yet, ENCO can ﬁnd the edge X 1 → X 3 by observing interventions on X 2 , since in this case, X 1 ⊥⊥ X 2 and X 1 (cid:54)⊥⊥ X 3 | X 2 .

Original: The remaining edges can be learned in the same manner.
Revised: The remaining edges can be learned in the same manner.

Original: We also emperically verify this by running ENCO on the graph structure of Figure 18 with the three variables being binary.
Revised: We also emperically verify this by running ENCO on the graph structure of Figure 18 with the three variables being binary.

Original: We set p ( X 2 | X 1 ) =
Revised: We set p ( X 2 | X 1 ) =

Original: δ [ X 2 = X 1 ] for
Revised: δ [ X 2 = X 1 ] for

Original: [Table]
Revised: [Table]

Original: canceling the two paths
Revised: canceling the two paths

Original: [Table]
Revised: [Table]

Original: and the remaining distributions are randomly initialized.
Revised: and the remaining distributions are randomly initialized.

Original: ENCO reconstructs the graph without errors
Revised: ENCO reconstructs the graph without errors

Original: [Table]
Revised: [Table]

Original: showing that it also works in practice.
Revised: showing that it also works in practice.

Original: D. C ONTINUOUS DATA
Revised: D. C ONTINUOUS DATA

Original: We verify that ENCO works just as well with continuous data by performing the experiments on datasets from Brouillard et al.
Revised: We verify that ENCO works just as well with continuous data by performing the experiments on datasets from Brouillard et al.

Original: (2020) that contained interventions on all variables.
Revised: (2020) that contained interventions on all variables.

Original: In these datasets, the graphs consist of 10 variables with an average of one edge per variable, and deploy three different causal mechanisms: linear, nonlinear additive noise models
Revised: In these datasets, the graphs consist of 10 variables with an average of one edge per variable, and deploy three different causal mechanisms: linear, nonlinear additive noise models

Original: [Table]
Revised: [Table]

Original: and nonlinear models with non-additive noise using neural networks.
Revised: and nonlinear models with non-additive noise using neural networks.

Original: The datasets contain 909 observational samples and 909 samples per intervention.
Revised: The datasets contain 909 observational samples and 909 samples per intervention.

Original: All results of GIES, IGSP, and DCDI have been taken from Brouillard et al.
Revised: All results of GIES, IGSP, and DCDI have been taken from Brouillard et al.

Original: (2020) (Appendix C.7, Table 22-24).
Revised: (2020) (Appendix C.7, Table 22-24).

Original: We follow the setup of Brouillard et al.
Revised: We follow the setup of Brouillard et al.

Original: and compare two different neural network setups.
Revised: and compare two different neural network setups.

Original: First, we use MLPs that model a Gaussian density by predicting a mean and variance variable (denoted by sufﬁx G ).
Revised: First, we use MLPs that model a Gaussian density by predicting a mean and variance variable (denoted by sufﬁx G ).

Original: The second setup uses normalizing ﬂows, more speciﬁcally a two-layer deep sigmoidal ﬂow (Huang et al., 2018), which is ﬂexible enough to model more complex distributions (denoted by sufﬁx DSF ).
Revised: The second setup uses normalizing ﬂows, more speciﬁcally a two-layer deep sigmoidal ﬂow (Huang et al., 2018), which is ﬂexible enough to model more complex distributions (denoted by sufﬁx DSF ).

Original: The rest of the experimental setup in ENCO is identical to the categorical case.
Revised: The rest of the experimental setup in ENCO is identical to the categorical case.

Original: Results are shown in Table 16, and the observations are the same as with categorical data.
Revised: Results are shown in Table 16, and the observations are the same as with categorical data.

Original: ENCO outperforms all other methods in all settings, especially for the more complex distributions.
Revised: ENCO outperforms all other methods in all settings, especially for the more complex distributions.

Original: The higher error rate for the DSF setup is mostly due to overﬁtting of the ﬂow models.
Revised: The higher error rate for the DSF setup is mostly due to overﬁtting of the ﬂow models.

Original: We conclude that
Revised: We conclude that

Original: ENCO works as accurately for both continuous and categorical data.
Revised: ENCO works as accurately for both continuous and categorical data.

Original: D. S KELETON LEARNING WITH OBSERVATIONAL BASELINE
Revised: D. S KELETON LEARNING WITH OBSERVATIONAL BASELINE

Original: To show the beneﬁt of learning a graph from observational and interventional data jointly, we compare
Revised: To show the beneﬁt of learning a graph from observational and interventional data jointly, we compare

Original: ENCO to a simple observational baseline.
Revised: ENCO to a simple observational baseline.

Original: This baseline ﬁrst learns the skeleton of the graph by applying greedy equivalence search (GES)
Revised: This baseline ﬁrst learns the skeleton of the graph by applying greedy equivalence search (GES)

Original: (Chickering, 2002) on the observational data.
Revised: (Chickering, 2002) on the observational data.

Original: Then, for each interventional dataset, we apply GES as well and use those skeletons to orientate the edges of the original one.
Revised: Then, for each interventional dataset, we apply GES as well and use those skeletons to orientate the edges of the original one.

Original: This can be done by checking for each undirected edge X − Y whether X → Y is in the skeleton of interventions on X or not.
Revised: This can be done by checking for each undirected edge X − Y whether X → Y is in the skeleton of interventions on X or not.

Original: As a reference implementation of GES, we have used the one provided in the Causal Discovery Toolbox (Kalainathan et al., 2020).
Revised: As a reference implementation of GES, we have used the one provided in the Causal Discovery Toolbox (Kalainathan et al., 2020).

Original: The results on continuous data are shown in Table 16.
Revised: The results on continuous data are shown in Table 16.

Original: Since GES assumes linear mechanisms and gaussianity of the data, it is unsurprising that it performs better on the linear Gaussian dataset than on the non-linear datasets.
Revised: Since GES assumes linear mechanisms and gaussianity of the data, it is unsurprising that it performs better on the linear Gaussian dataset than on the non-linear datasets.

Original: However, on all the three datasets, it constitutes the lowest performance compared to the other methods, including ENCO.
Revised: However, on all the three datasets, it constitutes the lowest performance compared to the other methods, including ENCO.

Original: This highlights the beneﬁts of incorporating interventional data in the learning of the skeleton and graph structure.
Revised: This highlights the beneﬁts of incorporating interventional data in the learning of the skeleton and graph structure.

Original: To gain further insights in comparison to the constraint-based baseline, we repeat the experiments with smaller sample sizes.
Revised: To gain further insights in comparison to the constraint-based baseline, we repeat the experiments with smaller sample sizes.

Original: The original dataset has 909 samples for observational data and per intervention, and we sub-sampleand 100 of those respectively for simulating smaller dataset sizes.
Revised: The original dataset has 909 samples for observational data and per intervention, and we sub-sampleand 100 of those respectively for simulating smaller dataset sizes.

Original: The results of those experiments can be found in Table 17.
Revised: The results of those experiments can be found in Table 17.

Original: It is apparent that the results of GES on the linear dataset get considerably worse with fewer data samples being available, while ENCO-G is able to reconstruct most graphs still without errors.
Revised: It is apparent that the results of GES on the linear dataset get considerably worse with fewer data samples being available, while ENCO-G is able to reconstruct most graphs still without errors.

Original: Especially for the small dataset of 100 samples, we noticed that the skeletons
Revised: Especially for the small dataset of 100 samples, we noticed that the skeletons

Original: [Table]
Revised: [Table]

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

Original: found by GES on observational data already contained couple of mistakes.
Revised: found by GES on observational data already contained couple of mistakes.

Original: This shows that for small datasets, observational data alone might not be sufﬁcient to ﬁnd the correct skeleton while by jointly learning from observational and interventional data, we can yet ﬁnd the graph up to minor errors.
Revised: This shows that for small datasets, observational data alone might not be sufﬁcient to ﬁnd the correct skeleton while by jointly learning from observational and interventional data, we can yet ﬁnd the graph up to minor errors.

Original: Further
Revised: Further

Original: [Table]
Revised: [Table]

Original: we also apply GES on the categorical data with an additional hyperparameter search over the penalty discount.
Revised: we also apply GES on the categorical data with an additional hyperparameter search over the penalty discount.

Original: The results in Table 18 give a similar conclusion as on the continuous data.
Revised: The results in Table 18 give a similar conclusion as on the continuous data.

Original: While the baseline attains good scores for chains
Revised: While the baseline attains good scores for chains

Original: [Table]
Revised: [Table]

Original: it makes considerably more errors on all other graph structures than ENCO.
Revised: it makes considerably more errors on all other graph structures than ENCO.

Original: This shows that ENCO is much more robust by jointly learning from observational and interventional data
Revised: This shows that ENCO is much more robust by jointly learning from observational and interventional data

Original: [Table]
Revised: [Table]

Original: D. N ON - NEURAL BASED DATA SIMULATORS
Revised: D. N ON - NEURAL BASED DATA SIMULATORS

Original: Using neural networks to generate the simulated data might give SDI, DCDI and ENCO an advantage in our comparisons since they rely on similar neural networks to model the distribution.
Revised: Using neural networks to generate the simulated data might give SDI, DCDI and ENCO an advantage in our comparisons since they rely on similar neural networks to model the distribution.

Original: To verify that ENCO works for other simulated data similarly well, we run experiments on categorical data with other function forms for the causal mechanisms instead of neural networks.
Revised: To verify that ENCO works for other simulated data similarly well, we run experiments on categorical data with other function forms for the causal mechanisms instead of neural networks.

Original: Since there is no straightforward way of deﬁning ’linear’ mechanisms for categorical data, we instead express a conditional distribution as a product of independent, single conditionals:
Revised: Since there is no straightforward way of deﬁning ’linear’ mechanisms for categorical data, we instead express a conditional distribution as a product of independent, single conditionals:

Original: [Equation]
Revised: [Equation]

Original: with p ( X i | X j ) =
Revised: with p ( X i | X j ) =

Original: exp( α Xi,Xj ) (cid:80)
Revised: exp( α Xi,Xj ) (cid:80)

Original: Xi exp( α Xi,Xj ) , α · , · ∼ N (0 , 2) .
Revised: Xi exp( α Xi,Xj ) , α · , · ∼ N (0 , 2) .

Original: Hence, the effect of each variable in the parent set is independent of all others, similar to linear functions in the continuous case.
Revised: Hence, the effect of each variable in the parent set is independent of all others, similar to linear functions in the continuous case.

Original: The individual probability densities represent a softmax distribution over variables sampled from a
Revised: The individual probability densities represent a softmax distribution over variables sampled from a

Original: Gaussian distribution.
Revised: Gaussian distribution.

Original: We apply GIES, IGSP, DCDI, SDI and ENCO to the same set of synthetic graph structures with these new causal mechanisms.
Revised: We apply GIES, IGSP, DCDI, SDI and ENCO to the same set of synthetic graph structures with these new causal mechanisms.

Original: Similar to the previous experiments, we provide 200 samples per intervention and 5k observational samples to the algorithms, and repeat the experiments withindependently sampled graphs.
Revised: Similar to the previous experiments, we provide 200 samples per intervention and 5k observational samples to the algorithms, and repeat the experiments withindependently sampled graphs.

Original: The results in Table 19 give the same conclusion as the experiments on neural-based causal mechanisms, namely that ENCO outperforms all baselines.
Revised: The results in Table 19 give the same conclusion as the experiments on neural-based causal mechanisms, namely that ENCO outperforms all baselines.

Original: Most methods experience a decrease in performance since the average causal effect of each parent is lower than in the neural case where more complex interactions between parents can be modeled.
Revised: Most methods experience a decrease in performance since the average causal effect of each parent is lower than in the neural case where more complex interactions between parents can be modeled.

Original: Still, ENCO only shows minor decreases, having less than one mistake on average for every graph structure when applying the orientation heuristic.
Revised: Still, ENCO only shows minor decreases, having less than one mistake on average for every graph structure when applying the orientation heuristic.