Original: Truncated Matrix Power Iteration for Differentiable DAG Learning
Revised: Truncated Matrix Power Iteration for Differentiable DAG Learning

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AfﬁliationAddress email
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: Recovering underlying Directed Acyclic Graph structures (DAG) from observational data is highly challenging due to the combinatorial nature of the DAGconstrained optimization problem.
Revised: Recovering underlying Directed Acyclic Graph structures (DAG) from observational data is highly challenging due to the combinatorial nature of the DAGconstrained optimization problem.

Original: Recently, DAG learning has been cast as acontinuous optimization problem by characterizing the DAG constraint as a smoothequality one, generally based on polynomials over adjacency matrices.
Revised: Recently, DAG learning has been cast as acontinuous optimization problem by characterizing the DAG constraint as a smoothequality one, generally based on polynomials over adjacency matrices.

Original: Existingmethods place very small coefﬁcients on high-order polynomial terms for stabilization, since they argue that large coefﬁcients on the higher-order terms are harmfuldue to numeric exploding.
Revised: Existingmethods place very small coefﬁcients on high-order polynomial terms for stabilization, since they argue that large coefﬁcients on the higher-order terms are harmfuldue to numeric exploding.

Original: On the contrary, we discover that large coefﬁcients onhigher-order terms are beneﬁcial for DAG learning, when the spectral radiuses ofthe adjacency matrices are small, and that larger coefﬁcients for higher order termscan approximate the DAG constraints much better than the small counterparts.
Revised: On the contrary, we discover that large coefﬁcients onhigher-order terms are beneﬁcial for DAG learning, when the spectral radiuses ofthe adjacency matrices are small, and that larger coefﬁcients for higher order termscan approximate the DAG constraints much better than the small counterparts.

Original: Based on this, we propose a novel DAG learning method with efﬁcient truncatedmatrix power iteration to approximate geometric series based DAG constraints.
Revised: Based on this, we propose a novel DAG learning method with efﬁcient truncatedmatrix power iteration to approximate geometric series based DAG constraints.

Original: Empirically, our DAG learning method outperforms the previous state-of-the-artsin various settings, often by a factor of 3 or more in terms of structural Hammingdistance.
Revised: Empirically, our DAG learning method outperforms the previous state-of-the-artsin various settings, often by a factor of 3 or more in terms of structural Hammingdistance.

Original:  Introduction
Revised:  Introduction

Original: Recovery of Directed Acyclic Graphs (DAGs) from observational data is a classical problem inmany ﬁelds, including bioinformatics [25, 40], machine learning [14] and causal inference [32].
Revised: Recovery of Directed Acyclic Graphs (DAGs) from observational data is a classical problem inmany ﬁelds, including bioinformatics [26, 42], machine learning [15] and causal inference [33].

Original: Thegraphical model produced by a DAG learning algorithm allows one to decompose the joint distributionover the variables of interest in a compact and ﬂexible manner and under certain assumptions [22, 32],these graphical models can have causal interpretations.
Revised: Thegraphical model produced by a DAG learning algorithm allows one to decompose the joint distributionover the variables of interest in a compact and ﬂexible manner and under certain assumptions [23, 33],these graphical models can have causal interpretations.

Original: DAG learning methods can be roughly categorized into constraint-based and score-based methods.
Revised: DAG learning methods can be roughly categorized into constraint-based and score-based methods.

Original: Most constraint-based approaches, e.g., PC [30], FCI [10, 31], rely on conditional independencetest and thus may require a large sample size [27, 34].
Revised: Most constraint-based approaches, e.g., PC [31], FCI [10, 32], rely on conditional independencetest and thus may require a large sample size [28, 36].

Original: The score-based approaches, including exactmethods based on dynamic programming [13, 28, 29], A* search [37, 38], and integer programming[11], as well as greedy methods like GES [8], model the validity of a graph according to some scorefunction and are often formulated and solved as a discrete optimization problem.
Revised: The score-based approaches, including exactmethods based on dynamic programming [14, 29, 30], A* search [40, 41], and integer programming[12], as well as greedy methods like GES [8], model the validity of a graph according to some scorefunction and are often formulated and solved as a discrete optimization problem.

Original: A key challenge forscore-based methods is the combinatorial search space of DAGs [7, 9].
Revised: A key challenge forscore-based methods is the combinatorial search space of DAGs [7, 9].

Original: There is a recent interest in developing continuous optimization methods for DAG learning in the pastfew years.
Revised: There is a recent interest in developing continuous optimization methods for DAG learning in the pastfew years.

Original: In particular, Zheng et al.
Revised: In particular, Zheng et al.

Original: [41] develops an algebraic characterization of DAG constraintbased on a continuous function, speciﬁcally matrix exponential, of the weighted adjacency matrixof a graph, and applies it to estimate linear DAGs with equal noise variances using continuousconstrained optimization [3, 4].
Revised: [43] develops an algebraic characterization of DAG constraintbased on a continuous function, speciﬁcally matrix exponential, of the weighted adjacency matrixof a graph, and applies it to estimate linear DAGs with equal noise variances using continuousconstrained optimization [3, 4].

Original: The resulting algorithm is called NOTEARS.
Revised: The resulting algorithm is called NOTEARS.

Original: Yu et al.
Revised: Yu et al.

Original: [36] proposesan alternative DAG constraint based on powers of a binomial to improve practical convenience.
Revised: [38] proposesan alternative DAG constraint based on powers of a binomial to improve practical convenience.

Original: Weiet al.
Revised: Weiet al.

Original: [35] uniﬁes these two DAG constraints by writing them in terms of a general polynomial one
Revised: [37] uniﬁes these two DAG constraints by writing them in terms of a general polynomial one

Original: These DAG constraints have been applied to more complex settings, e.g., with nonlinear relationships[15, 20, 36, 42], time series [21], and confounders [5], and have been shown to produce competitiveperformance.
Revised: These DAG constraints have been applied to more complex settings, e.g., with nonlinear relationships[16, 21, 38, 44], time series [22], and confounders [5], and have been shown to produce competitiveperformance.

Original: Apart from the polynomial based constraints, Lee et al.
Revised: Apart from the polynomial based constraints, Lee et al.

Original: [16], Zhu et al.
Revised: [17], Zhu et al.

Original: [43] presentalternative formulations of the DAG constraints based on the spectral radius of the weighted adjacencymatrix.
Revised: [45] presentalternative formulations of the DAG constraints based on the spectral radius of the weighted adjacencymatrix.

Original: Instead of solving a constrained optimization problem,Yue Yu and Ji [39] proposes NOCURLwhich employs an algebraic representation of DAGs such that continuous unconstrained optimizationcan be carried out in the DAG space directly.
Revised: Instead of solving a constrained optimization problem, Yu et al. [39] proposes NOCURLwhich employs an algebraic representation of DAGs such that continuous unconstrained optimizationcan be carried out in the DAG space directly.

Original: However, the focus of these works is to improve thescalability, and the performance of the resulting methods may degrade.
Revised: However, the focus of these works is to improve thescalability, and the performance of the resulting methods may degrade.

Original: To avoid potential numerical exploding that may be caused by high-order terms in the polynomialbased DAG constraints, previous works use very small coefﬁcients for high terms in DAG constraints[36, 41].
Revised: To avoid potential numerical exploding that may be caused by high-order terms in the polynomialbased DAG constraints, previous works use very small coefﬁcients for high terms in DAG constraints[38, 43].

Original: More precisely, the coefﬁcients decrease quickly to very small values for the relativelyhigher order terms that are still crucial for capturing the useful higher-order information.
Revised: More precisely, the coefﬁcients decrease quickly to very small values for the relativelyhigher order terms that are still crucial for capturing the useful higher-order information.

Original: This causesdifﬁculties for the DAG constraints to capture the higher order information for enforcing acyclicity.
Revised: This causesdifﬁculties for the DAG constraints to capture the higher order information for enforcing acyclicity.

Original: With such DAG constraints, a possible solution to maintain higher-order information for acyclicityis to apply an even larger penalty weight [36, 41], which, however, causes an ill-conditioning issueas shown in [19].
Revised: With such DAG constraints, a possible solution to maintain higher-order information for acyclicityis to apply an even larger penalty weight [38, 43], which, however, causes an ill-conditioning issueas shown in [20].

Original: Moreover, these small coefﬁcients will also be multiplied on the gradients ofhigher-order terms, leading to gradient vanishing.
Revised: Moreover, these small coefﬁcients will also be multiplied on the gradients ofhigher-order terms, leading to gradient vanishing.

Original: In this paper, we show that, on the contrary, largercoefﬁcients can be safely used to construct more informative higher-order terms without introducingnumeric exploding.
Revised: In this paper, we show that, on the contrary, largercoefﬁcients can be safely used to construct more informative higher-order terms without introducingnumeric exploding.

Original: The reason is that the weighted adjacency matrix of a DAG must be nilpotent;thus the candidate adjacency matrix often has a very small spectral radius.
Revised: The reason is that the weighted adjacency matrix of a DAG must be nilpotent;thus the candidate adjacency matrix often has a very small spectral radius.

Original: As a result, the higherorder power of the matrix is very close to zero matrix, and larger coefﬁcients can be safely usedwithout causing numeric exploding.
Revised: As a result, the higherorder power of the matrix is very close to zero matrix, and larger coefﬁcients can be safely usedwithout causing numeric exploding.

Original: We thus propose to use geometric series-based DAG constraintwithout small coefﬁcients for more accurate and robust DAG learning.
Revised: We thus propose to use geometric series-based DAG constraintwithout small coefﬁcients for more accurate and robust DAG learning.

Original: To relieve the computationalburden from the geometric series, we propose an efﬁcient Truncated Matrix Power Iteration (TMPI)algorithm with a theoretically guaranteed error bound.
Revised: To relieve the computationalburden from the geometric series, we propose an efﬁcient Truncated Matrix Power Iteration (TMPI)algorithm with a theoretically guaranteed error bound.

Original: We summarize our contributions as thefollowing:
Revised: We summarize our contributions as thefollowing:

Original: • We demonstrate that the key challenge for DAG learning is the gradient vanishing issueinstead of gradient and numeric exploding.
Revised: • We demonstrate that the key challenge for DAG learning is the gradient vanishing issueinstead of gradient and numeric exploding.

Original: For sparse graphs, the gradient vanishing issuecan be severe.
Revised: For sparse graphs, the gradient vanishing issuecan be severe.

Original: • We design a DAG constraint based on ﬁnite geometric series which is an order- d polynomialover adjacency matrices to escape from gradient vanishing.
Revised: • We design a DAG constraint based on ﬁnite geometric series which is an order- d polynomialover adjacency matrices to escape from gradient vanishing.

Original: We show that the relatively large
Revised: We show that the relatively large

Original: coefﬁcients on the higher order terms would not result in gradient exploding in practice.
Revised: coefﬁcients on the higher order terms would not result in gradient exploding in practice.

Original: • We show that there exists some constant k 6 d , s.t.
Revised: • We show that there exists some constant k 6 d , s.t.

Original: our order- d polynomial constraint
Revised: our order- d polynomial constraint

Original: can
Revised: can

Original: be reduced to order- k polynomial without expanding its feasible set.
Revised: be reduced to order- k polynomial without expanding its feasible set.

Original: Though ﬁnding suchvalue of k requires solving the NP-hard longest simple path problem, we develop a simpleheuristic to ﬁnd such k with bounded error to the exact, order- d , DAG constraint.
Revised: Though ﬁnding suchvalue of k requires solving the NP-hard longest simple path problem, we develop a simpleheuristic to ﬁnd such k with bounded error to the exact, order- d , DAG constraint.

Original: Based
Revised: Based

Original: on this result, we propose a DAG constraint based on efﬁcient Truncated Matrix PowerIteration
Revised: on this result, we propose a DAG constraint based on efﬁcient Truncated Matrix PowerIteration

Original: (TMPI), and conduct experiments to demonstrate that the resulting DAG learningmethod outperforms previous methods by a large margin.•
Revised: (TMPI), and conduct experiments to demonstrate that the resulting DAG learningmethod outperforms previous methods by a large margin.•

Original: We provide a systematical empirical comparison of the existing DAG constraints, including the polynomial and spectral radius based constraints, as well as the algebraic DAGrepresentation of NOCURL.
Revised: We provide a systematical empirical comparison of the existing DAG constraints, including the polynomial and spectral radius based constraints, as well as the algebraic DAGrepresentation of NOCURL.

Original:  Preliminaries
Revised:  Preliminaries

Original: DAG Model and Linear SEM Given a DAG model (a.k.a.
Revised: DAG Model and Linear SEM Given a DAG model (a.k.a.

Original: Bayesian network) deﬁned over randomvector x =
Revised: Bayesian network) deﬁned over randomvector x =

Original: [ x 1 , x 2 , . . .
Revised: [ x 1 , x 2 , . . .

Original: , x d ] > and DAG G , the distribution P ( x ) and DAG G are assumed to satisfythe Markov assumption [22, 32].
Revised: , x d ] > and DAG G , the distribution P ( x ) and DAG G are assumed to satisfythe Markov assumption [23, 33].

Original: We say that x follows a linear Structural Equation Model (SEM) if
Revised: We say that x follows a linear Structural Equation Model (SEM) if

Original: [Equation]
Revised: [Equation]

Original: where B is the weighted adjacency matrix representing the DAG G , and e = [ e 1 , e 2 , . .
Revised: where B is the weighted adjacency matrix representing the DAG G , and e = [ e 1 , e 2 , . .

Original: . , e d ] >denotes the exogenous noise vector consisting of d independent random variables.
Revised: . , e d ] >denotes the exogenous noise vector consisting of d independent random variables.

Original: With slight abuseof notation, we denote by G ( B ) the graph induced by weighted adjacency matrix B .
Revised: With slight abuseof notation, we denote by G ( B ) the graph induced by weighted adjacency matrix B .

Original: Moreover, we donot distinguish between random variables and vertices or nodes, and use these terms interchangeably.
Revised: Moreover, we donot distinguish between random variables and vertices or nodes, and use these terms interchangeably.

Original: Our goal is to estimate the DAG G from n i.i.d.
Revised: Our goal is to estimate the DAG G from n i.i.d.

Original: examples of x , indicated by the matrix X 2 R n ⇥ d .
Revised: examples of x , indicated by the matrix X 2 R n ⇥ d .

Original: Ingeneral, the DAG G can only be identiﬁed up to its Markov equivalence class under the faithfulness
Revised: Ingeneral, the DAG G can only be identiﬁed up to its Markov equivalence class under the faithfulness

Original: [32] or sparsest Markov representation [24] assumption.
Revised: [33] or sparsest Markov representation [25] assumption.

Original: It has been shown that for linear SEMs withhomoscedastic errors, from which the noise terms are speciﬁed up to a constant [17], and for linearnon-Gaussian SEMs, from which no more than one of the noise term is Gaussian, the true DAG canbe fully identiﬁed.
Revised: It has been shown that for linear SEMs withhomoscedastic errors, from which the noise terms are speciﬁed up to a constant [18], and for linearnon-Gaussian SEMs, from which no more than one of the noise term is Gaussian, the true DAG canbe fully identiﬁed.

Original: In this work, we focus on the linear SEMs with equal noise variances [23], whichcan be viewed as an instance of the former.
Revised: In this work, we focus on the linear SEMs with equal noise variances [24], whichcan be viewed as an instance of the former.

Original: DAG Constraints for Differentiable Structure Learning Recently, Zheng et al.
Revised: DAG Constraints for Differentiable Structure Learning Recently, Zheng et al.

Original: [41] reformulatedthe DAG learning problem as the following continuous optimization problem with convex leastsquares objective but non-convex constraint,
Revised: [43] reformulatedthe DAG learning problem as the following continuous optimization problem with convex leastsquares objective but non-convex constraint,

Original: [Equation]
Revised: [Equation]

Revised: subject

Original: [Equation]
Revised: [Equation]

Original: where k · k F and k · k 1 denote the Frobenius norm and elementwise ` 1 norm, respectively, ⌘ > 0 is 97 the regularization coefﬁcient that controls the sparsity of B , and  denotes the Hadamard product.
Revised: where k · k F and k · k 1 denote the Frobenius norm and elementwise ` 1 norm, respectively, ⌘ > 0 is 97 the regularization coefﬁcient that controls the sparsity of B , and  denotes the Hadamard product.

Original: Give any ˜ B 2 R d ⇥ d > 0 , the function h exp is deﬁned as
Revised: Give any ˜ B 2 R d ⇥ d > 0 , the function h exp is deﬁned as

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

Original: ) and can be efﬁciently computed using various different approaches [1, 2].
Revised: and can be efﬁciently computed using various different approaches [1, 2].

Original: The formulation (2) [Figure] involves a hard constraint, which is solved using the augmented Lagrangian method [3, 4] speciﬁcallydeveloped for constrained optimization.
Revised: The formulation (2)involves a hard constraint, which is solved using the augmented Lagrangian method [3, 4] speciﬁcallydeveloped for constrained optimization.

Original: To improve practical convenience, Yu et al.
Revised: To improve practical convenience, Yu et al.

Original: [36] proposed a similar DAG constraint using an order- dpolynomial based on powers of a binomial, given by
Revised: [38] proposed a similar DAG constraint using an order- dpolynomial based on powers of a binomial, given by

Original: [Equation]
Revised: [Equation]

Original: where ↵ > 0 is a hyperparameter and is often set to 1 /d , e.g., in the implementation of DAG-GNN[36].
Revised: where ↵ > 0 is a hyperparameter and is often set to 1 /d , e.g., in the implementation of DAG-GNN[38].

Original: This function can be evaluated using O (log d ) matrix multiplications using the exponentiationby squaring algorithm.
Revised: This function can be evaluated using O (log d ) matrix multiplications using the exponentiationby squaring algorithm.

Original: Both h exp ( ˜ B ) and h bin ( ˜ B ) are formulated by [35] as a uniﬁed one
Revised: Both h exp ( ˜ B ) and h bin ( ˜ B ) are formulated by [37] as a uniﬁed one

Original: [Equation]
Revised: [Equation]

Original:  DAG Learning using Truncated Matrix Power Iteration
Revised:  DAG Learning using Truncated Matrix Power Iteration

Original: In this section, we ﬁrst show that gradient vanishing is one main issue for previous DAG constraintsand propose a truncated geometric series based DAG constraints to escape from the issue in Section3.1.
Revised: In this section, we ﬁrst show that gradient vanishing is one main issue for previous DAG constraintsand propose a truncated geometric series based DAG constraints to escape from the issue in Section3.1.

Original: Then in Section 3.2 the theoretic properties of our DAG constraints are analyzed.
Revised: Then in Section 3.2 the theoretic properties of our DAG constraints are analyzed.

Original: In Section 3.3we provide an efﬁcient algorithm for our DAG constraints.
Revised: In Section 3.3we provide an efﬁcient algorithm for our DAG constraints.

Original: In Section 3.4, we discuss the relationbetween our algorithm and previous works.
Revised: In Section 3.4, we discuss the relationbetween our algorithm and previous works.

Original: 3.1 Truncated Geometric DAG Constraints
Revised: 3.1 Truncated Geometric DAG Constraints

Original: Theoretically, any positive values of c i sufﬁce to make the DAG constraint term h poly ( ˜ B ) in Eq.(5) a valid one, i.e., it equals zero if and only if the graph G ( ˜ B ) is acyclic.
Revised: Theoretically, any positive values of c i sufﬁce to make the DAG constraint term h poly ( ˜ B ) in Eq.(5) a valid one, i.e., it equals zero if and only if the graph G ( ˜ B ) is acyclic.

Original: Thus previously smallcoefﬁcients for higher order terms are preferred since they are known to have better resistance tonumerical exploding of higher order polynomials [36, 41].
Revised: Thus previously smallcoefﬁcients for higher order terms are preferred since they are known to have better resistance tonumerical exploding of higher order polynomials [38, 43].

Original: These small coefﬁcients forbid informativehigher order terms and cause severe gradient vanishing, as nilpotent properties of DAG result incandidate adjacency matrix with very small spectral radius (see empirical results in Figure 1 Left).
Revised: These small coefﬁcients forbid informativehigher order terms and cause severe gradient vanishing, as nilpotent properties of DAG result incandidate adjacency matrix with very small spectral radius (see empirical results in Figure 1 Left).

Revised: In this case, the weight updates during the optimization process can hardly utilize this higher-orderinformation from the vanishingly small gradients.

Revised: As a consequence, only lower-order informationcan be effectively captured, which may be detrimental to the quality of learned DAG.

Original: To avoid the severe vanishing gradient issue in previous DAG constraint ( e.g. , Eq.
Revised: To avoid the severe vanishing gradient issue in previous DAG constraint ( e.g. , Eq.

Original: (3),(4)), we propose to discard the coefﬁcients and use a ﬁnite geometric series based function:
Revised: (3),(4)), we propose to discard the coefﬁcients and use a ﬁnite geometric series based function:

Original: [Equation]
Revised: [Equation]

Original: The gradient of h geo w.r.t.
Revised: The gradient of h geo w.r.t.

Original: ˜ B denoted by r ˜ B h geo ( ˜ B ) is to an order- ( d  1) polynomial of ˜ B .
Revised: ˜ B denoted by r ˜ B h geo ( ˜ B ) is to an order- ( d  1) polynomial of ˜ B .

Original: With the same d , Eq.
Revised: With the same d , Eq.

Original: (6) is a more precise DAG constraint escaped from from severe gradientvanishing.
Revised: (6) is a more precise DAG constraint escaped from from severe gradientvanishing.

Original: Also it is notable that due to the nilpotent properties of weighted adjacency matricesof DAGs, the higher order terms in (6) and (7) often comes with small norms and thus our DAGconstraint would not suffer from numerical exploding.
Revised: Also it is notable that due to the nilpotent properties of weighted adjacency matricesof DAGs, the higher order terms in (6) and (7) often comes with small norms and thus our DAGconstraint would not suffer from numerical exploding.

Original: In Figure 1 we compared the norm of gradients on higher order terms of different DAG constraints,where we can see that for previous exponential-based (3) and binomial-based (4) DAG constraints,the gradients of higher order terms converges to 0 very quickly as order increases.
Revised: In Figure 1 we compared the norm of gradients on higher order terms of different DAG constraints,where we can see that for previous exponential-based (3) and binomial-based (4) DAG constraints,the gradients of higher order terms converges to 0 very quickly as order increases.

Original: Meanwhile, thegradient of higher order terms in (6) converges to 0 with a much slower speed and thus informativehigher order terms are reserved.
Revised: Meanwhile, thegradient of higher order terms in (6) converges to 0 with a much slower speed and thus informativehigher order terms are reserved.

Original: We further show that those non-informative higher-order terms that are close to zero in (6) can besafely dropped to form a more efﬁcient DAG constraint with some k ⌧ d :
Revised: We further show that those non-informative higher-order terms that are close to zero in (6) can besafely dropped to form a more efﬁcient DAG constraint with some k ⌧ d :

Original: [Equation]
Revised: [Equation]

Original: The DAG constraints (8) requires O ( k ) matrix multiplication which is far less than O ( d ) for (6), andunder quite mild conditions the error of (8) is tightly bounded (See Proposition 3 for details).
Revised: The DAG constraints (8) requires O ( k ) matrix multiplication which is far less than O ( d ) for (6), andunder quite mild conditions the error of (8) is tightly bounded (See Proposition 3 for details).

Original: In Figure1 right we show that our DAG constraint (8) (named TMPI) attains far more better performance thanprevious ones as it escapes from gradient vanishing.
Revised: In Figure1 right we show that our DAG constraint (8) (named TMPI) attains far more better performance thanprevious ones as it escapes from gradient vanishing.

Original: 3.2 Theoretic Properties for Truncated Geometric DAG Constraint
Revised: 3.2 Theoretic Properties for Truncated Geometric DAG Constraint

Original: We analyze the properties of the power of adjacency matrices, which play an important role inconstructing DAG constraints.
Revised: We analyze the properties of the power of adjacency matrices, which play an important role inconstructing DAG constraints.

Original: First notice that, for the k th power of an adjacency matrix ˜ B
Revised: First notice that, for the k th power of an adjacency matrix ˜ B

Original: 2 R d ⇥ d > 0 ,its entry ( ˜ B k ) ij is nonzero if and only if there is a length- k path (not necessarily simple) from node
Revised: 2 R d ⇥ d > 0 ,its entry ( ˜ B k ) ij is nonzero if and only if there is a length- k path (not necessarily simple) from node

Original: X i to node X j in graph G ( ˜ B ) .
Revised: X i to node X j in graph G ( ˜ B ) .

Original: This straightforwardly leads to the following property.
Revised: This straightforwardly leads to the following property, which is acorollary of [37, Lemma 1].

Original: Proposition 1.
Revised: Proposition 1.

Original: Let ˜ B 2 R d ⇥ d > 0 be the weighted adjacency matrix of a graph G with d vertices.
Revised: Let ˜ B 2 R d ⇥ d > 0 be the weighted adjacency matrix of a graph G with d vertices.

Original: G is a
Revised: G is a

Original: DAG
Revised: DAG

Original: if and only if ˜ B d = 0 .
Revised: if and only if ˜ B d = 0 .

Original: Proposition 1 can be used to construct a single-term DAG constraint term, given by
Revised: Proposition 1 can be used to construct a single-term DAG constraint term, given by

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: where k · k p denotes an elementwise ` p norm.
Revised: where k · k p denotes an elementwise ` p norm.

Original: However, as the candidate adjacency matrix gets closerto acyclic, the spectral radius of ˜ B d will become very small.
Revised: However, as the candidate adjacency matrix gets closerto acyclic, the spectral radius of ˜ B d will become very small.

Original: As a result, the value h single ( ˜ B ) , as wellas its gradient r ˜ B h single ( ˜ B ) may become too small to be represented in limited machine precision.
Revised: As a result, the value h single ( ˜ B ) , as wellas its gradient r ˜ B h single ( ˜ B ) may become too small to be represented in limited machine precision.

Original: This suggests the use of the polynomial based constraints for better performance.
Revised: This suggests the use of the polynomial based constraints for better performance.

Original: Without any prior information of the graph, one has to compute the d th power of matrix ˜ B , eitherusing the DAG constraint terms in Eq.
Revised: Without any prior information of the graph, one has to compute the d th power of matrix ˜ B , eitherusing the DAG constraint terms in Eq.

Original: (9) or (5).
Revised: (9) or (5).

Original: However, if the length of the longest simple path inthe graph is known, we can formulate a lower order polynomial based DAG constraint.
Revised: However, if the length of the longest simple path inthe graph is known, we can formulate a lower order polynomial based DAG constraint.

Original: Proposition 2.
Revised: Proposition 2.

Original: Let ˜ B 2 R d ⇥ d > 0 be the weighted adjacency matrix of a graph G with d vertices, and kbe the length of the longest simple path in graph G .
Revised: Let ˜ B 2 R d ⇥ d > 0 be the weighted adjacency matrix of a graph G with d vertices, and kbe the length of the longest simple path in graph G .

Original: The following three conditions are equivalent: (1) G is a DAG, (2)
Revised: The following three conditions are equivalent:(1) G is a DAG, (2)

Original: ˜ B k +1 = 0 , and (3) h k trunc ( ˜ B ) = 0 .
Revised: ˜ B k +1 = 0 , and (3) h k trunc ( ˜ B ) = 0 .

Original: The intuition of the above proposition is as follows.
Revised: The intuition of the above proposition is as follows.

Original: For the graph whose largest cycle has a length ofk , it sufﬁces to compute the k th power of its adjacency matrix, since computing any power larger thank is simply traveling in the cycles repeatedly.
Revised: For the graph whose largest cycle has a length ofk , it sufﬁces to compute the k th power of its adjacency matrix, since computing any power larger thank is simply traveling in the cycles repeatedly.

Original: Therefore, Proposition 2 suggests that h k trunc ( ˜ B ) =
Revised: Therefore, Proposition 2 suggests that h k trunc ( ˜ B ) =

Original: isa valid DAG constraint.
Revised: isa valid DAG constraint.

Original: Unfortunately, the longest simple path problem for general graphs is known to be NP-hard [26], andthus we have to ﬁnd an alternative way to seek the proper k .
Revised: Unfortunately, the longest simple path problem for general graphs is known to be NP-hard [27], andthus we have to ﬁnd an alternative way to seek the proper k .

Original: For a candidate adjacency matrix B , asimple heuristic is to iterate over the power of matrices to ﬁnd some k s.t. matrix ˜ B k is close enoughto a zero matrix.
Revised: For a candidate adjacency matrix B , asimple heuristic is to iterate over the power of matrices to ﬁnd some k s.t. matrix ˜ B k is close enoughto a zero matrix.

Original: Recall that Proposition 1 implies that the adjacency matrix of a DAG is nilpotent;therefore, when a candidate adjacency matrix ˜ B is close to being acyclic, its spectral radius would beclose to zero, which means that the ˜ B k would converge quickly to zero as k increases.
Revised: Recall that Proposition 1 implies that the adjacency matrix of a DAG is nilpotent;therefore, when a candidate adjacency matrix ˜ B is close to being acyclic, its spectral radius would beclose to zero, which means that the ˜ B k would converge quickly to zero as k increases.

Original: Thus, if weuse the above heuristic to ﬁnd k , it is highly possible
Revised: Thus, if weuse the above heuristic to ﬁnd k , it is highly possible

Original: we ﬁnd some k ⌧ d .
Revised: we ﬁnd some k ⌧ d .

Original: By using an approximate k instead of the exact one, the DAG constraints in Eq.
Revised: By using an approximate k instead of the exact one, the DAG constraints in Eq.

Original: (8) becomes anapproximate one with the following bounds.
Revised: (8) becomes anapproximate one with the following bounds.

Revised: Denoting by k · k 1 the element-wise inﬁnity norm (i.e., 174 maximum norm), we have the following proposition.

Original: Proposition 3.
Revised: Proposition 3.

Original: Given ˜ B 2 R d ⇥ d > 0 , if there exists k < d such that k ˜ B k k 1 6 ✏< 1 , then we have
Revised: Given ˜ B 2 R d ⇥ d > 0 , if there exists k < d such that k ˜ B k k 1 6 ✏ , then we have

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: Remark 1. Given the weighted adjacency matrix ˜ B and k < d deﬁned in Proposition 3, the termsh geo ( ˜ B )  h k trunc ( ˜ B ) and kr ˜ B h geo ( ˜ B )  r ˜ B
Revised: Proposition 3 provides an efﬁcient way to bound the error of truncated constraints h k trunc ( ˜ B ) to thegeometric series based DAG constraints h geo ( ˜ B ) .

Original: h k trunc ( ˜ B ) k 2 are upper bounded by O ( d ✏ 1+1 /k ) and

Original: O ( d ✏ ) , respectively.

Original: Proposition 3 provides an efﬁcient way to bound the error of truncated constraints h k trunc ( ˜ B ) to thegeometric series based DAG constraints h geo ( ˜ B ) .

Original: However, it also demonstrates a possible limitationfor all polynomial based DAG constraints.
Revised: However, it also demonstrates a possible limitationfor all polynomial based DAG constraints.

Original: As we know, when a solution is close to DAG, its spectralradius must be close to zero.
Revised: As we know, when a solution is close to DAG, its spectralradius must be close to zero.

Original: In this situation, if the longest simple path in a graph is very long, it isunavoidable to use very high order terms to capture this information while such higher order mightbe very close to 0.
Revised: In this situation, if the longest simple path in a graph is very long, it isunavoidable to use very high order terms to capture this information while such higher order mightbe very close to 0.

Original: In the worst case, the higher terms may be underﬂow due to limited machineprecision.
Revised: In the worst case, the higher terms may be underﬂow due to limited machineprecision.

Original: Thus some speciﬁc numerically stable method may be required, and we leave this forfuture work.
Revised: Thus some speciﬁc numerically stable method may be required, and we leave this forfuture work.

Original: 3.3 Dynamic Truncated Matrix Power Iteration
Revised: 3.3 Dynamic Truncated Matrix Power Iteration

Original: From Proposition 3, we know that it would be safe to ignore those matrix powers with index largerthan k .
Revised: From Proposition 3, we know that it would be safe to ignore those matrix powers with index largerthan k .

Original: The DAG constraint and its gradient, become order- k polynomials of B .
Revised: The DAG constraint and its gradient, become order- k polynomials of B .

Original: A straightforwardway to compute the DAG constraint as well as the gradient is provided in Algorithm 1, which wecall TMPI.
Revised: A straightforwardway to compute the DAG constraint as well as the gradient is provided in Algorithm 1, which wecall TMPI.

Original: For general polynomial based DAG constraints in (5), it requires O ( d ) times matrixmultiplication, and our Algorithm 1 can reduce the number of multiplication to O ( k ) (in practice k ismuch smaller than d ).
Revised: For general polynomial based DAG constraints in (5), it requires O ( d ) times matrixmultiplication, and our Algorithm 1 can reduce the number of multiplication to O ( k ) (in practice k ismuch smaller than d ).

Original: For example, when recovering 100 -node Erd˝os–Rényi DAG with expecteddegree 4 , k is often smaller than 20 .
Revised: For example, when recovering 100 -node Erd˝os–Rényi DAG with expecteddegree 4 , k is often smaller than 20 .

Original: Fast Truncated Matrix Power Iteration
Revised: Fast Truncated Matrix Power Iteration

Original: The speciﬁc structure of geometric series can be used tofurther accelerate the TMPI algorithm from O ( k ) to O (log k ) matrix multiplications, based on thefollowing results (the proof is provided in Appendix A.4).
Revised: The speciﬁc structure of geometric series can be used tofurther accelerate the TMPI algorithm from O ( k ) to O (log k ) matrix multiplications, based on thefollowing results (the proof is provided in Appendix A.5).

Original: Proposition 4.
Revised: Proposition 4.

Original: Given any d ⇥ d real matrix ˜ B , let f i ( ˜ B ) =
Revised: Given any d ⇥ d real matrix ˜ B , let f i ( ˜ B ) =

Original: ˜ B + ˜ B 2 + · · · + ˜ B i and h i ( ˜ B )
Revised: ˜ B + ˜ B 2 + · · · + ˜ B i and h i ( ˜ B )

Original: = tr( f i ( ˜ B )) ,we have the following recurrence relations:
Revised: = tr( f i ( ˜ B )) ,we have the following recurrence relations:

Original: [Equation]
Revised: [Equation]

Original: It is straightfoward to obtain the following corollary by setting i =
Revised: It is straightfoward to obtain the following corollary by setting i =

Original: j in Proposition 4.
Revised: j in Proposition 4.

Original: Corollary 1.
Revised: Corollary 1.

Original: Given any matrix d ⇥ d real matrix ˜ B , let f i ( ˜ B ) =
Revised: Given any matrix d ⇥ d real matrix ˜ B , let f i ( ˜ B ) =

Original: ˜ B + ˜ B 2 + · · · + ˜ B i and h i ( ˜ B )
Revised: ˜ B + ˜ B 2 + · · · + ˜ B i and h i ( ˜ B )

Original: =tr( f i ( ˜ B )) , we have have the following recurrence relations:
Revised: =tr( f i ( ˜ B )) , we have have the following recurrence relations:

Original: [Equation]
Revised: [Equation]

Original: This suggests that instead of increasing the index i (referring to the power of ˜ B ), arithmeticallyby 1 , as in line 9 of Algorithm 1, we can use the recurrence relations in Corollary 1 to increasethe index i geometrically by a factor of 2 , until the condition k ˜ B i k 1 6 ✏ is satisﬁed.
Revised: This suggests that instead of increasing the index i (referring to the power of ˜ B ), arithmeticallyby 1 , as in line 9 of Algorithm 1, we can use the recurrence relations in Corollary 1 to increasethe index i geometrically by a factor of 2 , until the condition k ˜ B i k 1 6 ✏ is satisﬁed.

Original: The fullprocedure named Fast TMPI is available in Algorithm 2, which requires O (log k ) matrix multiplications and O ( d 2 ) additional storage.
Revised: The fullprocedure named Fast TMPI is available in Algorithm 2, which requires O (log k ) matrix multiplications and O ( d 2 ) additional storage.

Original: Similar strategy can be applied to the original geometricseries based constraint (6) to form a fast algorithm that requires O (log d ) matrix multiplications.
Revised: Similar strategy can be applied to the original geometricseries based constraint (6) to form a fast algorithm that requires O (log d ) matrix multiplications.

Original: TMPI)
Revised: TMPI)

Original: Input ˜ B 2 R d ⇥ d > 0 , ✏
Revised: Input ˜ B 2 R d ⇥ d > 0 , ✏

Original: Output h (˜ B ) , r ˜ B h (˜ B ) 1: i 2 , h (˜ B ) tr(˜ B ) , r ˜ B h (˜ B )
Revised: Output h (˜ B ) , r ˜ B h (˜ B ) 1: i 2 , h (˜ B ) tr(˜ B ) , r ˜ B h (˜ B )

Original: I 2: while i 6 d do 3: r ˜ B h (˜ B ) r ˜ B h (˜ B ) + i (˜ B i  1 ) > 4: ˜ B i ˜ B
Revised: I 2: while i 6 d do 3: r ˜ B h (˜ B ) r ˜ B h (˜ B ) + i (˜ B i  1 ) > 4: ˜ B i ˜ B

Original: ˜ B i 5: h (˜ B ) h (˜ B )
Revised: ˜ B i 5: h (˜ B ) h (˜ B )

Original: + tr(˜ B i ) 6:
Revised: + tr(˜ B i ) 6:

Original: if k ˜ B
Revised: if k ˜ B

Original: i k 1 6 ✏ then 7: break 8: end if 9:
Revised: i k 1 6 ✏ then 7: break 8: end if 9:

Original: i i +10: end while 11: Output h (˜ B ) , r ˜ B h (˜ B )
Revised: i i +10: end while 11: Output h (˜ B ) , r ˜ B h (˜ B )

Original: Input ˜ B 2 R d ⇥ d > 0 , ✏
Revised: Input ˜ B 2 R d ⇥ d > 0 , ✏

Original: Output h ( ˜ B ) , r h ( ˜ B ) 1: i 2: ˜ B f ˜ B , ˜ B g
Revised: Output h ( ˜ B ) , r h ( ˜ B ) 1: i 2: ˜ B f ˜ B , ˜ B g

Original: I , ˜ B p ˜ B { ˜ B f stores f i , ˜ B
Revised: I , ˜ B p ˜ B { ˜ B f stores f i , ˜ B

Original: f stores r tr f i , and ˜ B p stores ˜ B i } 3: while i 6 d do 4: ˜ B oldf ˜ B f , ˜ B oldg ˜ B g , ˜ B oldp
Revised: f stores r tr f i , and ˜ B p stores ˜ B i } 3: while i 6 d do 4: ˜ B oldf ˜ B f , ˜ B oldg ˜ B g , ˜ B oldp

Original: ˜ B p 5: ˜ B f ˜ B oldf + ˜ B ⇥ ˜ B oldf , ˜ B p ˜ B oldp ⇥ ˜ B oldp 6: ˜ B g ˜ B oldg + ˜ B > p ˜ B oldg + i (˜ B f  ˜ B oldf + ˜ B oldp  ˜ B p ) 7: if k ˜ B p k 1 6 ✏ break 8: i 2 i 9: end while 10: Output tr h ˜ B
Revised: ˜ B p 5: ˜ B f ˜ B oldf + ˜ B ⇥ ˜ B oldf , ˜ B p ˜ B oldp ⇥ ˜ B oldp 6: ˜ B g ˜ B oldg + ˜ B > p ˜ B oldg + i (˜ B f  ˜ B oldf + ˜ B oldp  ˜ B p ) 7: if k ˜ B p k 1 6 ✏ break 8: i 2 i 9: end while 10: Output tr h ˜ B

Original: f
Revised: f

Original: i , ˜ B g
Revised: i , ˜ B g

Original: The Full Optimization Framework For NOTEARS, we apply the same augmented Lagrangianframework [3, 4] as Zheng et al.
Revised: The Full Optimization Framework For NOTEARS, we apply the same augmented Lagrangianframework [3, 4] as Zheng et al.

Original: [41] to convert the constrained optimization problem into thefollowing unconstrained problem
Revised: [43] to convert the constrained optimization problem into thefollowing unconstrained problem

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

Original: where the parameters ⇢ and ↵ are iteratively updated using the same strategy as Zheng et al.
Revised: where the parameters ⇢ and ↵ are iteratively updated using the same strategy as Zheng et al.

Original: Wealso apply the proposed DAG constraint to GOLEM [18] that adopts likelihood-based objective withsoft constraints, leading to the unconstrained optimization problem
Revised: Wealso apply the proposed DAG constraint to GOLEM [19] that adopts likelihood-based objective withsoft constraints, leading to the unconstrained optimization problem

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

Original: where ⌘ 1 , dynamically computed  updated  optimization iteration  Algorithm 1 or 2.
Revised: where ⌘ 1 , formulations value k isdynamically computed and updated during each optimization iteration based on Algorithm 1 or 2.

Original: 3.4 Relation to Previous Works
Revised: 3.4 Relation to Previous Works

Original: Our algorithm is closely related to the following DAG constraint considered by Zheng et al.
Revised: Our algorithm is closely related to the following DAG constraint considered by Zheng et al.

Original: Lemma 1 (Zheng et al.
Revised: Lemma 1 (Zheng et al.

Original: [41, Proposition 1]) .
Revised: [43, Proposition 1]) .

Original: Let ˜ B 2 R d ⇥ d > 0 with spectral radius small than one bethe weighted adjacency matrix of a graph G with d vertices.
Revised: Let ˜ B 2 R d ⇥ d > 0 with spectral radius small than one bethe weighted adjacency matrix of a graph G with d vertices.

Original: G is a DAG if and only if
Revised: G is a DAG if and only if

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: The inverse of I  ˜ B is just the geometric series as follows,
Revised: The inverse of I  ˜ B is just the geometric series as follows,

Original: [Equation]
Revised: [Equation]

Original: Zheng et al.
Revised: Zheng et al.

Original: [41] argue that (13) is not well deﬁned for ˜ B with spectral radius larger than 1, and inpractice the series (13) may explode very quickly Based on this argument, they choose to use thematrix exponential to construct the DAG constraints.
Revised: [43] argue that (13) is not well deﬁned for ˜ B with spectral radius larger than 1, and inpractice the series (13) may explode very quickly Based on this argument, they choose to use thematrix exponential to construct the DAG constraints.

Original: For the same reason Yu et al.
Revised: For the same reason Yu et al.

Original: [36] also use verysmall coefﬁcients for higher terms in polynomial based DAG constraints.
Revised: [38] also use verysmall coefﬁcients for higher terms in polynomial based DAG constraints.

Original: However, as we show in
Revised: However, as we show in

Original: Section 3, Yu et al.
Revised: Section 3, Yu et al.

Original: [36], Zheng et al.
Revised: [38], Zheng et al.

Original: [41] actually suffers more from gradient vanishing due to smallcoefﬁcients for higher terms.
Revised: [43] actually suffers more from gradient vanishing due to smallcoefﬁcients for higher terms.

Original: In fact, our constraints (8) provide a good approximate for (12) sincewe are actually ﬁnd an approximate inverse of I  ˜ B with a bounded error.
Revised: In fact, our constraints (8) provide a good approximate for (12) sincewe are actually ﬁnd an approximate inverse of I  ˜ B with a bounded error.

Original: Proposition 5.
Revised: Proposition 5.

Original: Given any d ⇥ d real matrix ˜ B , if for some k , k ˜ B k k 1 6 ✏ < 1  , then
Revised: Given ˜ B 2 R d ⇥ d > 0 , if for some k , k ˜ B k k 1 6 ✏ < 1 /d , then

Original: [Equation]
Revised: [Equation]

Original: Note that the inﬁnity norm k B k k 1 can be a loose upper bound for the spectral radius k B k k 2 , and 227 thus in practice the actual error can be even better than the above bound.

Original: Previous works [19, 35] pointed out that for constraints with form h ( B  B ) , the Hadamard productwould lead to vanishing gradients since r B h ( B  B )
Revised: Previous works [20, 37] pointed out that for constraints with form h ( B  B ) , the Hadamard productwould lead to vanishing gradients since r B h ( B  B )

Original: = 

Revised: =

Original: if and only if h ( B  B ) = 0 .
Revised: if and only if h ( B  B ) = 0 .

Original: Therefore,this may lead to ill-conditioning issue [19].
Revised: Therefore,this may lead to ill-conditioning issue [20].

Original: One may replace the Hadamard product with absolutevalue, as in Wei et al.
Revised: One may replace the Hadamard product with absolutevalue, as in Wei et al.

Original: [35], but the non-smoothness of absolute value makes the optimization problemdifﬁcult to solve and may lead to poorer performance [19].
Revised: [37], but the non-smoothness of absolute value makes the optimization problemdifﬁcult to solve and may lead to poorer performance [20].

Original: As pointed out by Ng et al.
Revised: As pointed out by Ng et al.

Original: [19], QuasiNewtown methods, e.g., L-BFGS [6], still perform well even when faced with the ill-conditioningissue caused by vanishing gradients of constraints with form h ( B  B ) .
Revised: [20], QuasiNewtown methods, e.g., L-BFGS [6], still perform well even when faced with the ill-conditioningissue caused by vanishing gradients of constraints with form h ( B  B ) .

Original: In this work, we identifyanother type of gradient vanishing issue caused by (1) the higher order terms and small coefﬁcientsin h itself and (2) the small spectral radius of candidate B , and provide an effective solution byconstructing a Truncated Matrix Power Iteration based DAG constraint.
Revised: In this work, we identifyanother type of gradient vanishing issue caused by (1) the higher order terms and small coefﬁcientsin h itself and (2) the small spectral radius of candidate B , and provide an effective solution byconstructing a Truncated Matrix Power Iteration based DAG constraint.

Original:  Experiments
Revised:  Experiments

Original: We compare the performance of different DAG learning algorithms and DAG constraints to demonstrate the effectiveness of the proposed Truncated Matrix Power Iteration (TMPI)-based DAGconstraints.
Revised: We compare the performance of different DAG learning algorithms and DAG constraints to demonstrate the effectiveness of the proposed Truncated Matrix Power Iteration (TMPI)-based DAGconstraints.

Original: We give an outline of the experimental settings in this section, and more details includingthe parameter settings can be found in supplementary ﬁles 1 .
Revised: We give an outline of the experimental settings in this section, and more details includingthe parameter settings can be found in supplementary ﬁles 1 .

Original: 4.1 DAG Learning
Revised: 4.1 DAG Learning

Original: We ﬁrst conduct experiments on synthetic datasets using similar settings as previous works [36, 39, 41].
Revised: We ﬁrst conduct experiments on synthetic datasets using similar settings as previous works [38, 39, 43].

Original: In summary, random Erdös-Rényi graphs are generated with d nodes and kd expected edges (Denotedby ER k ), and edge weights generated from uniform distribution over the union of two intervals[  2 ,  0 .
Revised: In summary, random Erdös-Rényi graphs are generated with d nodes and kd expected edges (Denotedby ER k ), and edge weights generated from uniform distribution over the union of two intervals[  2 ,  0 .

Original: [ [0 . 5 , 2 . 0] are assigned to each edge to form a weighted adjacency matrix B .
Revised: [ [0 . 5 , 2 . 0] are assigned to each edge to form a weighted adjacency matrix B .

Original: Then nrandom samples are generated from model x
Revised: Then nrandom samples are generated from model x

Original: = B x + e to form an n ⇥ d data matrix X , where thenoise e are i.i.d.
Revised: = B x + e to form an n ⇥ d data matrix X , where thenoise e are i.i.d.

Original: sampled from Gaussian, Exponential, or Gumbel distribution.
Revised: sampled from Gaussian, Exponential, or Gumbel distribution.

Original: In our experiments,we set n = 1000 , and 4 different d = 10 , 30 , 50 , 100 are considered.
Revised: In our experiments,we set n = 1000 , and 4 different d = 10 , 30 , 50 , 100 are considered.

Original: For each setting, we generate100 different graphs to obtain an average performance.
Revised: For each setting, we generate100 different graphs to obtain an average performance.

Original: Models and Baselines We consider DAG learning framework based on both Mean Square Error(MSE) loss (see Eq.
Revised: Models and Baselines We consider DAG learning framework based on both Mean Square Error(MSE) loss (see Eq.

Original: and likelihood-based (see Eq.
Revised: and likelihood-based (see Eq.

Original: loss from GOLEM [18].
Revised: loss from GOLEM [19].

Original: For bothlosses, we include the following DAG constraints: (1) MSE/Likelihood loss+Exponential DAG constraint (3), i.e. NOTEARS [41]/GOLEM[18] ; (2) MSE/Likelihood loss+Binomial DAG constraint(4) from DAG-GNN [36]; (3) MSE/Likelihood loss+Spectral radius based DAG constraint from
Revised: For bothlosses, we include the following DAG constraints: (1) MSE/Likelihood loss+Exponential DAG constraint (3), i.e. NOTEARS [43]/GOLEM[19] ; (2) MSE/Likelihood loss+Binomial DAG constraint(4) from DAG-GNN [38]; (3) MSE/Likelihood loss+Spectral radius based DAG constraint from

Original: NOBEARS [16]; (4) MSE/Likelihood loss+Single
Revised: NOBEARS [17]; (4) MSE/Likelihood loss+Single

Original: Term DAG constraint (9); (5) MSE/Likelihoodloss+Truncated Matrix Power Iteration (TMPI)
Revised: Term DAG constraint (9); (5) MSE/Likelihoodloss+Truncated Matrix Power Iteration (TMPI)

Original: based DAG constraint (8).
Revised: based DAG constraint (8).

Original: For the NOCURL [39]
Revised: For the NOCURL [39]

Original: DAG constraint, since it is designed as a two stage methods where the initialization stage rely on
Revised: DAG constraint, since it is designed as a two stage methods where the initialization stage rely on

Original: MSE loss and Binomial DAG constraint, we include it as a baseline, but do not combine it withlikelihood loss.
Revised: MSE loss and Binomial DAG constraint, we include it as a baseline, but do not combine it withlikelihood loss.

Original: The other baselines that we are comparing to including GES [8] and MMPC [33].
Revised: The other baselines that we are comparing to including GES [8] and MMPC [35].

Original: For MSE loss based DAG learning algorithms (10)), we set ⌘ = 0 .
Revised: For MSE loss based DAG learning algorithms (10)), we set ⌘ = 0 .

Original: 1 and use the same searchstrategy as Zheng et al.
Revised: 1 and use the same searchstrategy as Zheng et al.

Original: [41] to update the parameters ⇢ and ↵ .
Revised: [43] to update the parameters ⇢ and ↵ .

Original: For likelihood loss based DAG learning algorithms (11), we set ⌘ 1 = 0 .
Revised: For likelihood loss based DAGlearning algorithms (11), we set ⌘ 1 = 0 .

Original: 02 and ⌘ 2 = 5 . 

Revised: 02 and ⌘ 2 = 5 .

Original: as Ng et al.
Revised: as Ng et al.

Original: For the Binomial DAGconstraints (4), the parameter ↵ is set to 1 /d as [36].
Revised: For the Binomial DAG constraints (4), the parameter ↵ is set to 1 /d as [38].

Original: For our TMPI DAG constraint, we set theparameter ✏ = 10  6 .
Revised: For our TMPI DAG constraint, we set theparameter ✏ = 10  6 .

Original: For the NOCURL algorithm, we use their public available implementationand recommended parameter setting.
Revised: For the NOCURL algorithm, we use their public available implementationand recommended parameter setting.

Original: For GES and MMPC, we use the implementation and defaultparameter setting in Kalainathan and Goudet [12].
Revised: For GES and MMPC, we use the implementation and defaultparameter setting in Kalainathan and Goudet [13].

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

Original: Results
Revised: Results

Original: The result for MSE/likelihood based DAG learning algorithm is shown in Figure 2/3 ,where we compared the Structural Hamming Distance (SHD) of different algorithms.
Revised: The result for MSE/likelihood based DAG learning algorithm is shown in Figure 2/3 ,where we compared the Structural Hamming Distance (SHD) of different algorithms.

Original: The Original
Revised: The Original

Original: NOTEARS ( i.e. MSE+Exponential) / GOLEM ( i.e. GOLEM+Exponential) has similar performanceas MSE/GOLEM+Binomial.
Revised: NOTEARS ( i.e. MSE+Exponential) / GOLEM ( i.e. GOLEM+Exponential) has similar performanceas MSE/GOLEM+Binomial.

Original: This is as expected since the coefﬁcients of each order in Exponentialand Binomial constraints are actually very close (See Figure 1).
Revised: This is as expected since the coefﬁcients of each order in Exponentialand Binomial constraints are actually very close (See Figure 1).

Original: The performance of algorithmsusing single-term loss is often not comparable to the other polynomial-based constraints since itsuffers more from gradient vanishing.
Revised: The performance of algorithmsusing single-term loss is often not comparable to the other polynomial-based constraints since itsuffers more from gradient vanishing.

Original: The performance of algorithms using the spectral radius basedconstraints (BEARS) are also worse than the ones using Exponential and Binomial constraints.
Revised: The performance of algorithms using the spectral radius basedconstraints (BEARS) are also worse than the ones using Exponential and Binomial constraints.

Original: Thealgorithms using our TMPI constraints outperforms all others in almost 
Revised: Thealgorithms using our TMPI constraints outperforms all others in almost all cases, which demonstratesthe effectiveness of our DAG constraint.

Original: 4.2 Extension to Nonlinear Case
Revised: 4.2 Extension to Nonlinear Case

Original: Our algorithm are mainly designed for linear cases but it can be straightforwardly extended tononlinear cases.
Revised: Our algorithm are mainly designed for linear cases but it can be straightforwardly extended tononlinear cases.

Original: In this section, we extend our DAG constraint to nonlinear case by replacing thebinomial DAG constraint in DAG-GNN [36], NOTEARS-MLP[42] and GRAN-DAG[15] with ours.
Revised: In this section, we extend our DAG constraint to nonlinear case by replacing thebinomial DAG constraint in DAG-GNN [38], NOTEARS-MLP[44] and GRAN-DAG[16] with ours.

Original: Synthetic data We conducted an experiment to compare our algorithm and DAG-GNN on 50node ER1 graph with nonlinear SEM x = B cos( x +1) + e over 5 simulations.
Revised: Synthetic data We conducted an experiment to compare our algorithm and DAG-GNN on 50node ER1 graph with nonlinear SEM x = B cos( x +1) + e over 5 simulations.

Original: Our algorithmachieves SHD of 22 . 2 ± 4 . 2 while DAG-GNN achieves SHD of 25 . 2 ± 4 .
Revised: Our algorithmachieves SHD of 22 . 2 ± 4 . 2 while DAG-GNN achieves SHD of 25 . 2 ± 4 .

Original: We simulated a nonlineardataset with 50 -node ER1 graphs and each function being an MLP as Zheng et al.
Revised: We simulated a nonlineardataset with 50 -node ER1 graphs and each function being an MLP as Zheng et al.

Original: The original
Revised: The original

Original: NOTEARS-MLP has SHD 16 .
Revised: NOTEARS-MLP has SHD 16 .

Original: 9 ± 1 . 5 , while NOTEARS-MLP with our TMPI constraint achieves an
Revised: 9 ± 1 . 5 , while NOTEARS-MLP with our TMPI constraint achieves an

Original: SHD of 14 . 9 ± 1 . 3 .
Revised: SHD of 14 . 9 ± 1 . 3 .

Original: Real Data We also run an experiment on the Protein Signaling Networks (PSN) dataset [25]consisting of n = 7466 samples with d = 11 nodes.
Revised: Real Data We also run an experiment on the Protein Signaling Networks (PSN) dataset [26]consisting of n = 7466 samples with d = 11 nodes.

Original: The PSN dataset are widely used to evluate theperformance of nonlinear DAG recovery algorithms, however since the identiﬁability of the DAG isnot guaranteed we compare both SHD and SHDC (SHD of CPDAG) as [15].
Revised: The PSN dataset are widely used to evluate theperformance of nonlinear DAG recovery algorithms, however since the identiﬁability of the DAG isnot guaranteed we compare both SHD and SHDC (SHD of CPDAG) as [16].

Original: The results is shown in
Revised: The results is shown in

Original: Table 1, and we can see that by replacing the DAG constraint with our TMPI constraint there is aperformance improvement on SHD and SHDC (SHD of CPDAG) for almost all cases.
Revised: Table 1, and we can see that by replacing the DAG constraint with our TMPI constraint there is aperformance improvement on SHD and SHDC (SHD of CPDAG) for almost all cases.

Original: 4.3 Ablation Study
Revised: 4.3 Ablation Study

Original: We use the ER3 dataset in previous section to conduct an ablation study to compare the performanceof the geometric constraints (6) and our TMPI constraints (8), with both naive implementationsand the fast implementations based on Proposition 4 and Corollary 1.
Revised: We use the ER3 dataset in previous section to conduct an ablation study to compare the performanceof the geometric constraints (6) and our TMPI constraints (8), with both naive implementationsand the fast implementations based on Proposition 4 and Corollary 1.

Original: The results are shown in
Revised: The results are shown in

Original: Figure 4.
Revised: Figure 4.

Original: The SHD between Geometric constraint and TMPI constraints in both the naive and fastimplementations are similar, and our Fast TMPI algorithm runs signiﬁcantly  than the other three. [Figure]
Revised: The SHD between Geometric constraint and TMPI constraints in both the naive and fastimplementations are similar, and our Fast TMPI algorithm runs signiﬁcantly faster than the otherthree.

Original: [Figure]
Revised: [Figure]

Original:  Conclusion and Future Work
Revised:  Conclusion and Future Work

Original: [Figure] While polynomial constraints-based DAG learning algorithms achieve moderate accuracy, theyoften suffer from gradient vanishing.
Revised: While polynomial constraints-based DAG learning algorithms achieve moderate accuracy, theyoften suffer from gradient vanishing.

Original: We identiﬁed one important source of gradient vanishing:improper small coefﬁcients for higher order terms, and propose to use the geometric series based
Revised: We identiﬁed one important source of gradient vanishing:improper small coefﬁcients for higher order terms, and propose to use the geometric series based

Original: DAG constraints to escape from gradient vanishing.
Revised: DAG constraints to escape from gradient vanishing.

Original: We further proposed an efﬁcient algorithm thatcan evaluate the constraint with bounded error in O (log k ) complexity, where k issome constant thatis often far smaller than the number of nodes.
Revised: We further proposed an efﬁcient algorithm thatcan evaluate the constraint with bounded error in O (log k ) complexity, where k is often far smallerthan the number of nodes.

Original: By replacing the previous DAG constraints with ours,the performance of DAG learning can often be signiﬁcantly improved in various settings.
Revised: By replacing the previous DAG constraints with ours, the performance of DAG learning can often be signiﬁcantly improved in various settings.

Original: A clear next step would be ﬁnding strategies to adaptively adjust the coefﬁcients for higher terms inpolynomial based constraints.
Revised: A clear next step would be ﬁnding strategies to adaptively adjust the coefﬁcients for higher terms inpolynomial based constraints.

Original: In a nut-shell, polynomial-based constraints may explode on candidategraphs that are far from acyclic, and also suffer from gradient vanishing problem when the graph isnearly acyclic.
Revised: In a nut-shell, polynomial-based constraints may explode on candidategraphs that are far from acyclic, and also suffer from gradient vanishing problem when the graph isnearly acyclic.

Original: Fixed coefﬁcients can not solve both problems at the same time.
Revised: Fixed coefﬁcients can not solve both problems at the same time.

Original: Another possibledirection is to use the sparse properties in DAGs to make DAG learning more efﬁcient.
Revised: Another possibledirection is to use the sparse properties in DAGs to make DAG learning more efﬁcient.

Original: Since theadjacency matrix of DAG must be nilpotent, there might exists more compact representation of theadjacency matrix, maybe a low-rank CANDECOMP/PARAFACChecklist For all authors...
Revised: Since theadjacency matrix of DAG must be nilpotent, there might exists more compact representation of theadjacency matrix, maybe a low-rank CANDECOMP/PARAFAC

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