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

Original: Anonymous Author(s)

Original: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: Recovering underlying Directed Acyclic Graphstructures (DAG)  from observational data is highly challenging due to the combinatorial nature of the DAGconstrained optimization problem.
Revised: Recovering underlying Directed Acyclic Graph (DAG) structures 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 a continuous optimization problem by characterizing the DAG constraint as a smooth equality 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: Existing methods place very small coefﬁcients on high-order polynomial terms for stabilization, since they argue that large coefﬁcients on the higher-order terms are harmful due 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 on higher-order terms are beneﬁcial for DAG learning, when the spectral radiuses of the adjacency matrices are small, and that larger coefﬁcients for higher-order terms can 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 truncated matrix 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-arts in various settings, often by a factor of 3 or more in terms of structural Hamming distance.

Original:  Introduction
Revised:  Introduction

Original: Recovery of Directed Acyclic Graphs (DAGs) from observational data is a classical problem inmany ﬁelds, including bioinformatics [27, 43], machine learning [16] and causal inference [34].
Revised: Recovery of Directed Acyclic Graphs (DAGs) from observational data is a classical problem in many ﬁelds, including bioinformatics [31, 49], machine learning [19], and causal inference [39].

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 [24, 34],these graphical models can have causal interpretations.
Revised: The graphical model produced by a DAG learning algorithm allows one to decompose the joint distribution over the variables of interest in a compact and ﬂexible manner, and under certain assumptions [27, 39], 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 [32], FCI [10, 33], rely on conditional independencetest and thus may require a large sample size [29, 37].
Revised: Most constraint-based approaches, e.g. , PC [37], FCI [11, 38], rely on conditional independence test and thus may require a large sample size [33, 43].

Original: The score-based approaches, including exactmethods based on dynamic programming [15, 30, 31], A* search [41, 42], 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.
Revised: The score-based approaches, including exact methods based on dynamic programming [18, 35, 36], A* search [47, 48], and integer programming [13], as well as greedy methods like GES [9], model the validity of a graph according to some score function 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 for score-based methods is the combinatorial search space of DAGs [8, 10].

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 past few years.

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

Original: [44] 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: [50] develops an algebraic characterization of DAG constraint based on a continuous function, speciﬁcally matrix exponential, of the weighted adjacency matrix of a graph, and applies it to estimate linear DAGs with equal noise variances using continuous constrained optimization [4, 5].

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

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

Original: [39] proposesan alternative DAG constraint based on powers of a binomial to improve practical convenience.
Revised: [45] proposes an alternative DAG constraint based on powers of a binomial to improve practical convenience.

Original: Weiet al.
Revised: Wei et al.

Original: [38] uniﬁes these two DAG constraints by writing them in terms of a general polynomial one
Revised: [44] 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[17, 22, 39, 45], time series [23], 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 [20, 25, 45, 51], time series [26], confounders [6], and ﬂexible score functions [53], and have been shown to produce competitive performance.

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

Original: [18], Zhu et al.
Revised: [21], Zhu et al.

Original: [46] presentalternative formulations of the DAG constraints based on the spectral radius of the weighted adjacencymatrix.
Revised: [52] present alternative formulations of the DAG constraints based on the spectral radius of the weighted adjacency matrix.

Original: Instead of solving a constrained optimization problem, Yu et al.
Revised: Instead of solving a constrained optimization problem, Yu et al.

Original: [40] proposes NOCURLwhich employs an algebraic representation of DAGs such that continuous unconstrained optimizationcan be carried out in the DAG space directly.
Revised: [46] proposes NOCURL which employs an algebraic representation of DAGs such that continuous unconstrained optimization can 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 [21, 46, 52] is to improve the scalability, 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[39, 44].
Revised: To avoid potential numeric exploding that may be caused by high-order terms in the polynomial-based DAG constraints, previous works use very small coefﬁcients for high-order terms in DAG constraints [45, 50].

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 relatively higher order terms that may contain useful higher-order information.

Original: This causesdifﬁculties for the DAG constraints to capture the higher order information for enforcing acyclicity.
Revised: This may cause difﬁculties 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 [39, 44], which, however, causesan ill-conditioning issueas shown in [21].
Revised: With such DAG constraints, a possible solution to maintain higher-order information for acyclicity is to apply an even larger penalty weight [45, 50], which, however, causes ill-conditioning issues as demonstrated by Ng et al.

Original: Moreover, these small coefﬁcients willalso be multiplied on the gradients ofhigher-order terms, leading to gradient vanishing.
Revised: Moreover, these small coefﬁcients will be multiplied on the gradients of higher-order terms during the optimization process, 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, larger coefﬁcients may be safely used to construct more informative higher-order terms without introducing numeric 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 higher-order power of the matrix may be very close to zero matrix, and larger coefﬁcients may be safely used without 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 constraint without 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 computational burden 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 the following:

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 issue instead of gradient and numeric exploding.

Original: For sparse graphs, the gradient vanishing issuecan be severe.
Revised: For sparse graphs, the gradient vanishing issue can 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 polynomial over adjacency matrices to escape from gradient vanishing.

Original: We show that the relatively large coefﬁcients on the higher order terms would not result in gradient exploding in practice.
Revised: We show that the relatively large 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 can be reduced to order- k polynomial without expanding its feasible set.
Revised: our order- d polynomial constraint can 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 such value of k requires solving the NP-hard longest simple path problem, we develop a simple heuristic to ﬁnd such k with bounded error to the exact, order- d , DAG constraint.

Original: Based on this result, we propose a DAG constraint based on efﬁcient Truncated Matrix PowerIteration (TMPI), and conduct experiments to demonstrate that the resulting DAG learningmethod outperforms previous methods by a large margin.•
Revised: Based on this result, we propose a DAG constraint based on efﬁcient Truncated Matrix Power Iteration (TMPI), and conduct experiments to demonstrate that the resulting DAG learning method 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 DAG representation 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 random vector 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 [24, 34].
Revised: , x d ] > and DAG G , the distribution P ( x ) and DAG G are assumed to satisfy the Markov assumption [27, 39].

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 abuse of 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 do not 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: samples 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 [34] or sparsest Markov representation [26] assumption.
Revised: In general, the DAG G can only be identiﬁed up to its Markov equivalence class under the faithfulness [39] or sparsest Markov representation [30] assumption.

Original: It has been shown that for linear SEMs withhomoscedastic errors, from which the noise terms are speciﬁed up to a constant [19], 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 with homoscedastic errors, from which the noise terms are speciﬁed up to a constant [22], and for linear non-Gaussian SEMs, from which no more than one of the noise term is Gaussian [34], the true DAG can be fully identiﬁed.

Original: In this work, we focus on the linear SEMs with equal noise variances [25], whichcan be viewed as an instance of the former.
Revised: In this work, we focus on the linear SEMs with equal noise variances [28], which can be viewed as an instance of the former.

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

Original: [44] reformulatedthe DAG learning problem as the following continuous optimization problem with convex leastsquares objective but non-convex constraint,
Revised: [50] reformulated the DAG learning problem as the following continuous optimization problem with convex least-squares objective but non-convex constraint,

Original: [Equation]
Revised: [Equation]

Original: subject 
Revised: subject to h

Original: [Equation]
Revised: [Equation]

Original:  where k · k F and k · k 1 denote the Frobenius norm and elementwise ` 1 norm, respectively, ⌘ > 0 is97 the regularization coefﬁcient that controls the sparsity of B , and  denotes the Hadamard product.
Revised: (2) where k · k F and k · k 1 denote the Frobenius norm and elementwise ` 1 norm, respectively, ⌘ > 0 is the regularization coefﬁcient that controls the sparsity of B , and  denotes the Hadamard product, which is used to map B to a positive weighted adjacency matrix with the same structure.

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

Original: [Equation]
Revised: [Equation]

Original: [Equation] and can be efﬁciently computed using variousdifferent approaches [1, 2]. 
Revised: and can be efﬁciently computed using various approaches [1, 2]. The key here is that h exp ( ˜ B ) =

Original:  The formulation (2)involves a hard constraint, which is solved using the augmented Lagrangian method [3, 4] speciﬁcallydeveloped for constrained optimization.
Revised: if and only if G ( ˜ B ) is a DAG. The formulation (2) involves a hard constraint, and Zheng et al. [50] adopted the augmented Lagrangian method [4, 5] to solve the constrained optimization problem.

Revised: Ng et al.

Revised: [24] demonstrated that the augmented Lagrangian method here behaves similarly to the classical quadratic penalty method [15, 29] in practice, and showed that it is guaranteed to return a

Revised: DAG under certain conditions.

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

Original: [39] proposed a similar DAG constraint using an order- dpolynomial based on powers of a binomial, given by
Revised: [45] proposed a similar DAG constraint using an order- d polynomial 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[39].
Revised: where ↵ > 0 is a hyperparameter and is often set to 1 /d , e.g. , in the implementation of DAG-GNN [45].

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 exponentiation by squaring algorithm.

Original: Both h exp ( ˜ B ) and h bin ( ˜ B ) are formulated by [38] as a uniﬁed one
Revised: Both h exp ( ˜ B ) and h bin ( ˜ B ) are formulated by Wei et al. [44] as a uniﬁed one

Original: [Equation]
Revised: [Equation]

Original:  DAG Learning using Truncated Matrix Power Iteration
Revised:  DAG Learning with 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 constraints and propose a truncated geometric series based DAG constraints to escape from the issue in Section 3.1.

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

Original: In Section 3.3we provide an efﬁcient algorithm for our DAG constraints.
Revised: In Section 3.3 we 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 relation between our algorithm and previous works.

Original: 3.1  Truncated Geometric DAG Constraints
Revised: 3.1 Gradient Vanishing Issue and Truncated Geometric DAG Constraint

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 e 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 [39, 44].
Revised: previously small [Figure] coefﬁcients for higher-order terms are preferred since they are known to have better resistance to numeric exploding of higher-order polynomials [45, 50].

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 informative higher-order terms and cause severe gradient vanishing, as nilpotent properties of DAG result in candidate adjacency matrix with very small spectral radius (see empirical analysis in Figure 1 left ).

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

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

Original: To avoid the severe vanishing gradient issue in previous DAG constraint ( e.g. , Eq.
Revised: To avoid the severe vanishing gradient issue in existing DAG constraints ( 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 ) isto an order- ( d  1) polynomial of ˜ B .
Revised: ˜ B denoted by r ˜ B h geo ( ˜ B ) is an order- ( d  1) polynomial of ˜ B .

Original: With the same d , Eq.
Revised: Compared to (3) and (4), Eq.

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

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 matrices of DAGs, the higher-order terms in (6a) and (6b) often comes with small norms; thus, our DAG constraint may not suffer from numeric exploding in practice, based on the observations in

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: Section 4. In Figure 1 we compared the norm of gradients on higher-order terms of different DAG constraints, where we observe that for previous exponential-based (3) and binomial-based (4) DAG constraints, the gradients of higher-order terms converges to 0 very quickly as the 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, the gradient of higher-order terms in (6a) converges to 0 with a much slower speed and thus informative higher-order terms are preserved.

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 (6a) can be safely dropped to form a more efﬁcient DAG constraint with some k 6 d , where we often have k ⌧ d in practice:

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: Naively, the DAG constraint (7) requires O ( k ) matrix multiplication which is far less than O ( d ) for (6a), and under mild conditions the error of (7) is tightly bounded (see Proposition 3 for details). In

Original: In Figure1 right we show that our DAG constraint (8) (named TMPI) attains farmore better performance thanprevious ones as it escapes from gradient vanishing.
Revised: Figure 1 right we show that our DAG constraint (7) (named TMPI) attains far better performance than previous ones as it escapes from gradient vanishing.

Original: 3.2 Theoretic Properties for Truncated Geometric DAG Constraint
Revised: 3.2 Theoretical Properties of 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 in constructing 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, which is acorollary of [38, Lemma 1].
Revised: This straightforwardly leads to the following property, which is a corollary of [44, 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]

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 closer to 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 well as 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 , either using the DAG constraint terms in Eq.

Original: (9) or (5).
Revised: (8) 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 in the 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 1 in graph G .
Revised: Let ˜ B 2 R d ⇥ d > 0 be the weighted adjacency matrix of a graph G with d vertices, and k be the length of the longest simple path 1 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 [Equation]

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 simple cycle has a length of k , it sufﬁces to compute the k th power of its adjacency matrix, since computing any power larger than k 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: is a valid DAG constraint.

Original: Unfortunately, the longest simple path problem for general graphs is known to be NP-hard [28], 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 [32], and thus 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 , a simple heuristic is to iterate over the power of matrices to ﬁnd some k s.t. matrix ˜ B k is close enough to 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 be close to zero, which implies 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  we ﬁnd some k ⌧ d .
Revised: Thus, if we use the above heuristic to ﬁnd k , it is highly possible that 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 constraint in Eq.

Original: (8) becomes anapproximate one with the following bounds.
Revised: (7) becomes an approximate one with the following bounds.

Original: Denoting by k · k 1 the element-wise inﬁnity norm (i.e., 174 maximum norm), we have the following proposition.
Revised: Denoting by k · k 1 the elementwise inﬁnity norm ( i.e. , 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 ( k +1)
Revised: Given ˜ B 2 R d ⇥ d > 0 , if there exists k < d such that k ˜ B k k 1 6 ✏ < 1 ( k +1)

Original: d , then wehave
Revised: d , then we have

Original: [Equation]
Revised: [Equation]

Original: and

Original: [Equation]

Original: Remark 1.
Revised: Remark 1.

Original: The bound ✏ < 1 ( k +1)
Revised: The condition ✏ < 1 ( k +1)

Original: dis needed in the proposition above  such that the inequalities arenon-vacuous, speciﬁcally since we must have kr 
Revised: d in the proposition above is such that the inequalities are non-vacuous, speciﬁcally since we must have kr ˜ B h geo

Original: [Equation]
Revised: [Equation]

Original: B h geo
Revised: B [Equation]  r [Equation]

Revised: k

Original: [Equation]
Revised: [Equation]

Revised: kr

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 ) .
Revised: [Equation] k F [Equation] Proposition 3 provides an efﬁcient way to bound the error of truncated constraints h k trunc ( ˜ B ) to the geometric 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 limitation for 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 spectral radius will be close to zero.

Original: In this situation, if the longest simple path in a graph isvery 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 long, it is unavoidable to use high-order terms to capture this information while such high-order terms might be close to 0 .

Original: In the worst case, the higher terms maybe underﬂow due to limited machineprecision.
Revised: In the worst case, the higher-order terms may underﬂow due to limited machine precision.

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 for future 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: Proposition 3 indicates that it may be safe to discard those matrix powers with index larger than k .

Original: The DAG constraint and its gradient, become order- k  polynomials of  B .
Revised: In this case, the DAG constraint and its gradient correspond to order- k and order- ( k  1) polynomials of ˜ B , respectively.

Original: A straightforwardway to compute the DAG constraint as well as the gradient is provided in Algorithm 1, which wecall TMPI.
Revised: A straightforward way to compute the DAG constraint as well as the gradient is provided in Algorithm 1, which we call 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: The general polynomial based DAG constraint in (5) requires O ( d ) matrix multiplication, while Algorithm 1 reduces the number of multiplication to O ( k ) . In practice, k can be much smaller than d ; for example, when recovering 100 -node Erd˝os–Rényi

Original: For example, when recovering 100 -node Erd˝os–Rényi DAG with expecteddegree 4 , k is often smaller than 20 . 
Revised: DAG with expected degree 4 , k is often smaller than 20 . Input: ˜ B 2 R d ⇥ d > 0 , ✏ Output: h (˜ B ) , r ˜ B h (˜ B ) 1: i 2 , h (˜ B ) tr(˜ B ) , r ˜ B h (˜ B ) 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: Fast Truncated Matrix Power Iteration
Revised: ˜ B i 5: h (˜ B ) h (˜ B ) + tr(˜ B i ) 6: if k ˜ B i k 1 6 ✏ then break 7: i

Revised: i +8: end while 9: Output h (˜ B ) ,

Revised: r ˜ B h (˜ B )

Revised: Input: ˜ B 2 R d ⇥ d > 0 , ✏

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

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

Revised: ˜ B p 5: ˜ B f ˜ B oldf + ˜ B oldp ˜ 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 ✏ then break 8: i 2 i 9: end while 10: Output tr(˜ B f ) , ˜ B g

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.5).
Revised: Fast Truncated Matrix Power Iteration We exploit the speciﬁc structure of geometric series to further accelerate the TMPI algorithm from O ( k ) to O (log k ) matrix multiplications. This is acheived by leveraging the following recurrence relations, with a proof provided in Appendix A.4.

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 )) . Then we have the following recurrence relations:

Original: [Equation]
Revised: [Equation]

Original: It is straightfoward to obtain the following corollary by setting i = j in Proposition 4.
Revised: This straightfowardly implies the following corollary, with a proof given in Appendix A.5.

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 )) . Then 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: Instead of increasing the index i ( i.e. , power of ˜ B ) arithmetically by 1 , as in line 9 of Algorithm 1, Corollary 1 suggests using its recurrence relations to increase the 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.Similar strategy can be applied to the original geometricseries based constraint (6) to form a fast algorithm that requires O (log d ) matrix multiplications. [Table] d > 0 , ✏
Revised: The full procedure named Fast TMPI is described in Algorithm 2, which requires O (log k ) matrix multiplications and O ( d 2 ) additional storage.

Original: Output h (˜ B ) , r ˜ B h (˜ B ) 1: i 2 , h (˜ B ) tr(˜ B ) , r ˜ B h (˜ B ) I 2: while i 6 d do 3: r ˜ B h (˜ B ) r ˜ B h (˜ B ) + i (˜ B i  1 ) > 4: ˜ B i ˜ B ˜ B i 5: h (˜ B ) h (˜ B ) + tr(˜ B i ) 6:
Revised: An example implementation is given in Listing 1 in Appendix B.1.

Original: if k ˜ B i k 1 6 ✏ then 7: break 8: end if 9: i i +10: end while 11: Output h (˜ B ) , r ˜ B h (˜ B ) Input ˜ B 2 R d ⇥ d > 0 , ✏
Revised: Similar strategy can be applied to the original geometric series based constraint (6a) to form a fast algorithm that requires O (log d ) matrix multiplications.

Original: Output h (˜ B ) , r h (˜ B ) 1: i 2: ˜ B f ˜ B , ˜ B g I , ˜ B p ˜ B { ˜ B f stores f i , ˜ B 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 ˜ 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: A comparison of the time complexity between our proposed constraint and the existing ones is provided in Appendix B.2.

Original: f i , ˜ B g 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 Lagrangian framework [4, 5] adopted by Zheng et al.

Original: [44] to convert the constrained optimization problem into thefollowing unconstrained problem
Revised: [50] to convert the constrained optimization problem into the following unconstrained problem

Original: [Equation]
Revised: [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 [20] that adopts likelihood-based objective withsoft constraints, leading to the unconstrained optimization problem
Revised: We also apply the proposed DAG constraint to GOLEM [23] that adopts likelihood-based objective with soft constraints, leading to the unconstrained optimization problem

Original: [Equation]
Revised: [Equation]

Original: where ⌘ 1 , ⌘ 2 > 0 are the regularization coefﬁcients.
Revised: where ⌘ 1 , ⌘ 2 > 0 are the regularization coefﬁcients.

Original: In both formulations above,  the value of k isdynamically computed and updated during each optimization iteration based on Algorithm 1 or 2.
Revised: In both formulations above, we use Algorithmor 2 to dynamically compute h k trunc ( B  B ) (and the value of k ) during each optimization iteration based on a predeﬁned tolerance ✏ .

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: [44, Proposition 1]) .
Revised: [50, 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 smaller than one be the 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:  ˜

Original: The inverse of I  ˜ B is just the  geometric series as follows,
Revised: [Equation]  [Equation] When the spectral radius is smaller than one, the inverse of I  ˜ B is simply the following inﬁnite geometric series:

Original: [Equation]
Revised: [Equation]

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

Original: [44] 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: [50] argued that (12) is not well deﬁned for ˜ B with spectral radius larger than 1, and in practice the series (12) may explode very quickly. Based on this argument, they adopted the matrix exponential for constructing the DAG constraint.

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

Original: [39] also use verysmall coefﬁcients for higher terms in polynomial based DAG constraints.
Revised: [45] also used very small coefﬁcients for higher-order terms in polynomial based DAG constraints.

Original: However, as we show in Section 3,  Yu et al.
Revised: However, as we illustrate in Section 3, the DAG constraints proposed by Yu et al.

Original: [39], Zheng et al.
Revised: [45], Zheng et al.

Original: [44] actually suffers more from gradient vanishing due to smallcoefﬁcients for higher terms.
Revised: [50] may suffer from gradient vanishing owing to the small coefﬁcients for higher-order 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 constraint (7) serves a good approximation for (11) since we are actually looking for an approximate inverse of I  ˜ B with a bounded error.

Original: Proposition 5.
Revised: Proposition 5.

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

Original: [Equation]
Revised: [Equation]

Original:  Previous works [21, 38] pointed out that for  constraints with form h ( B  B ) , the Hadamard productwould lead to vanishing gradients since r B h ( B  B ) {+= 
+}
Revised:  [Equation] Previous works [24, 44] pointed out that for DAG constraints with form h ( B  B ) , the Hadamard product would lead to vanishing gradients in the sense that the gradient r B h ( B  B ) {+= 
+}

Original: [-= 
-] if and only if h ( B  B ) = 0 . 
Revised: [-= 
-] if and only if h ( B  B ) = 0 . Ng et al.

Original: Therefore,this may lead to ill-conditioning issue [21].
Revised: [24] further showed that this may lead to ill-conditioning issue when solving the constrained optimization problem.

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

Original: [38], but the non-smoothness of absolute value makes the optimization problemdifﬁcult to solve andmay lead to poorer performance [21].
Revised: [44], but the non-smoothness of absolute value may make the optimization problem difﬁcult to solve and lead to poorer performance [24].

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

Original: [21], 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: [24], QuasiNewtown methods, e.g. , L-BFGS [7], still perform well even when faced with the ill-conditioning issue 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 identify another type of gradient vanishing issue caused by (1) the higher-order terms and small coefﬁcients in function h itself and (2) the small spectral radius of candidate matrix B , and further provide an effective solution by constructing 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 DAG constraint.

Original: We give an outline of the experimental settings in this section, and more details includingthe parameter settings can be found in supplementary ﬁles 2 .
Revised: We give an outline of the experimental settings in this section, and more details including the parameter settings can be found in supplementary ﬁles.

Original: 4.1 DAG Learning
Revised: 4.1 DAG Learning

Original: We ﬁrst conduct experiments on synthetic datasets using similar settings as previous works [39, 40, 44].
Revised: We ﬁrst conduct experiments on synthetic datasets using similar settings as previous works [23, 46, 50].

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 [14] are generated with d nodes and kd expected edges (denoted by ER k ). 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 n samples are generated from the linear SEM 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 the noise 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: We set the sample size n = 1000 and consider 4 different number of nodes d = 10 , 30 , 50 , 100 unless otherwise stated.

Original: For each setting, we generate100 different graphs to obtain an average performance.
Revised: For each setting, we conduct 100 random simulations 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: Methods and Baselines We consider continuous DAG learning methods based on both Mean Squared Error (MSE) loss (see Eq.

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

Original:  loss from GOLEM [20].
Revised: (10)) loss from GOLEM [23].

Original: For bothlosses, we include the following DAG constraints: (1) MSE/Likelihood loss+ExponentialDAG constraint (3), i.e. NOTEARS
Revised: For both losses, we include the following DAG constraints: (1) MSE/Likelihood loss+Exponential

Original: [44]/GOLEM[20] ; (2) MSE/Likelihood loss+Binomial DAG constraint(4) from DAG-GNN [39]; (3) MSE/Likelihood loss+Spectral radius based DAG constraint from NOBEARS [18]; (4) MSE/Likelihood loss+Single
Revised: DAG constraint (3), i.e. NOTEARS [50]/GOLEM[23] ; (2) MSE/Likelihood loss+Binomial DAG constraint (4) from DAG-GNN [45]; (3) MSE/Likelihood loss+Spectral radius based DAG constraint from NOBEARS [21]; (4) MSE/Likelihood loss+Single

Original: Term DAG constraint (9); (5) MSE/Likelihood loss+Truncated Matrix Power Iteration (TMPI)
Revised: Term DAG constraint (8); (5) MSE/Likelihood loss+Truncated Matrix Power Iteration (TMPI)

Original: based DAG constraint (8).
Revised: based DAG constraint (7).

Original: For the NOCURL [40] DAG constraint, since it isdesigned as a two stage methods where the initialization stage rely on 
Revised: For the NOCURL method [46], since it is a two stage methods where the initialization stage rely on MSE loss and

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

Original: The other baselines that we are comparing to including GES [8] and MMPC [36].
Revised: The other baselines include GES [9] and MMHC [41, 42] 3 . [Figure]

Original: For MSE loss based DAG learning algorithms (10)), we set ⌘ = 0 .
Revised: For MSE loss based DAG learning methods (9), we set ⌘ = 0 .

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

Original: [44] to update the parameters ⇢ and ↵ .
Revised: [50] to update the parameters ⇢ and ↵ .

Original: For likelihood loss based DAGlearning algorithms (11), we set ⌘ 1 = 0 .
Revised: For likelihood loss based DAG learning methods (10), 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 [39].
Revised: For the Binomial DAG constraints (4), the parameter ↵ is set to 1 /d as Yu et al.

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

Original: For the NOCURL algorithm, we use their public available implementationand recommended parameter setting.
Revised: For the NOCURL method, we use their public available implementation and default parameter setting 4 .

Original: For GES and MMPC, we use the implementation and defaultparameter setting in Kalainathan and Goudet [13]. [Figure]
Revised: For GES and MMPC, we use the implementation and default parameter setting in the CausalDiscoveryToolbox package [16].

Original: [Figure]
Revised: [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 results for MSE/likelihood based DAG learning algorithm are shown in Figure 2/3, where we compare the Structural Hamming Distance (SHD) of different algorithms.

Original: The Original
Revised: The complete results for other metrics are reported in Appendix C.1.

Original:  NOTEARS ( i.e. MSE+Exponential) / GOLEM ( i.e. GOLEM+Exponential) has similar performanceas MSE/GOLEM+Binomial.
Revised: The original NOTEARS ( i.e. MSE+Exponential) / GOLEM ( i.e. GOLEM+Exponential) has similar performance as 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 may not be surprising because the coefﬁcients of each order term in the Exponential and Binomial constraints are actually very close (see Figure 1 for an example).

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 algorithms using single-term DAG constraint is often not comparable to the other polynomial-based constraints since it suffers 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 based constraints (BEARS) are also worse than the ones using Exponential and Binomial constraints.

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

Revised: Additional Experiments We conduct additional experiments with scale-free graphs, smaller sample size, larger graphs ( i.e. , 500 -node graphs), and denser graphs ( i.e. , ER6 graphs).

Revised: Due to the space limit, the results of these experiments are provided in Appendices C.2, C.3, C.4, and C.6, respectively, which further demonstrate the effectiveness of our proposed DAG constraint.

Revised: We also compare our method with NOFEARS [44] in Appendix C.5.

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: Although the description of our algorithm in Section 3 focuses on the linear case, it can be straightforwardly extended to the nonlinear case.

Original: In this section, we extend our DAG constraint to nonlinear case by replacing thebinomial DAG constraint in DAG-GNN [39], NOTEARS-MLP[45] and GRAN-DAG[17] with ours.
Revised: In this section, we extend our DAG constraint to nonlinear case by replacing the binomial DAG constraint in DAG-GNN [45], NOTEARS-MLP [51] and GRAN-DAG [20] with our TMPI constraint.

Original: Synthetic data Weconducted 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 compare our algorithm and DAG-GNN on 50 -node 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 algorithm achieves 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 simulate a nonlinear dataset with 50 -node ER1 graphs and each function being an MLP as Zheng et al.

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

Original: 9 ± 1 . 5 , while NOTEARS-MLP with our TMPI constraint achieves an SHD of 14 . 9 ± 1 . 3 .
Revised: 9 ± 1 . 5 , while NOTEARS-MLP with our TMPI constraint achieves an SHD of 14 . 9 ± 1 . 3 .

Original: Real Data We also run an experiment on the Protein Signaling Networks(PSN) dataset [27]consisting of n = 7466 samples with d = 11 nodes.
Revised: Real Data We also run an experiment on the Protein Signaling Networks dataset [31] consisting of n = 7466 samples with d = 11 nodes.

Original: ThePSN dataset are widely used to evluate theperformance ofnonlinear DAG recovery algorithms, however since the identiﬁability of the DAG isnot guaranteed we compare both SHD and SHDC (SHD of CPDAG) as [17].
Revised: The dataset is widely used to evaluate the performance of DAG learning algorithms; however, since the identiﬁability of the DAG is not guaranteed, we compare both SHD and SHDC (SHD of CPDAG), following Lachapelle et al.

Original: The results is shown in 
Revised: In this experiment, we replace the original DAG constraints in DAG-GNN [45] and Gran-DAG [20] with our TMPI DAG constraints.

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: The results are shown in Table 1, and it is observed that by replacing the DAG constraint with our TMPI constraint, there is a performance 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 bothnaive 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 performance of the Geometric constraint (6a) and our TMPI constraint (7), with both the naive implementation ( i.e. , linear complexity) and fast implementation ( i.e. , logarithmic complexity) from Algorithmsand 2, respectively.

Original: The results are shown in Figure 4.
Revised: The results are shown in Figure 4.

Original: The SHD between Geometric constraint and TMPI constraints in both the naive and fastimplementations are similar, and our Fast TMPI
Revised: The SHD between Geometric constraint and TMPI constraint in both the naive and fast implementations are similar, and our Fast TMPI

Original: algorithm runs signiﬁcantly faster than the otherthree.
Revised: algorithm runs signiﬁcantly faster than the other three, especially when the graph size is large.

Revised: We also conduct further empirical study to illustrate the numeric difference between different k for our

Revised: DAG constraint; due to limited space, a detailed discussion is provided in Appendix C.7.

Original: [Figure]
Revised: [Table]

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 promising performance, they often suffer from gradient vanishing in practice.

Original: We identiﬁed one important source of gradient vanishing:improper small coefﬁcients for higher order terms, and propose to use the geometric series based DAG constraints to escape from gradient vanishing.
Revised: We identiﬁed one important source of gradient vanishing: improper small coefﬁcients for higher-order terms, and proposed to use the geometric series based 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 is often far smallerthan the number of nodes.
Revised: We further proposed an efﬁcient algorithm that can evaluate the constraint with bounded error in O (log k ) complexity, where k is often far smaller than 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-order terms in polynomial 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 nutshell, polynomial-based constraints may explode on candidate graphs that are far from acyclic, and also suffer from gradient vanishing problem when the graph is nearly acyclic.

Original: Fixed coefﬁcients can not  solve both problems at the same time.
Revised: Fixed coefﬁcients are not able to 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 possible direction 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/PARAFAC 
Revised: Since the adjacency matrix of DAG must be nilpotent, there might exist more compact representation of the adjacency matrix, such as a low-rank CANDECOMP/PARAFAC (CP) decomposition, which can potentially be used to derive an efﬁcient algorithm for very large graphs. Acknowledgements We thank the anonymous reviewers, especially Reviewer iGwL, for their helpful comments. ZZ, DG,

Revised: YL, EA and JQS were partially supported by Centre for Augmented Reasoning at the Australian

Revised: Institute for Machine Learning.

Revised: IN and KZ were partially supported by the National Institutes of

Revised: Health (NIH) under Contract R01HL159805, by the NSF-Convergence Accelerator Track-D award #2134901, by a grant from Apple Inc., and by a grant from KDDI Research Inc.. MG was supported by ARC DE

Revised: Checklist

Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?

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Revised: [Yes] See lines 73 and(c)

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Revised: [Yes] (b) Did you include complete proofs of all theoretical results?

Revised: [Yes] See Appendices 3.

Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Revised: [No] We are pleased to release code on our website after the work is accepted.

Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Revised: [Yes] See Appendices (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Revised: [Yes] See Section(d)

Revised: Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Revised: [Yes] See Appendices 4.

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Revised: [Yes] See Appendices (c) Did you include any new assets either in the supplemental material or as a URL?

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Revised: [Yes] We use public available datasets (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?

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