Meta-D2AG: Causal Graph Learning with Interventional Dynamic Data
Authors: Tian Gao
Songtao Lu
Junkyu Lee
Elliot Nelson
Debarun Bhattacharjya
Yue Yu
Miao Liu
Venue: NeurIPS 2025 poster ()
Keywords: intervention
dynamic
meta learning
Primary area: optimization
TL;DR: learning causal graphs from dynamic interventional data (i.e., RL data)

Abstract: Causal discovery in the form of a directed acyclic graph (DAG) for dynamic time series data has been widely studied in various applications. Much of the existing work has focused on observational, offline, and/or stationary settings. In this work, we propose a dynamic DAG discovery algorithm, Meta-D$^2$AG, based on online meta-learning. Meta-D$^2$AG is designed to learn dynamic DAG structures from potentially nonlinear and non-stationary times series datasets, accounting for changes in both parameters and graph structures. Notably, Meta-D$^2$AG explicitly treats data collected at different time points with distribution shifts as distinct domains, which is assumed to occur as a result of external interventions. Moreover, Meta-D$^2$AG contains a new online meta-learning framework to take advantage of the temporal transition among existing domains such that it can quickly adapt to new domains with few measurements. A first-order optimization approach is utilized to efficiently solve the meta-learning framework, and theoretical analysis establishes the identifiability conditions and the convergence of the learning process. We demonstrate the promising performance of our method through better accuracy and sample efficiency on benchmark datasets against state-of-the-art baselines.

Meta-D'AGi Causal Graph Learning with Interventional Dynamic Data
Tian Gao*
Songtao Lu" The Chinese University of Hong Kong
Junkyu Lee IBM Research
Elliot Nelson Independent
IBM Research
Debarun Bhattacharjya
Yue Yu Lehigh University
Miao Liu IBM Research
IBM Research
Abstract
Causal discovery in the form of directed acyclic graph (DAG) for dynamic time series data has been widely studied in various applications. In this work, we propose dynamic DAG discovery algorithm, Meta-D2Ape AG, based online meta- learning. Meta-D"AGi designed to learn dynamic DAG structures from potentially nonlinear and non-stationary time series datasets, accounting for changes in both parameters and graph structures. Unlike most of the existing work focusing on observational, offline, and/or stationary settings, Meta-D2 AG explicitly treats data collected at different time points with distribution shifts as distinct domains, which is assumed occur result of external interventions. Moreover, Meta- D2 AG involves a new online meta-learning framework to take advantage the temporal transition among existing domains such that it can quickly adapt new domains with few measurements. A first-order optimization approach is utilized to efficiently solve the meta-learning framework. and theoretical analysis establishes the identifiability conditions and the convergence of the learning process. We demonstrate the promising performance of the proposed meta learning framework through better accuracy on benchmark datasets against state-of of--the-rt baselines.
Introduction
Probabilistic graphical models [50 32 and causal graphical models [51] have been an active area of research in machine learning. Dynamic probabilistic graphical models 33 are particularly suitable to capture dynamics in temporal data like time series by explicitly modeling how variables change over time. These probabilistic models have been successfully applied to real-world problems, such as for neuroscience [55], molecular biology [35] and computer vision 42 tasks.
Dynamic Bayesian networks (DBNs) 12 are among the most popular dynamic graphical models in the literature, initially proposed for discrete-time models with discrete variables. Later, plenty of subsequent work extended these models consider continuous variables that are more suitable for time series data since they often involve continuous-valuee measurements. For example, DBNs are used for structured vector auto-regressive (SVAR) models in the statisties and econometrics literature [56 57
66 65). Parameter learning and directed acyclic graph (DAG) structure learning from observational time series data are the typical learning tasks for DBNs. One can categorize standard structure learning methods into score-based 233
34
30
8 5. constraints-
60), or more recently, neural-based methods [41. A more tractable continuous optimization framework has been proposed based the algebraic characterization of the DAG [77) to address exponenial computational complexity in exact structure learning. This
based (62
67 10
17
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39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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continuous optimization formulation [46 70, [4] has been successfully extended to nonlinear neural
models 73 33
26
78
72 and dynamic DAG structure learning 49
25
Despite its success, typical DAG learning has one major limitation. Often, the data distribution may change over time and the model parameters even graph structures may fluctuate. For instance, in applications such as monitoring sensor networks, different sensors may be active at different times due operating costs or weather conditions. Changes in city policies, urban development, or even major events (such as festivals or construction projects) can lead to temporal shifts in the data distribution. In retail, the sales of certain products may exhibit seasonal variations, such as increased demand for winter clothing during colder months. Standard DAG learning algorithms assume datasets involve samples that are independent and identically distributed (i.i.d.). ignoring correlations across time steps. Moreover, structure learning in DBNs over time series data generally ignores the distribution shift and assumes stationarity, which may affect the accuracy of the learned graphs. Datasets collected over long period of time could contain distribution changes, such as monthly data or even yearly data. When these changes are significant, one may have to re-learn the graph from scratch, which loses relevant historical information and reduces learning efficiency, especially if some parameters or graph edges remain constant over time.
As a step toward addressing such non-stationary problems, we propose a meta learning based dynamic DAG learning approach, Meta-D AG, to model such distribution shifts over time. We consider the data collected over different periods as different domains under the distribution shifts, and utilize historical data to learn set of shared parameters across the whole domain and private parameters that adapt to domain-specife changes per domain. In Meta-D² AG, we view distribution shifts as caused by external interventions, e.g., updating the public health policy would influence the health record. Accordingly, we utilize the explicit intervention distribution when intervention targets are known, or the observational distribution when those targets are unknown.
The proposed framework employs two-level optimization process to separate the invariant and domain-specific subspaces, and can quickly adapt a new data distribution and achieve higher graph learning accuracy with a few samples. This procedure is particularly beneficial when the data collector may not be able capture a sufficient amount of time series data due to con-staiionarity making a low sample requirement crucial. Moreover, the chosen meta-learning framework is especially suitable for those non-stationary environments where some aspects of the environment evolve slowly or change periodically. This characteristic can make learning stationary components easier, as data across different interventions or domains are observed.
There have been a few works using a transfer-learning approach to handle invariant causal structures across domains, but they are mainly for reinforcement learning or observational time series data 11 45 37 29). There is also a line of works for multi-task DAG learning, but they require additional
order information to be provided (6) and do not consider new domains in an online setting (2. 25
considers a potential heterogeneous distribution, but it assumes fixed graph structure where only parameters are assumed to change.
The major contributions of our work are as follows:
We propose novel meta-learning framework via bilevel optimization for time series DAG learn- ing that explicitly models intervention as the source of distribution shift. It can handle both observational and interventional data distributions with nonlinear relationships by formulating non-stationary collection of dynamic datasets as different domains.
We propose an efficient first-order optimization approach to solve the proposed bilevel optimization framework with constraints in the lower level.
Our theoretical analysis establishes the identifiability conditions in the batch setting and provides convergence rates in the online setting.
2 Background and Related Work
Consider a set of realizations of potentially non-stationary time series, with each individual realization of size in the form of X. := [=r,)] E Rd. Here i E (0, .., T} represents the time index, X, represents the observed values of all d number of variables in an observational or interventional time series dataset, and I4,s denotes the i-th component of X,. For simplicity of notation, consider one sequence X, without loss of generality.

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A causal Bayesian network [51] is defined by a distribution Px over set of random variables X IR and DAG G (V, with nodes V and edges E. Each node V; E V {Vi,... V23 is associated with random variable z; and each edge (i, E E represents a direct causal relation from variable x; to 23-- With a slight abuse of notation, we will use X and V interchangeably. We assume the distribution Px is Markov with respect to graph G, which enables the factorized joint distribution as P(X)= = I (1 where nf is the set of parents of node j in the graph G and B denotes the instantiations of a subset of X whose indices are B C V. We also assume causal sufficiency, i.e., there are no hidden common causes between any pair of variables in X [52]. As typically assumed in such models, there may exist instantaneous or contemporaneous influences as well as a time-delayed impact among variables.
Among many possible ways to model Px in time series dataset, we follow the typical setting in 49 characterizing X through a standard structured vector autoregressive (SVAR) model 13 64 30
X, = xx=;, W + X, W +... X,,, Zi,
where ł E {p...., 7 with horizon T, is the autoregressive order, and Z, E RRd is vector of noise variables drawn from any continuous distribution. We assume that Z, is independent of Z and of X₁' for all t' < t. The d x d matrices W and W i E {l.....p) represent weighted adjacency matrices for the intra-slice and inter-slice edges in G. respectively, and they model the contemporaneous and time-lagged causal relations. Eq. [ can be written in the matrix form as p + Z, where X E Rnnd is matrix whose rows are Xt, Z Raxd is matrix formed similarly by Zr, and Yj. E [l, (I,....P). are time lagged versions of X. The number n is the effective sample size, which is equal to T - -p+1
X = XW + Y, Y. + + W:
The goal of typical causal structure learning tasks is to recover the DAG 9 using samples from Px and/or from the interventional distributions. We follow a continuous constrained optimization re-formulation for DAG learning [75] that uses continuous DAG constraint, h h(W) 0 on the weighted adjacency matrix W to avoid the combinatorial search on the feasible solutions W:
min Cx(X; W) + XΩ W) s.t. h h(W)= 0, 0,W
where Lg the loss function and 0 indicates parameters other than W, 200, W) is a regularization term model parameters and/or the edge complexity in W with tunable regularization parameter A. (75) proposed (W) = Tr W) d. where Tr represents the matrix trace, and showed that the graph is acyclic if and only if the constraint h h(W)= = O. Typically, the loss fe can be the least square loss 775) in linear structured equation models (SEM), or evidence lower bound [73, among other losses 26 The problem can be approximately solved using the classic augmented Lagrangian
method.
For time series datasets, DYNOTEARS 4 extends the continuous optimization framework to DBNs by explicitly modeling the intra-slice and inter-slice adjacency matrices separately with linear SEM model:
min (X: W w*) + AQ W s.t. (W") = =0, 0,W.A
(3)
where Wa is the matrix for intra-slice edges and w is the matrix for inter-slice connections, and Lo (X; Wa W - XW" Y Here n is the total sample size, and || ||1 is the Frobenius norm. To distinguish with the Frobenius norm, we use || 1l2 to denote the (vector) fo-nrrm.
Beyond observational time series data, learning causal graph structure in Markov Decision Process (MDP) requires additional machinery to model actions, and this can be seen as interventional dynamic DAG learning. Let binary indicator matrix R [ržy] e {0, R encode the interventional family I such that kij = 1 when Ij is an intervened target in IN. and 0 otherwise. For each intervention family k. we denote the weighted intra-slice adjacency matrix by We. Rddd and the weighted p-order inter-slice adjacency matrix by W (k) E &gdxd
Let W()) (Wo (1)) W and W- (W (I...., W(K)) } represent the collection of all matrices over K interventional families. Then, IDYNO 18] formulates the continuous optimization as
1 K d
min W,G.8 n
"ΣX
M
[Co (X, MLP(X:0,, W (())) La, (LPXXXXX W())) s.t. h (WW) = 0
1=1
3

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where fg is the parameter of one MLP that predicts the value of each variable Xj.
The above learning procedure has several limitations: 1) it needs offline data, meaning that the data needs to be saved prior to the learning process and they cannot adapt to new data when distributions change, and 2) the interventional family is fixed and limited to known fixed number K, while K changes over time in practice. Ultimately, the method neglects the sequential and non-stationary aspects of time series problems. This problem is also relevant in reinforcement learning, where the policies are learned and updated online, and hence the distributions and underlying MDP may change particularly around the action variables.
3 Meta-Learning for Dynamic DAG Learning
We focus DAG learning for times series data, which can quickly adapt to non-stationary distribu- tions and graphs. First, we propose generic meta DAG learning framework via bilevel optimization for time series data when all the data are given under the batch offline setting. Then, we extend the framework to the online setting, where new data from different domains are constantly arriving. Recall that existing work on online learning of DAG structures [|] parameter learning settings or assumed i.idd data. Finally, we present an efficient online Meta-D2 AG an algorithm based on the single-level penalty-base maaed bilevel gradient descent.
31 have largely focused on
3.1 Problem Formulation with Meta DAG Learning
Consider a non-stationary time series setting where data can be divided into Q disjoint domains {I...., Q}. In each domain q, time series X, = [X,,,]==1 is stationary: however, one or more variables may form different distributions across the domains. The goal is to recover a set of causal DAGs, represented with weighted adjacency matrices W, (We: q: wh J==1:
governing the underlying data generation process. Given the potential for variation across domains, a desired property of effective learning algorithms is the ability to use minimal samples at each domain. This can be achieved by leveraging historical data from previous domains that share similaritics with the current domain. A meta-learning approach is well-suited to meet this requirement.
Meta Learning. A common view of meta-learning (24 is learn a general purpose algorithm that can generalize across a set of tasks M with shared parameters, and learn a new task more efficiently given the previous tasks. The goal to evaluate the expected loss of data over a distribution over tasks M. In general, meaanlen algorithms require a set of meta-traini veea-training and meta-testing tasks drawn from the distribution p(T). This way we can learn new task- susk-specific parameters quickly with few samples. Meta-learning has been applied to sequential data and in particular reinforcement tasks 16
54 79.
and continuous learning 59 20
In a standard two-step meta-learning procedure, domains Q are divided into source M and target U datasets, [Q] |M + [U,, for meta-train and meta-test steps. The meta-training step uses a set of M tasks with data D Doource = (Dirrain D ) Let d and ºp be the shared parameters over all tasks and the private parameters specific each particular task, respectively. The meta-training stage aims to optimize loss function L by
x- arg min log L( 2,
The meta-test stage finds the optimal parameters o for each specific task with data D Drarget ((Dirain Diest)) by
T arg min log L( Dearget/ Dh D Daource).
4,
However, our proposed setting slightly different. The goal of each task to learn private parameters Ppmm and corresponding graph structure Wm with few samples Drource and shared parameters 4, in meta-training. One can indeed adapt to data from a new domain u and infer @ meta-testing, but since DAG learning is an unsupervised learning task, the meta-test stage is optional. Note that none of the existing meta-learning algorithms focuses on learning causal graph that is applicable to general interventional time series datasets.

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Training Data
Validation Data
2
3
2
3
M
1) Estimate: prituaae including Wprivare.m
Estimate: Sshared including W Wshared
OFFLINE
OFFLINE ONLINE
Graph: 9m
W shared
W private,m Vm
ONLINE
1) Estimate: 4 shared: including Wahared
Estimate:
1b privatam 1 including W privaae.m111
772
m
2
m
Figure 1: Overview of the proposed Meta-D2AG framework, with offline and online settings. Upper Half: In the offline setting, Meta-D² AG uses different sets of data to iteratively learns the private and shared parameters of each domain to infer the causal graph Am for each domain; Lower Half: Meta-D2 AG can also adapt to the online setting by using data of incoming domains to infer domain- specific private graphs, given the shared parameters learned from history.
3.2 Meta-Learning for Dynamic DAG
We propose novel meta-learning framework 39 for dynamic causal DAG learning in the presence of the non-stationarity of parameters and graphs by treating each segment of the time series different domain. We define the private parameters Φ = {W p w/b Op). and the shared parameters
&.- = (W", Wh }, where Op or O, include other model parameters such as linear or nonlinear SEMs. Since DAG learning is an unsupervised task,
should modify the typical meta-learning setting accordingly. The source dataset contains a set of training data and potentially a set of validation data, i.e., Dsource ((Xtraii X v m=1- Xtrain is used for training all M private task parameters, and Drall is to train shared parameters 4. This batch or offline setting is illustrated by the upper half of Figure [
When desired, the target dataset contains a new dataset to learn a DAG in new domains is then used to learn private parameters & p.u in new task
U, i.e.. D Darget {(Xtraii J==1 X u, given shared parameters &..
As such, we formulate our meta gradient- gradient-based dynamic DAG learning framework (Meta-D' ADA as a bilevel optimization problem. In meta-training, the objective with the upper level loss and the lower level 9 can be written as:
min
mm (2,,
D p.m X val
(4a)
M
11=1
s.t. O+
(1.) arg min Jm s, O p.m. x. Xtrain), Vm, pp.m
s.t. h(Wa. W Wp.m) =
(4b)
where i indicates the shared parameters over different training domains and & Pp.m indicates the private parameter evolving at each domain. n is the sample size of each domain. We use é regularized least square loss for ey and fn each domain, i.e., 2/ ||,,, f' ( ( p x,, with either +> or +n| P. 1l1. where f. is the model function for each domain m, such as SEMs, and contains other trainable domain-spedne data, and we will drop it to simplify notations as lm (Φs, Φ p) and 9m ( & p.
parameters. X,, is either the train or validation
In meta-testing, one can solve the inner loop to obtain a new th* p.m- Recall that this bilevel formulation is different from the standard meta-learning formulation in that the lower level (or the inner loop) optimization inside the constraint contains another constraint h( W., Wp.m) = O. This poses more complex meta-learning formulation with additional difficulties satisfy constraints. Note that this formulation can also apply to DAG learning from i.i.d. data with multiple domains. Note also that we only enforce the acyclicity constraint h at the lower level, as the upper-level constraints would be redundant.
Remark 3.1. The meta formulation can reduce to the vanilla continuous optimization approach 49 with single domain, when the outer-loop optimization is not needed. Hence, the formulation is more general and covers existing continuous DAG learning frameworks.
5

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3.3 Online Meta-Learning for Dynamic DAG
In dynamic datasets, domains in time series datasets may not be independent of each other, given the innate dependency overtime, even if there exists non-stationarity. Hence, assume that domain parameters are function of the new time sequences D, and the parameters in the previous tasks, i.c. Namely, all previous datasets with index < q should be used to train the shared parameters for dataset for the maximal utilization of information. The source dataset contains Dsource ((Drrain, D. j= with Dirsin is used for training private domain-specifie parameters for each domain m. Similarly, D is used to train shared parameters. Since the online sequence can be arbitrarily long, we can determine the total number of the historical set M to train the shared parameters. On the other hand, the target dataset Darrget ((D)ram)) }}== contains Dirain to learn private parameters in new domains given the shared parameters e. and U can be a set of sequential domains sampled directly after M domains. To further refine the model in online fashion, we could consider some Dyal for the additional meta- vetaa-raiinng procedure to update Dg. This online setting illustrated by the buttom half of Figure [T
2, f(. 1).
As such, the modified online meta gradient-based dynamic DAG learning framework at the current domain can be written
min 8,
Σ L (1 p..) 1=1
(5a)
m
s.t. P.. = arg min gi T P,,,, Vi, s.t. h(W,, wp.) 0,
(5b)
Ppii
Relation between 9 and (W,, W.). To instantiate the bilevel optimization framework to learn Wa W p.a with linear SEMs. In
graph G. we can choose to use Hadamard product nonlinear SEMs with neural networks, we adopt an existing approach 78 by including W. and W, the first layer parameters. Putting everything together with a two-layer feedforward neural network with domain-specific parameter pp. as an example and using the least square loss, the proposed formulation has the following forms:
1 m A
min W; m
Σ
X;
(Ws) (Ws) Ws) X,) + Wsll
(6a)
2n
1
s.t. W P (Ws), (Ws)) arg min - X; (p Ws)x
(6b)
Wr.: 2n
s.t. h(Ws o Wp.i) = 0
(6c)
Interventional loss can be used when such targets are known, following equation
Remark 3.2. The choice of multiplying shared and private parameters is inspired by existing meta- learning approaches and the masking approach in DAG learning [48]. As a result, in order to recover correct graphs in each domain, We has to include the edges of all W, hence it unlikely to be DAG alone. An alternative formulation G; = W.+ W which is also typical in many meta-learning literature (15), could be used, but it may be slightly matrices share similar weights.
restrictive as it assumes all adjacency
3.4 Penelty-based Bilevel Proximal Gradient Descent
Solving the above bilevel optimization is not straightforward, as there is an additional constraint in the inner loop. We propose penalty- based bilevel gradient descent inspired by 66 solve the problem. The problem can be written in general form of
min p.: m
L( + p..)
(7a)
8.L. C p.i E S((.))
== arg min gi p.,
Φ B..)) Vi,
s.b. h(D %,,)) =
(7b)
where S,(4,) denotes the optimal solution set of the ith lower-level problem.

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Towards this end, by moving both the objectives and constraints in the inner loop different penalty terms the outer loop, we can write the penalized objective as:
1
L( & p..) + 7/ (:: P..) -5 (b.)) + v, pp.)),
(8)
where {nr, } and 9, ( min...: 9: (+s p.)) penalty coefficients. The second term of equation [ penalizes the constraints directly in the objective. One issue with equation 7a that m can be large and computation would become expensive for long-history domains. As in typical online algorithms, it prudent to use a sliding window to capture the historical influence as follows.
min d 2.. P..
Σ
(1.. pp.s)
(9a)
s.t. Pp,t
S,(4)) == arg min gi S. p,,)). Vi, s.t. h( pp,s) = 0,
(9b)
p.
where w denotes the size of the sliding window.
Sliding Window Formulation. Due the inherent nonlinearity of neural networks, the corre- sponding objective functions exhibit non-convex properties. [22) has shown that it is possible to measure the regret using local regret metric, allowing us establish regret bound based smooth nonconvex objective function. Then, the penalty-based sliding average is:
Sc, (P (D.i)
((s,, +pa)) (9 pp.) - (..)) vish (eh( pp.))
(10)
As such, the optimization becomes single level and adopt proximal gradient descent method with a penalty function to optimize tb = ($,, pp.) together. For a more detailed discussion of the algorithm and proofs on the following convergence property, refer Appendix A.I]
Algorithm 1 Online Meta-1>2 AG Algorithm
Require: Data Xm, m E fl, 3,, w, 7 Output: Learned shared weighted adjacency ma- trix W, and private W p.m. Vm initialize W. and Wp.a
6:
97 (+.) = mine, p.c 9: (#., Pp.s) [Minimize
I:
inner loop)
7: Compute S;, with equation 10
2:
8:
+- prox,, 17 St.
for each domain do
9: end while
4: Initialize pp.m 2 ppm
10:
end for
5: while equation holds do
As there is non-smooth term in the objective, adopt the proximal residual, i.c., Pf (f d)
"7 --'(x pr nd)),, Vn 0 (21) in the optimality gap quantify the local regret, where prox,,(") == min, nf _ || - e112 denotes the proximal operator, and is any vector, f(x) = ||>||| and p (v) the soft-thresholding operator. To summarize, we show the online algorithm in Algorithm where V v..3..u. denotes approximate evaluation of V (T)) obtained by optimizing mine,.. 9i( p,,, pp.)) using any optimization algorithm practically, rather than evaluating
exactly at pp. E (..)..
Theoretical Results
Convergence Before presenting theoretical convergence results, first introduce set of mathematical assumptions regarding the problem.
Assumptions 1. We assume that g(,) h(,)) ), and l,(,) for all i are differentiable and Lipschitz continuous. Moreover, for every 9: ( is gradient Lipschitz continuous with constant I,, and satisfies the proximal Polyak- Lojasiewicz (PL) error bound.

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Note that these assumptions are standard and quite moderate. For example, in the applications of DAG learning, the loss functions are typically score functions, which are smooth. In addition, the causal structure constraint h(,,) for DAG is also smooth. Since the non-smooth component involves the normm, effectively penalizes variable magnitudes, which in turn ensures the boundedness of the iterates. This observation consequently implies the Lipschitz continuity of the loss functions.
The assumption of the proximal PL error bound [27 is primarily employed here make the optimization process of the lower-level (or inner loop) more tractable, a condition that holds for broad spectrum of non-convex loss functions. Specifically, when the neural network size large, i.e, overparameterized, the associated loss function satisfies the PL condition. In this context, due to the presence of the non-smooth term, we utilize the proximal PE error bound. Under this condition, a few steps of gradient descent can approximately find the global optimal solution, meaning that the last iterates of the gradient descent can be used to approximate o O), allowing S))) to be easily evaluated, denoted as 5, (D), where & is the concatenation of all the model parameters. This approach is both practical and can be rigorously quantified theoretically.
The convergence criterion in Line 5 of Algorithm[] is based on the sum of gradients in the sliding window:
IP, (f;+1, (. > 6/w, where d is a constant, which means that if the proximal residual is not sufficiently small, we continue optimizing the model parameters using the current domain data.
(11)
Theorem 4.1. Regret Bound. Under these assumptions, suppose that 4 { pp. |
m W + +11, 1...., m) is generated by the algorithm. Then, the following holds:
Reg (m)
Σ IP/ U.PP,VS,,ce (2.))) O
m7
(12)
Remark 4.2. Since there are two levels in the optimization process, quantifying convergence requires solving the lower-level problems, unlike classic single-level problems. Thus, our proof must account for errors in solving both levels. Due to the nonconvexity of the loss function, we use the sliding window concept demonstrate the stationarity of the solutions obtained.
The above regret indicates that the changes in the Biddnng-windon task-averaged functions converge to a small value when the sliding window size is large, meaning that E, I (#;, (.. |I2 Reg..(m) (m))m 0 Ow²). It is worth noting that this algorithm achieves the optimal regret bound, as equation 12 matches the optimal regret the time-smoothed online proximal gradient method for solving classic single-level nonconvex minimization problems [22 21]
Other Assumptions We also make few standard assumptions for dynamic DAG learning [49], such as causal sufficiency (no latent variables), following an SEM model (linear or nonlinear), and sampling frequency of the event sequence is at least as high as the fluctuations in the underlying causal process. However, we relax the typical assumption that the graph structure is fixed over time, and assume that it is only fixed within certain time steps (as fixed within a domain in the meta formulation).
Identifiability In Appendix A.2
we demonstrate that, under a mild condition on the shared parameters W. estimated from previous domains, the earlier identifiability results from [4] 18 can be extended to our method for learning the graph with adjacency matrix W. Wp.m within a given domain m.
Complexity Our general meta-learnig framework has the same computation complexity, with respect to the number of nodes, as existing continuous optimization approaches. The additional computation in our formulation mainly comes from the lower-level optimization, where we need to learn DAG for each of M domains. The upper level can be solved faster without any constraints.
5 Empirical Evaluation
We evaluate our proposed meta dynamic DAG learning algorithm against some of the existing DAG learning algorithms, designed for dynamic time series datasets and/or with multi-domain capabilities.
8

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120
GOLEM
120
IDYNO-Soft
IDYNO Soft
100
DYNOTEARS
100
NTS ATS-ND
NTS-NOTEARS
80
JPCMC
Meta D2AG
60
40
40
0
10
15
20
10
15
20
(a) SHD results v.s. different variable size of different algorithms synthetic datasets (batch setting).
(b) SHD results v.s. different variable size of different algorithms synthetic datasets (online setting).
test the algorithms on synthetic datasets as well as a simulated reinforcement learning (RL) environment, Sprites World, with intervention datasets. The experiments are run machine with 3.2 GHz CPU and 64 GB of memory.
Baselines We compare our method with the following baselines with available code: continuous DAG learning methods GOLEM 477, dynamic DAG learning algorithm DYNOTEARS 49,, NTS- NOTEARS [63], offline interventional dynamic DAG learning algorithms IDYNO [8], and multi- domain dynamic DAG algorithm J-PCMCI+ [19. We repeat each experiment times and compare the structural Hamming distance (SHD) between the estimated graph and the ground truth graph (the lower, the better). baselines except J-PCMCI+ are designed to handle one domain, so we learn each domain separately and report the average SHD with standard deviation. J- J-PCMCI+ only finds one unique graph across all domains so we use as the graph each domain.
5.1 Synthetic Datasets
We first evaluate different approaches synthetic time series datasets. We simulate the data according to the SEM from equation mostly following the setup and code from 44) for ease of comparison. The generating process consists of the following three steps: 1) Generating weighted graphs in the form W and Wb. Directed graphs are generated first as a lower triangular matrix, and then randomly permuted, 2) Generating data matrices X and corresponding Generating interventional data with random targets and different distributions. For a more detailed generation and training process, we refer readers to Appendix B.2 [.2
per W and W and 3)
We show results for the batch setting where all 10 domains are learned jointly per equation 4 and sliding window size of previous time domains to meta-train
for the online setting where and the current domain as the meta-test dataset. For the batch setting, since all domains are used in meta-training and there is new meta-test data, report the SHD of training accuracy of each domain. For the online setting, we report SHD from the meta-test domain at each time step. given the previous 3 domains as meta-training (padding in the beginning with the data from the first domain). For baseline methods in the online setting, to distinguish from the batch setting where each domain are learned separately, we feed all domain data up the current time stamp each method and produce graph.
from Figure 2a in the batch setting, our method (colored in red) has the lowest SHD. Since each domain has different graphs, hurts the performance of J-PCMCI+ For the online setting shown in Figure 2b M AGs again superior to baselines. In appendix, an ablation study with two variants our method, one only optimizing the shared parameters and one with linear loss function, showing the advantage of the proposed formulation. Moreover, in Meta-D 2 AG the DAG constraint follows the matrix exponential form as used previously by [49) Other and better DAG constraints can also be used, such as 76
As
also conduct
Ablation Study In the appendix, we have included further results of many ablation studies of our approach, including the necessity of the lower level optimization private parameters and the impact

---

of sequence lengths and variable variance. Overall, the proposed Meta-D&AG shows promising performance in these settings.
5.2 Sprites World RL Environment
Sprites world 669 is Python-based RL environment in which objects of various shapes interact with each other created
Table 1: SHD performance different
in 2-dimensional space. In the experiment, 5 and 10 objects, each associated with a continuous variable that assigns values following a linear structural equation model or non-linear structural equation model similar to causal discovery benchmarks (28 [53] For
methods in Sprites World.
Method
O 5
0 10
further details about the data generation and training
GOLEM IDYNO DYNOTEAR JPCMCI+
6.0 6.0+1.3
2.4
process, we refer readers Appendix [B.3]
5.8 1.7
12 + 3.0
As shown in Table comparing with the baselines, our method achieves the lowest SHD performance for 5 and 10 objects, again demonstrating promising graph discovery performance in RL environment.
1.3
39 + 1.3
127
31 + 0.6
Meta-D'AG
0.2
9.1 1.5
6 Conclusion
We propose a new Meta-D2AGG method for causal graph discovery in dynamic datasets. Leveraging the meta-learning framework to capture temporal transitions between different time domains is well-suited for causal graph discovery in dynamic datasets. We make three major contributions. First, we formulate meta-learning framework tailored for dynamic DAG learning, accommodating slightly varied distributions in different domains over time. We further extend the framework an online setting, where a windowed formulation is proposed to improve learning efficiency in scenarios with extensive historical data. Second, recognizing the challenges posed by acyclicity constraints in inner-level optimization, we propose an efficient first-order method to solve the bilevel framework. Finally, theoretical analysis and experimental evaluations are provided to show the convergence of the learning process and the superiority in empirical performances. Future research could explore methods for the automatic detection of potential distribution shifts without the need for such a priori information. For example, distribution shift detection is an active area of research, and we can adopt various strategies, such as measuring prediction error, using distribution distance, and/or hypothesis testing to detect shifts (38) and form a new domain. It would also be interesting to combine the proposed approach with irregular 7 or sub-sampled time series [36].
Limitation and Impact Statement One limitation of our approach is its reliance on predefined domains in time series datasets, which may be restrictive when distribution shifts occur irregularly. Obtaining such partitions of temporal data when no domain labels are provided would be a useful future direction. This paper presents work whose goal is to advance the field of causal discovery algo- rithms. Although there are no substantial potential socictal consequences of our work, misuse could result in wrong conclusions about the causal relationships, which need to be carefully empirically validated in a controlled environment.
Acknowledgments and Disclosure of Funding
The authors thank anonymous reviewers for their constructive feedback. The work Songtao Lu is supported in part by the Chinese University of Hong Kong (CUHK) Direct Research Grant (Project No. 4055259). Yue Yu is partially supported by the National Science Foundation (NSF) under award DMS- 2436624 and the Defense Advanced Research Projects Agency (DARPA) under award HR001 12590032. This research was, in part, funded by the U.S. Government by an agreement with Cornell University. The views and conclusions contained in this document are those of the authors and do not necessarily reflect the views Cornell University, and should not be interpreted representing the official policies, either expressed or implied of the U.S. Government.
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21

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A Technical Appendix
A.1 Algorithm and Convergence
As the optimization algorithm is general, for case of reading, we simplify the notation here. Instead of using Dg and pp, we use (z, y).
Let define
(x) = fi(;;a) W. W p.a (W.), P. (W.) + 7| w1
(13)
and
1
g( vi)
| train pp o X )) + 7l W p.ili..
(14)
2n
Then, the problem can concisely written as
min Σ (x + (x)
(15a)
s.t. Vi E S(x) == arg min g. vi) + nf())) s.t. h(x, y1) =0
(15b)
9i
where in this case f0 = || I| but it can generally represent any cnojection-friendly non-smooth term.
To solve this problem, we can apply the penalized method. Towards this end, we first write the
penalized objective follows
Σ ((0 + (gi(x, vi) (x)) + wh(x, vi) + (f) + (1s)
Ci(2, (unki,
(16)
where f7,1) v) are penalizers, and g7 (x) ming gi y).
Recall the definition of the proximal operator:
prox, (x nd)
arg min fz) + / || - bll z||
(17)
arg min nf + ,ydd, = - x) + 2 22112= -1===11²
(18)
Then, can have the proximal residual defined as
(f, d) (r prox,, (x - -jd)),
(19)
which will serve as the optimality gap for measuring the convergence of the algorithm.
For simplicity of the notations, define = = (x, [ui.j i + 1,
,i)). l,).., Then, we can construct
the window average as follows
(2)
Σ j=i-w+1
(z) (17 u;) - (3())) vhy(x, m1).
(20)
Due to the non-convexity and non-smoothness of the loss function, follow the existing notation of the local regret of policy z up to time m with window length w, which is defined
Reg (m)
Σ
IPp/(22, (x)))1
(21)
1=1
From the definition of () we can have
(z = (z)) + w
(es( yi) - fi- (x,, yi- vi-
+ 7 (a: vi) ge (x) - (gi- (2, wi-w) - G )i (x)))
+ (h,(I, y:) h (x, Bi vi-w))
(22)
22

---

However, due to the bilevel structure of the problem, we can only achieve an estimate of 9 (æ) through the inner loop update y, which is denoted as (i(2) g(z, ()..). Then, we can use [P (z++1, V (2(+1)) determine whether we need update the model.
Recall our algorithm: when [ (+++((++1) > 8/w, we implement the proximal gradient descent. Otherwise, we update our model, where
(2) = 1.)) + (L: vi) - ( vi-
+ (0 vi) 0((x)) - (gi- w (2, vi-w) - Di (i-w(x)))
(h,(21, Wi) (2, (x,
(23)
Next, we will find an upper bound of [P ( First, we can have
IP (z)())))
IP/ (z., [+(=)))]++PP//) VS.(z:)) - (s; (z)))
(24)
1 (2), (
2 Va (x;) (Vgi Vi Vgi (2:))]
+ [Vi1(T,, Vh, (7,, Vi m )| + V4 vi) - V ( (i
2 (o)(a.) - v.,(=:)] + 2 V (x:i)
Vzi-w(z;)
(25)
16
+ [VVg((.)) - (g.(2:) 1 V (z:) - Vgh (x:)ll + Vi.w
(26)
w
L. (T:) + 2 Lyds ( (i-w) + Vi..
(27)
where (a) is due the triangle inequality, (b) holds by following the condition of updating the model and Vi is defined as
Vi Y V (a vi)
(ol((.) (V (Tit (it: (
+
[h,( vi)
Vi (Mi,, Di w + Vii v:) Vii (2i, yi wi-w)]
(28)
and in (c) we apply the gradient Lipschitz continuity of g: (x, -), Vi and ds(v) arg miny'es I| w|l.
When the lower level problem satisfies the proximal PE error bound, i.e.
y - prox,, / (v V yg v)) 2 (1)
(29)
where is constant and y here is generic, representing any ys, then according 22 Appendix G] we have the function also satisfies the proximal Pt. condition, which is
( 2 p (F(z, 9) F (x))
(30)
2
where F(x, y) = g(x.zz v) + nf (x) (=)
= (x) + nf nf(x)) and
x)(y;))= x) = 2Lg min Vg v). v) + Ly
1 21l2 + -
-
(31)
2
y'
By the Lipschitz gradient of g(x, v), we have ((4,v/+)) + ()(()+1)
g((x, vt) + (f(()) + (Vvg(z, Vh4 - vi) + L |1 1/112 + mf(((+1) (f(
(32)
2
=(. vl) + f(v)) - 2L, (1/1))))
(33)
23

---

where y/ denotes the iterates in the inner loop.
Substituting equation 30 equation 33 gives
F (x) = (1- (1-1)
Lg /
T - < ) (F(x,
L
F*(x)) (F(x.)))-F(())) (34)
where 7 denotes the number of iterations in the inner loop.
From [4] Corollary 3.6), know that the proximal PL error bound implies the quadratic growth property, meaning that there exists a constant j such that
1 1 ( H 2
Le )
S(x)
11 (1
(F(x,v() F (x))
(35)
where 0; = V/++1 As x is bounded, the term ds(x.) (Ti) shrinks to 0 at a linear rate.
Summing over i 1,..., m gives
Reg.. (m)
IPf (zi, V.S. (2)))1
(36)
1=1
4mn2²
+ 4L2g7 Σ Ls (())) (T:) )+)++ + 4Lzr² Σ Ss (T) (Di-w)
+
Σ w
4
m
Ve,
(37)
w²
1=
When we choose T> - log p (xx, F= we have Ei Ls (Ti) ~ mm/w"
The last term Σ; Vi,w is also referred to the sliding window variation. When the gradients of the loss function are bounded, we automatically have that this term is (m)) Therefore, we can conclude that our algorithm still can achieve the optimal regret bound Om/ww²)
A.2 Identifiability
In this section, we demonstrate that Theorem 3.2 from [8] (an extension of Theorem from 4)) can be applied in our online meta-learning setting. We denote the current domain with index m, and recall that a model estimate Wa of the shared, cross-domain part of the graph adjacency matrix can be formed from the data from the previous objective is to optimize the part of the adjacency matrix specific to the domain, Wp. (where k 1 indicates the observational graph, and k > 1 indexes interventions this graph, in (18).
1 domains. When encountering domain m,
We denote with 9m the DAG whose edges are given by the nonzero elements of the adjacency matrix W = W. o W p.1,k=1- Our score function for 9m can be written (following the notation of [4]
Sz. (Gm) = supe... W
E Σ 1=1
log (X: W9m (X;
R - 7/9mm
.0m,k)=
(38)
Pm
11 is the LO norm (which counts the number where the regularization term 1mmll = W. 1nm,,
(k) of edges in 9m), 6n (0m 1km1 with 6m, {6m,k.) 1 are neural network parameters, pm is the ground truth generating distribution of the data for domain m with intervention I₁ E I, and
K,d R = (rky) R.j= specifies the set of interventions, which influence the log-likelihood function as follows,
d
log (k) (X; W, R², 0m,k)
Σ
- log NNX,,NN(XOOO Km,k.j (1))) log ((X,NN(X Om, Wz))
where we again denote the observational distributions with k: 1, and sum over the model likelihood function ff (for intervention k) for each node j in the graph. Here, "NN" denotes a neural network function, and W := W. Wp.m, is the full adjacency matrix for domain m and intervention l (we assume W does not depend on the intervention k). Lastly, when taking the supremum in equation 38 we use [m indicate any values of the continuous matrix W, subject to the constraint that edges absent in Gm correspond zero entries in W.
24

---

The only difference between this setting and the setting of (18) is that instead of learning an adjacency matrix, we are learning factor p.m.kk of the full adjacency matrix W, W W.m.kk Thus, in the case where Ws is set to the all-ones matrix, we recover the setting of [8]] exactly. Theorem 3.2 from [88] (as well as Theorem from 4 involve the set of DAGs maximizing a score function Sz- (9), where I* is the ground truth set of interventions. Our score function, equation 38 is equivalent when Ws is set to the all-ones matrix.
For any given adjacency matrix W maximizing equation 38 (for a fixed graph 9m), as long as the elements of Ws s are nonzero wherever W is nonzero (wherever the graph 9n, has an edge), then there exists a matrix Wp
p satisfying W = W o W for which each element is determined uniquely as (Wp) (W*) k// (Wa)))) Thus, the set of W* maximizing equation 38 is in one-to-one correspondence with the set of W p maximizing the same quantity (given fixed Ws). Conditional
on given graph In and given any matrix W picked out by the supremum over W W there exists a matrix Wp p for which our score function, equation 38, is maximized with W = W o W
This allows to extend Theorem 3.2 from 18. with additional condition on W,:
Theorem A.1. For graph D. where DAG, if E a (9) with SI defined in equation 38 and furthermore (i) the density model has sufficient capacity to exactly represent the ground truth distributions, (ii) a given set of interventions I satisfies I I-faithfulness for the true graph and distributions ( (iii) the density models are strictly positive, (v) the ground truth densities p(h) (X) have finite differential entropy, and (vi) the elements of Wa are nonzero wherever the elements of the adjacency matrix of the true graph G nonzero, then & is (I, -MMMaroov equivalent to for small enough n3.
In practice, condition (vi) be supported enforced by adding additional regularization term or hard constraint to prevent Ws from becoming too sparse.
Remark A.2. Since learning W. in the upper learning is conducted with any constraints (in particular without an acyclicity constraint), the upper-level optimization is the same in any L1 Ci-regularized regression problem. Hence it is easy for the assumption on the nonzero elements of W, to be satisfied.
25

---

B Data Appendix
B.1 Dynamic DAG Learning and Data Normalization
Recently [58] have found there may be biases existing in the data generating process, and data normalization suggested. Here we would like to point out that time series DAG learning differs from the i.i.d. setting, and it is not obvious how the conclusion from [58) is directly applicable to time series data and what sort of standardization even makes sense to generate data in our setting. For example, standardization in i..dd data is across samples, which is different from typical time series standardization across time. Moreover, time series data like Brownian motion naturally increases variance over time, real-world datasets should indeed show higher variance for variables at later epochs. Yet another complexity is that the data in our setting is non non-stationary. It would be an interesting research question to investigate how standardization of data might affect time series DAG learning in general.
B.2 Implementation Details for Synthetic Datasets
We the following steps and hyper-parameters generate the data.
a with degree of 3 for total d nodes, from 5 to 20, with its weights sampled uniformly at random from 2([-2.0, -0.5] JJ [0.5, 2.0]). We use the same Erdos-Renyi model to generate inter-slice graph W with degree of 3, with its weights sampled uniformly random from 44[| -0.3] 3] U [0.3, 0.5)). Then for each domain m, we randomly remove or add 2 edges from the previous domain (randomly generated in the first domain), to ensure temporal smoothness. Then we generate weights of W p.m similarly as W. but with edge weight ranges of 5 times smaller, i.e., 4[[ 0.4, - -0.1] U [0.1, 0.4)) for p and (([1-.11, -.006] U [0.1, 0.06]) for W.:
For step 1). we use the Erdos-Renyi model to generate shared intra-slice graph W
For step 2). we generate data with the first autoregressive order, where data only depends on the previous time slice. The underlying function between node and its parents becomes two-layer MLP with a sigmoid activation function. In the MLP setting, the weights of the MLPs are sampled uniformly from (([[[[0. -0.5] U 20.5, and then weights in the first layer are updated by the parental weights from a and W structures. The number of hidden neuron sizes is set to be 10. We generate 10 sequences with 100 time slices each with standard Gaussian noises. We estimate graph from each sequence and report the average SHD along with its standard deviation.
For step 3), to generate interventional data, we flip a fair coin to decide whether to intervene at any time slice, in which case we sample one node uniformly to be the intervened node. We sample intervened nodes' values from another distribution. We assume every node has a probability of 0.1 to be intervened upon at each time slice, and if node is chosen, the soft intervention comes from a different 2-layer MLP, depending on the values of its parent nodes per its graph and similarly generated as the first MLP. This soft intervention distribution can be very different from the observation distribution.
Hence, overall we generate 10 sequences for 10 domains, each sequence with 100 time steps. For the hyperparameter values, we use a separate validation dataset to choose the best per- forming hyperparameterrs for each method per SHD? We search for the best value of each of parameters sequentially, including two L, penalty coefficients for Wa and W5, the thresh- old to obtain Wi and W and the hidden neuron size. For Xa and Xw; we search over a value range of (10-b 5, 10 10-3, 10-2, 1002,10011 10--). } Graph threshold search range is set to be (0.001, 0.01, 0.05, 0.1, 0.2, 0.3). and neuron size range is searched over {8, 16, 32, 64). We use two-layer MIP architecture in the experiments: X, = WWo((W, (W.) W.) )), where W, is the MLP layer weight and is the ReLU activation function. For other baseline methods, we use the default parameter settings.
B.3 Implementation Details for Sprite World Datasets
To generate a meta-learning dataset, we first generated a random Erdos-Rent graph, replicated it to the next step, and added single edge for each variable at two consecutive time steps to introduce
2 Alternatively, if datasets do not contain ground truth graphs, we can use the reconstruction loss on data as alternative to measuring the quality of the learned graph
26

---

correlation across time steps. When we switch to different domain, we randomly add or remove an edge in the instantaneous graph and time-delayed graph. Since we are interested in structure learning, we used a random policy agent that randomly selects at most two objects and intervene the value with a truncated normal distribution at each time step. The details of hyper-parameter are given as follows. For non-linear SEM, we utilized a two-layer MLP with the weight in the first layer ranges between (0.5, or (-2.0, - and the second layer between (0.01, 0.5) or (-0.3, -0.01) ) and the bias follows the Gaussian distribution with scale I. We again generate 10 domains with 10 different sequences with 100 time steps, and report the SHD performances across sequences.
B.4 Ablation Study: Linearity and Private Parameters
We perform an ablation study with two new baselines, 1) a multi-task DAG learning where only shared parameters are learned (named MT-DAG), and linear meta-learning formulation (named Meta-D2AG-Lineart on synthetic datasets with scale-free (SF) graphs.
The data generation process is the as ER graphs discussed, except for the graph types. Our Meta-D2AG shows the best performance against these two variants of the proposed approaches, as well as the best performing IDYNO-Soft From Section 4.1, in both batch and online settings.
Var. Size
IDYNO
MT-DAG
Meta-D2AG-LLnear
Meta-D2AG
d==
11+3.9
9.3+0.9
21+0.5
7.2+1.4
d = 10
23+3.1
25:4.4
37+1.7
18+0.8
51+3.2
58+15
50+1.8
24:2.6
d = 20
4245.3
4629
67+4.1
35+1.9
Table 2: SHD Results for Scale-Free Graph (SF) the Batch Setting
Var. Size
IDYNO
MT-DAG Meta-D2AG-Lineat Meta-D2AG
d
8.8+0.3
1.+1.6
21+1.3
8.7
10
29+1.9
3741.6
46+2.7
18+0.7
d 15
5442.9
55+4.9
5342.4
27=0.4
20
4745.5
7746.1
6645.8
36:1.2
Table 3: SHD Results for Scale-Free Graph (SF) in the Online Setting
B.5 Ablation Study: Sequence Length
We conduct an experiment on the effect of sample sizes vs SHD. We use variable dimension d = 20 and 10 total sequences, and the rest setting the samw as in Section 5.1. When the sequence lengths are lower, all methods perform worse, but our method remain competitive.
Sequence Length
10
50
100
GOLEM
115 +
102
81 + 10
DYNOTEAR
187 + 23
150 - 20
89 + 16
IDYNO
129 + 16
97 14
97
81
JPCMCI+
132 + 21
1:0:200
119
NTS-NOTEARS
119 +
95 0
+
Meta-D2AG
101 + 10
80
46 16=0 6
Table 4: Performance across different sequence lengths in Online Setting, d = 20.
B.6 Ablation Study: Random Variance
To study the impact of variable variance, we repeat the synthetic dataset in Experiment 5. with random noises between 0.1 and 2 for each variable noise term, with 4 different variable sizes. Below are the results against baselines. The relative errors are higher than of fixed variance of but the method remains the best among methods tested.
27

---

15
GOLEM
5.9 0.9
1.
43.7 0..777 7.8
9.9
DYNOTEAR
1.1
3.0
6.
5.3
IDYNO
6.5
2.1
5.7
90 9.2
JPCMCI+
0.1
23.6
10.2
105 + 19.8
NTS-NOTEARS
12.3 +
3.1
55.7
104 + 10.2
Deta
0.1
22.9
46.7 - 4.6
82.7 = 8.9
Table SHD with random variable variance across varying dimensions
online setting.
28


Reviews:
[{"review_id": "sKNZ1jw2yK", "reviewer": "Reviewer_ufRx", "summary": "The authors propose Meta-D²AG, a dynamic DAG discovery algorithm for time series data, which is based on meta-learning. The model adapts to non-stationary distributions over time by considering different time windows as separate domains and uses meta-learning to learn shared parameters across domains while quickly adapting to new domains with few measurements.", "questions": "1. Could the authors provide a visual illustration or diagram of the overall model pipeline to help non-expert readers better understand the approach?\n\n2. Have the authors evaluated the model on any real-world datasets, and if not, can they comment on its expected performance and applicability beyond synthetic settings?\n\n3. Given that the method relies on predefined domain splits, have the authors considered ways to automatically detect distribution shifts or segment domains in practice when such labels are not available?", "limitations": "yes, limitation were adressed in the conclusion", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Weaknesses\n- The paper could benefit from clearer examples or visualizations to help readers intuitively understand how the method works\n- The proposed approach is technically complex and may be difficult for non-experts to understand or implement without significant background knowledge\n- Most experiments are done on synthetic or simulated data. It's unclear how the method performs on real-world datasets.\n\n### Strengths \n- The results show significant improvements over baseline methods\n- The paper addresses an important problem innovatively by combining methods from different research areas\n- The method is flexible and works with observational and interventional data", "was_revised": true, "final_justification": "The authors' responses addressed my main concerns, particularly around model clarity and real-world applicability. Combined with the positive feedback from other reviewers, this led me to raise my score, despite some remaining difficulty in fully assessing the technical details.", "rebuttal": "**Visual illustration or diagram of the overall model pipeline to help non-expert readers better understand the approach** : thank you for the suggestion, and we will add an overview figure to highlight our methods, complementary to our description in Section 3.2 and 3.3. It will consists of two sub-figures, each for offline and online versions of our methods. Each sub-figure will contain both meta-train and meta-test phrases. For meta-train, we will highlight different domain data $\\bf{X}$, which parameters are shared and private ($\\Phi_s$ and $\\Phi_{p}$'s), learned graphs $A_{m} = W_s \\circ W_{p,m}$ for each domain $m$, and its joint loss function. For meta-test, we will highlight how meta parameters are fixed and only need to solve for new $\\Phi_{p,m}$ for a new domain. \n\n**Have the authors evaluated the model on any real-world datasets, and if not, can they comment on its expected performance and applicability beyond synthetic settings**: Thank you for the comment. While we do not have public datasets with ground truth causal graphs to test our algorithm on, we emphasize that our algorithm should be same or better performance than existing methods, as our framework includes existing methods as inner problem. Our methods are well suited to real world datasets where stationarity is violated. \n\n**Given that the method relies on predefined domain splits, have the authors considered ways to automatically detect distribution shifts or segment domains in practice when such labels are not available** : Thank you for the comment. Distribution shift detection is an active area of research, and we can adopt various strategies, such as measuring prediction error, using distribution distance, and/or hypothesis testing to detect shift [1] and form a new domain. We will add such a discussion.\n\n[1] Lu et al, \"Learning under Concept Drift: A Review\", IEEE Transactions on Knowledge and Data Engineering."}, {"review_id": "f3twIlHsMj", "reviewer": "Reviewer_mYVS", "summary": "The paper proposes a new dynamic DAG discovery algorithm, Meta-DAG, based on online meta-learning. It can learn from potentially nonlinear and non-stationary time series. Meta-DAG involves a new online meta-learning framework to take advantage of the temporal transition among existing domains such that it can quickly adapt to new domains with few measurements. The authors also provide theoretical analysis of the convergence of the algorithm as well as identifiability results. Meta-DAG shows improved performance on the conducted tests.", "questions": "1. One of the main claims made in the introduction section was the ability of the proposed method to quickly adapt to new data distribution and achieve higher graph learning accuracy with a few samples. However, this claim was never experimentally validated. It would help to compare the baselines and the proposed method on the sample requirements to learn the parameters on new domain data. \n2. As detailed in Appendix B.2, the noise variance seems to be set to 1 for all the variables. What happens when the noise variance is not constant and is random? Does Meta-DAG exhibit the same trend event in this case? \n3. The conversion of (6a), (6b) to (7) seems a bit unclear. It would help to have a more elaborate discussion on this in the main paper or the appendix. \n\nIf the authors can satisfactorily address the questions above, I would be inclined to raise my overall recommendation.\n\n(Minor comment) There's a typo in line 220.", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Quality: The paper is technically sound with good theoretical results. The experimental validation showcases improved performance over the baseline. However, one of the main claims of the paper, Meta-DAG can \"quickly adapt to a new data distribution and achieve higher graph learning accuracy with a few samples\" hasn't been shown experimentally. \n\nClarity: The presentation is sound. However, the discussion in section 3.4 could benefit from further explanation. Especially around eq (7). \n\nSignificance: The proposed work addresses the difficult task of dynamic causal discovery from multiple domains. The experimental results show that the proposed method Meta-DAG outperforms the baselines in both synthetic and real-world experiments. However, it would help to have addition experiments testing the limits of the proposed method, see Questions. \n\nOriginality: The proposed work is novel and it introduces an online meta-learning based procedure for dynamic causal graph learning.", "was_revised": true, "final_justification": "The new experiments provided during the rebuttal addresses my concerns.", "rebuttal": "**Sample size v.s. performance**: Per request, we conducted following experiment on samples sizes vs SHD. We use variable dimension $d=20$ and $10$ total sequences. When sample is low, all methods perform worse, but our method remain competitive. \n\n| sequence length | 10  | 50 | 100|\n|---|---|----|----|\n| GOLEM  | $115\\pm20$  |  $102\\pm10$     |  $81\\pm 10$   |\n| DYNOTEAR  |  $187\\pm23$    |   $150\\pm20$  | $89\\pm16$   |\n| IDYNO  |  $129\\pm16$   | $97\\pm14$    |   $80\\pm21$   |\n| JPCMCI+ | $132\\pm21$     |   $130\\pm20$  | $119\\pm9$   |\n| NTS-NOTEARS   | $119\\pm15$    |   $95\\pm 5$   | $63\\pm13$   |\n| Meta-D2AG   | $101\\pm 10$   |  $80\\pm 7$    | $46 \\pm 6$   |\n\n\n\n**Random variance**: Per suggestion, we repeat the synthetic dataset in Experiment 1 with random noises between 0.1 and 2 for each variable noise term, with 4 different variable sizes $d=5, 10, 15,$ and $20$. Below are the results against baselines. The relative errors are higher than of fixed variance of 1, but the Meta-D2AG method remains the best among methods tested. \n\n| d | 5 | 10 | 15 | 20 |\n|---|---|----|----|----|\n| GOLEM  | $5.9\\pm0.9$  |  $26.6\\pm1.8$  |  $43.7\\pm7.8$    |  $93.5 \\pm 7.9$  |\n| DYNOTEAR  |  $6.7\\pm1.1$  |  $22.3\\pm3.0$  |   $60.4\\pm4.6$ | $95.6\\pm5.3$   |\n| IDYNO  |  $6.5\\pm1.1$ | $26.5\\pm1.1$   | $57.5\\pm5.7$   |   $90\\pm9.2$ |\n|  JPCMCI+ | $6.6\\pm0.1$   | $23.6\\pm1.1$   |   $85.3\\pm10.2$ | $105\\pm19.8$   |\n| NTS-NOTEARS | $12.3\\pm0.5$   | $33.3\\pm2.1$   |   $55.7\\pm 5.3$ | $104\\pm10.2$   |\n| Meta-D2AG  | $6.6\\pm0.1$  | $22.9\\pm 2.2$   |  $36.7\\pm 4.6$ | $82.7 \\pm 8.9$   |\n\n\n\n**The conversion of (6a), (6b) to (7) seems a bit unclear**: We move both the objective and constraints in the inner problem as different penalty terms to the outer problem, and thus form as a single objective. Thank you for your comments, and we will clarify this and other typos in the paper."}, {"review_id": "StpHHpGmaY", "reviewer": "Reviewer_fhwy", "summary": "This paper investigates causal discovery by proposes an online meta-learning based method to learn dynamic DAG structures from potentially nonlinear and non-stationary time series datasets. Different from existing work, this paper explicitly treats data collected at different time points with distribution shifts as distinct domains, which is assumed to occur as a result of external interventions. The authors design a first-order optimization approach to solve the meta-learning framework, and provide a detailed theoretical analysis to establish the identifiability conditions and the convergence of the learning process. Extensive experiments are conducted on benchmark datasets to demonstrate the promising performance of the proposed approach.", "questions": "N.A.", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\nS1. The proposed meta-learning framework can handle both observational and interventional data distribution with nonlinear relationships by formulating a non-stationary collection of dynamic datasets as different domains. \n\nS2. A theoretical analysis is provided to establish the identifiability conditions in the batch setting and provides convergence rates in the online setting.\n\nS3. Experimental results on benchmark datasets demonstrate the superiority of the proposed approach.\n\nWeaknesses\n\nW1. The authors only use SHD as the metric in the experimental study. Maybe more metrics like TPR could be used to report the performance of the proposed approach.\n\nW2. In Figure 1b, is IDYNO-Soft plotted with error bars? Why the other methods do not have error bars?\n\nW3. It would be easier for readers to understand the general idea if the authors could use some figure to show the overall framework.", "was_revised": true, "final_justification": "I’ll retain my score", "rebuttal": "**Metrics**: We use SHD as it is the standard metric. We will add more results using TPR, but for the Sprites World  dataset in Section 5.2 for example, our method has TPR = 0.73 while the other baselines are in range of 0.5 to 0.65.\n\n**Error bars**: the errors bars for some methods are not rendered correctly, and we will fix them. In general they are small related to the mean, as seen in Table 1, 2, and 3.\n\n**Overall figure**: thank you for the suggestion, and we will add an overview figure to highlight our methods, complementary to our description in Section 3.2 and 3.3. It will consists of two sub-figures, each for offline and online versions of our methods. Each sub-figure will contain both meta-train and meta-test phrases. For meta-train, we will highlight different domain data $\\bf{X}$, which parameters are shared and private ($\\Phi_s$ and $\\Phi_{p}$'s), learned graphs $A_{m} = W_s \\circ W_{p,m}$ for each domain $m$, and its joint loss function. For meta-test, we will highlight how meta parameters are fixed and only need to solve for new $\\Phi_{p,m}$ for a new domain."}, {"review_id": "wWjIK6lVVa", "reviewer": "Reviewer_gc1P", "summary": "This paper addresses causal discovery in dynamic, non-stationary time series, proposing Meta-D²AG—a meta-learning framework based on bilevel optimization that models both parameter and structural changes as domain shifts prompted by interventions. Unlike prior approaches, Meta-D²AG explicitly treats distribution shifts as domains and leverages both historical data and online learning to enable rapid adaptation with few samples, using a first-order optimization method for efficiency. Theoretical guarantees on identifiability and convergence are provided. Empirical results on synthetic data and a reinforcement learning environment show that the method outperforms recent baselines in terms of structural Hamming distance (SHD), under both batch and online evaluation.", "questions": "- The synthetic time-series experiments in the paper only consider non-linear structural equation models. To better assess the generality of the proposed method, it would be important to include experiments on linear vector autoregressive (VAR) models.\n- The work would benefit from comparisons with additional baselines to better demonstrate the performance advantage, such as NTS-NOTEARS[1] and NGC[2].\n- The authors mention using default hyperparameter settings for other methods (Line 1002). To ensure a fair comparison, it is important to conduct some level of hyperparameter tuning for these baselines.\n- The paper should report runtime comparisons between Meta-D²AG and other methods to clarify whether the performance gain comes at the cost of increased computational overhead.\n- The experiments are conducted under a fixed sample size. Evaluating performance across varying sample sizes would be helpful to understand the robustness and scalability of the method under different data regimes.\n- The method assumes that domain boundaries (e.g., change points) are predefined, which may limit its applicability to real-world time series with unknown or unobserved distribution shifts. Could the authors discuss how Meta-D²AG might be extended to incorporate automatic domain segmentation or change-point detection techniques?\n\nReference\n[1] Sun X, Schulte O, Liu G, et al. Nts-notears: Learning nonparametric dbns with prior knowledge[J]. arXiv preprint arXiv:2109.04286, 2021.\n[2] Tank A, Covert I, Foti N, et al. Neural granger causality[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021, 44(8): 4267-4279.", "limitations": "Nothing", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- Addresses a Practically Significant Problem: The challenge of learning causal structure from non-stationary, interventional time series is of high interest in machine learning, with broad applications.\n- Clear Framing and Motivation: The paper frames the problem well, connecting real-world distribution shifts with the technical approach; the definition of \"domains\" via interventions is clearly articulated in Section 1.\n- Methodological Contribution: Introduces a new bilevel meta-learning approach enabling separation and adaptation of shared versus domain-specific parameters (Section 3.2, “Meta-Learning for Dynamic DAG”).\n- Online and Efficient Learning: Offers a mechanism for online adaptation using a sliding window and efficient penalty-based bilevel gradient descent (Section 3.4); this is concrete and distinct from prior work.\n- Solid Theoretical Foundation: Provides theoretical convergence guarantees (Theorem 4.1, Section 4), with clear statement of assumptions. The regret bound and discussion around use of sliding window optimization are appropriate for the task.\n- Empirical Validation: The experiments compare against a broad set of contemporary baselines, covering both synthetic and RL datasets. Quantitative evidence—Strong empirical results are shown in Figure 1a and 1b (showing that Meta-D²AG achieves consistently lower SHD than competitors as variable numbers/d increases). Table 1 shows strong SHD performance in the Sprite World environment, with scores better than all baselines for both 5 and 10 objects.\n- Clarity of Exposition: Most sections are clearly written and logically organized; the relationship between conceptual formulation and empirical outcomes is well maintained.\n- Explicit Limitation Statement: Section 6 makes limitations explicit, notably the use of predefined domains, and discusses societal impact.\n\n### Weaknesses\n- Empirical Scope Is Limited: The main experimental validation focuses on synthetic datasets generated via SEMs and a simulated environment (Sprites World), lacking real-world datasets. This limits the demonstrated utility of the approach and may not fully capture practical challenges (Section 5).\n- Assumption of Known Domain Boundaries: The method assumes access to temporally segmented domains (Section 3), but in practice, detecting or segmenting domains (especially under gradual or unknown shifts) is non-trivial. This is acknowledged in Section 6, but the lack of any attempt to address it (even with heuristics) restricts applicability and significance.\n- Comparison Baselines and Fairness: The comparison in Section 5 does not fully leverage methods that are specifically tailored to online or meta-learning in dynamic graphs. Baselines appear limited to those with available code, potentially missing very recent or specialized meta-learning baselines. Furthermore, for online settings, all data up to the current domain is used for baselines, but it is not clear whether this is fully fair for all compared methods.\n- Reproducibility and Implementation Details: The experiments lack sufficient implementation details in the main paper (Section 5). \n- Hyperparameter settings, architectures, loss settings, and optimization strategies (e.g., window size, regularizer values) are only mentioned briefly or deferred to the appendix, affecting reproducibility and clarity.\n- Interpretability of Quantitative Gains: While SHD is a standard metric, it is not clear how sensitive the methodology is to parameter choices. Ablation and sensitivity analyses are reported as being in the appendix, so the main paper does not allow assessment of robustness (Section 5).\n- Computational Complexity Discussion Lacks Details: While complexity is discussed in Section 4, there is no empirical measurement or comparison of runtime/efficiency, especially as the window size or number of domains grows.", "was_revised": false, "final_justification": "", "rebuttal": "**Sample size v.s. performance**: Per request, we conducted following experiment on samples sizes vs SHD. We use variable dimension $d=20$ and $10$ total sequences. When sample is low, all methods perform worse, but our method remain competitive. \n\n| sequence length | 10  | 50 | 100|\n|---|---|----|----|\n| GOLEM  | $115\\pm20$  |  $102\\pm10$     |  $81\\pm 10$   |\n| DYNOTEAR  |  $187\\pm23$    |   $150\\pm20$  | $89\\pm16$   |\n| IDYNO  |  $129\\pm16$   | $97\\pm14$    |   $80\\pm21$   |\n| JPCMCI+ | $132\\pm21$     |   $130\\pm20$  | $119\\pm9$   |\n| NTS-NOTEARS   | $119\\pm15$    |   $95\\pm 5$   | $63\\pm13$   |\n| Meta-D2AG   | $101\\pm 10$   |  $80\\pm 7$    | $46 \\pm 6$   |\n\n\n**Run time**: We include the running time of each method below, for the experiment with random variance (Per Reviewer mYVS's quest).\n\n| d | 5 | 10 | 15 | 20 |\n|---|---|----|----|----|\n| GOLEM  | $139\\pm50$  |  $141\\pm12$  |   $241\\pm11$  |  $420 \\pm 10$  |\n| DYNOTEAR  |  $1.4\\pm0.1$  |  $2.9\\pm2.1$  |  $5.9\\pm3.3$ | $21\\pm5.3$   |\n| NTS-NOTEARS | $247\\pm 66$   | $170\\pm2$   |   $377 \\pm 33$ | $480\\pm29$   |\n| Meta-D2AG  | $232\\pm20$  | $265\\pm 30$   |  $260\\pm 4.6$ | $350 \\pm 8.9$   |\n\n\n**Assumption of Known Domain Boundaries: The method assumes access to temporally segmented domains (Section 3), but in practice, detecting or segmenting domains (especially under gradual or unknown shifts) is non-trivial**: Thank you for the comment. Distribution shift detection is an active area of research, and we can adopt various strategies, such as measuring prediction error, using distribution distance, and/or hypothesis testing to detect shift [1] and form a new domain. We will add such a discussion.\n\n[1] Lu et al, \"Learning under Concept Drift: A Review\", IEEE Transactions on Knowledge and Data Engineering.\n\n\n\n**Other comments**: We thank the reviewer for all the comments. Our reported results uses the general default hyperparameter settings, but during the experimental study, we conducted some hyperparameter search and find the default settings are generally robust and hence we choose to use them. We did not have time to finish linear experiments, but our method focuses on nonlinear settings due to more flexibility and modeling power. Per Suggestion, we also included NTS-NOTEARS results here. We will release codes."}]

Author rebuttal:

Meta-review: This paper introduces a novel framework for causal discovery in dynamic and non-stationary settings, combining methodological innovation with solid theoretical grounding and empirical evidence. The reviewers recognize the importance of the problem, the originality of the approach, and the overall soundness of the work. While some limitations were noted in clarity, scope of evaluation, and assumptions, the authors have provided satisfactory clarifications and additional results during the rebuttal. Overall, the strengths of the paper clearly outweigh the weaknesses, and I recommend acceptance.
Decision: Accept (poster)