Structure Learning from Time-Series Data with Lag-Agnostic Structural Prior
Authors: Taiyu Ban
Changxin Rong
Xiangyu Wang
Lyuzhou Chen
Yanze Gao
Xin Wang
Huanhuan Chen
Venue: ICLR 2026 Poster (2026)
Keywords: Continuous DAG structure learning
dynamic causal discovery
structure learning from time series data
Primary area: causal reasoning
TL;DR: This paper introduces how to use lag-agnostic prior, commonly available knowledge, to guide the discovery of lag-aware causal interactions from time-series data in the continuous optimization framework.

Abstract: Learning instantaneous and time-lagged causal relationships from time-series data is essential for uncovering fine-grained, temporally-aware interactions. Although this problem has been formulated as a continuous optimization task amenable to modern machine learning methods, existing approaches largely neglect the use of coarse-grained, lag-agnostic causal priors, an important form of prior knowledge that is often available in practice. To address this gap, we propose a novel framework for structure learning from time series to integrate lag-agnostic priors, enabling the discovery of lag-specific causal links without requiring precise temporal annotations. We introduce formulations to precisely characterize the lag-agnostic prior, and demonstrate their consequential and process-equivalence to priors, maintaining consistency with the intended semantics of the priors throughout optimization. We further analyze the challenge for optimization due to the increased non-convexity by lag-agnostic prior constraints, and introduce a data-driven initialization to mitigate this issue. Experiments on both synthetic and real-world datasets show that our method effectively incorporates lag-agnostic prior knowledge to enhance the recovery of fine-grained, lag-aware structures.

Published as a conference paper at ICLR 2026
STRUCTURE LEARNING FROM TIME-SERIES DATA WITH LAG-AGNOSTIC STRUCTURAL PRIOR
Taiyu Ban; Changxin Rong. Xiangyu Wangi Lyuzhou Chen
Yanze Gao, Xin Wang, Huanhuan Chen
School of Computer Science and Technology
University of Science and Technology of China, Hefei, China
ABSTRACT
Learning instantaneous and time-lagged causal relationships from time-series data is essential for uncovering fine-grained, temporally-awan interactions. Although this problem has been formulated as a continuous optimization task amenable modern machine learning methods, the integration of coarse- saars--grained lag-agnostic causal priors, an important and commonly available form of prior knowledge, remains largely unaddressed. To address this gap, we propose novel framework for structure learning from time series to integrate lag-agnostic priors, enabling the discovery of lag-spccific causal links without requiring precise information the exact lag of causality. We introduce formulations to precisely characterize the lag-agnostic priors, and demonstrate their consequential and process-equivalence priors, maintaining consistency with the intended semantics of the priors through- out optimization. We further analyze the challenge for optimization due the increased non-convexity by lag-agnostic prior constraints, and introduce a data- driven initialization mitigate this issue. Experiments on both synthetic and real-world datasets show that our method effectively incorporates aggoosisi prior knowledge to enhance the recovery of fine-grained, lag-aware structures.
INTRODUCTION
Time-series data, which captures the dynamic evolution of variables over time, are fundamental to research across both science and artificial intelligence (Sarropoulos et al.) 2022 Kim al. (024), where uncovering the underlying temporal causal mechanisms is a critical task (Li et al. 2023
Rajapakse and Zhou 2007). This temporal causal relationship is characterized by time lag, i.c., the causality between variables manifests after some delay. Such lag- -pe-sfic interactions are known as inter-slice causality, or intra-slice causality when the interaction instant (i.e., at zero lag).
Accordingly, uncovering time-series causal mechanisms be formalized task aimed at recovering the lag-specific structure from temporal data. In this formulation, inter-slice causality is represented by set of structures corresponding to different time lags, while intra-slice causality is captured by a directed acyclic graph (DAG). By expressing the DAG constraint through a smooth characterization (Zheng et al. 2018). the problem becomes a continuous optimization task, making it compatible with modern machine learning techniques (Pamfil et al. 2020).
structure learning
In practice, researchers usually possess prior knowledge about partial causal relationships, which is essential for recovering more interpretable and insightful structures than relying on data alone, and should be actively incorporated into structure learning. To this end, [Sun et al. (2023) proposed integrating causal knowledge with specific time lags into structure learning from time-series data. However, such precise lag-specific information is rarely available real-world settings Instead, prior knowledge is more commonly available in the form of summarized causal relationships, without
Equul contribution.
I 'Corresponding authors:
Xiangyu
Wang (sa31 1..312euscc.ee en) and Huanhuan Chen
(hchen@ustc.. edu..duucm cn).
'Lag- causality refers to causal links with known, specific time lags (including instant causality). This is reflected in the scarcity of publicly available time-series data with annotated lag-specific structures.

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Published as a conference paper at ICLR 2026
specifying exact time lags (Marbach et al.) 2010 Runge et al. 2019a). We refer this more accessible form of knowledge as a aagg-ggoottc prior due to its invisibility to time lags of causality
Incorporating lag-agnostic priors is to specify the presence (or absence) of causal edge without knowledge of its exact time lag. Due to this ambiguity, such constraints cannot be directly imposed on specific structural parameters. This complicates the precise representation of prior knowledge, posing a challenge to preserve the intended information without introducing unintended bias toward specific lags during optimization. Moreover, the ambiguity introduces additional non-convexity" making the optimization landscape more difficult and prone to poor local optima resulting from suboptimal lag selections. In summary, structure learning from time-series data with lag-agnostic priors is significant yet underexplored challenge in the field.
To address this gap, we propose novel continuous optimization framework for incorporating lag- agnostic structural priors into structure learning from time-series data. We first show the issue of
in an intuitive maximum- based characterization of lag-agnostic priors. Despite the consequential equivalence (i.e., it ultimately enforces the same structural conditions as the lag-agnostic priors), the maximum-bee
formulation unexpectedly introduces bias toward specific lags during the optimization process, which is misaligned with the prior that the time lag is unknown. Targeting this issue, we introduce formulations that are both consequentially equivalent and process-qquivalent, which are proven to maintain consistency with the intended semantics of the priors throughout optimization. This ensures that the prior constraints are not only satisfied at convergence but are also fully respected during the entire optimization trajectory.
For the optimization behavior, we show the increased non-convexity caused by incorporating lag- agnostic priors, which leads more suboptimal local optima compared the task without prior. Early optimization dynamics, and in particular, the initialization, can potentially bias which lag- specific edge satisfies the constraint, thus sticking into bad optima. To mitigate this, we introduce a data-driven initialization strategy that leverages unconstrained structure learning to guide early updates and promote convergence to favorable optima.
We conduct extensive experiments on both synthetic and real-world datasets. Results show that our method effectively leverages coarse- cearse-grained, agg-ososttic prior knowledge to significantly improve the recovery of fine-graid fine-grained, awge causal structures. This work provides practical insights and robust strategies for incorporating commonly available causal priors into dynamic causal discovery.
Our main contributions are listed as follows:
1. We introduce continuous optimization framework for time-series structure learning to incorporate lag-agnostic structural priors, enabling the use of widely available coarse coarse-grained knowledge to guide the recovery of yessoved causal mechanisms.
2. We analyze the process-ineqivalens issue in the characterization of lag-agnostic priors, and propose formulations that are both consequentially and process-equivalent to the intended priors throughout optimization.
3. We analyze the increased non-convexity caused by agg-agnottic constraints, and propose data-driven initialization strategy that facilitates the convergence to favorable optima is validated by comprehensive experiments on synthetic and real-world datasets.
Related work, theoretical details, and complete experimental results provided in the Appendix.
PRELIMINARIES
This section introduces the preliminaries of structure learning from series data and describes how to incorporate aag-specfiic edge constraints into the learning process.
Notations For matrix A, we use Asser, #2te2.... to denote slicing along multiple dimensions, where each (sx ex.) specifies an inclusive start and end index for the k-th dimension. If a dimension
See Appendix A.5 for discussion of the sources and reliability of lag-agnostic prior.
*See Example for details on the non-convexity induced by lag-agnostic priors.
Note that our method does not guarantee finding the global optimum.
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is omitted, it indicates selection of all elements along that axis. We write Σ A for the sum of all elements in matrix A, and |A| for the element-wise absolute value of A.
2.1 STRUCTURE LEARNING FROM TIME SERIES
Let X E Rd'xd be multivariate time series, where X; I denotes the observations of d variables at time t. Temporal dependencies are modeled using linear vector autoregressive (VAR) process:
X; Σ W W X Ets
where Wo captures intra-slice (instantaneous) dependencies, W, for capture inter-slice (lagged) dependencies, and E, is zero-centered noise. The maximum lag L defines the longest allowable delay the model. Each W, E Rdddd parameterizes directed edges from X to X,, defining a structure modeling the generation process of the time-series data.
Zheng et al. (2018) introduces a smooth constraint of acyclicity, which used ensure the acyclic intra-slice structure Wo by Pamfil et al. (2020):
h(W)): = Tr (e (w) = 0,
where denotes the Hadamard (element-wise) product.
The full modeling objective can be written as a regression task over time-series observations.
min L(X; (W)())
subject to h hW))(O)) O.
Here, W-( denotes the lag-r structure derived from parameters The model can be either linear, where 0 = [Wr), or nonlinear, where 0 determines the lagged structure (+) via predefined mapping (Sun et al. 2023] Zheng et al., 2020). Hence, constraints on the structure are directly applicable to the nonlinear settings with the defined mapping from 0 to W,.
The augmented Lagrangian approach (Zheng et al. 2018) is used to optimize this problem. After optimization, thresholding step extracts the edge structure:
WF I( 6) W,,
(2)
where 5 is the edge threshold, and I(-) the element wise indicator function, which values the inner condition holds and values otherwise.
2.2 INCORPORATING AAG-SPECIFIC EDGE CONSTRAINTS
To enforce the presence or absence specific lagged edge (Ws) there are two approaches, a hard constraint and soft penalty. The hard way directly constrain the corresponding parameters:
min
L (X; (0)))_)) subject to h(W)) = 0,
(3)
where ()). = (0 (0)) 2 8) enforces the presence of edge (y)is. or O.,(( = 8
(W, (W)())).g 0} enforces its absence. This strict constraint was used in the work by Sun et al. 2023
Alternatvelly, one can add loss term to softly encourage edge presence or absence:
min L <x: (0)+)) (8 ((,, subject to l(W)) = 0,
(4)
where the penalty ReLU(6-I(W,,()) alone can be used to encourage absence. The soft constraint allows balance between prior enforce- ment and data fitting, but does not guarantee strict adherence.
encourages the presence of edge (W.).s. while I(W, (0))
Discussion. While the hard constraint strictly enforces prior knowledge, the soft constraint allows trade-offss with data fit. However, in the case of lag-agnostic priors, the exact lag of a causal edge is unknown, making parameter-leve constraints ill-defined. Therefore, the hard constraint becomes unsuitable in this setting, and soft, flexible formulations are preferred.
Remark 1. For simplicity, we assume a linear setting where (+ = W, in the remainder of the paper. Nonetheless, the proposed methods and analysis also apply nonlinear parameterizations.
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3 TIME-SERIES STRUCTURE LEARNING WITH LAG-AGNOSTIC STRUCTURAL CONSTRAINTS
3.1 PROBLEM FORMULATION
Let X RTxd be multivariate time series, where T is the number of time steps and d the number of variables. Let W, E IRdxd denote the lag-specific structure matrix at lag T E {0, 1,.... L), where each entry (W,)s )ij represents the causal influence from variable time We use WoLr (or simply W) to denote the set of all lagged structure matrices.
variable time t.
Let Ca E {0, 1] ddd be binary masks encoding lag-agnostic structural priors: Cp specifies the presence of edges, and C. specifies their absence. A small threshold 5 >> O is used to determine whether an edge is considered present.
Definition (Lag-Agnostic Edge Presence). For node pair (i,j) with (Cp)s 1, the aag-agnostic presence constraint requires that edge (i.j) at least one lag T E {0,1...... L} exists, 1(W~)yl > 8.
Definition (Lag-Agnostic Edge Absence). For a node pair (i.j) with (Co)sy 1, the sug-agnostic absence constraint requires that (W+); 0 for all E (o......,1 I...
Given these constraints, the structure learning objective is formulated as:
min L(X; Wo.L), subject to h(Wo) = 0, Wo. F Ca, Cp. Wor
(5)
The adherence to lag-agnostic constraints is defined formally as:
WoLL } C. > V(i,j) with (Ca)aj = 1, (W+)ij = 0 for all T,
(6)
WO.L } Cp + V(i,j) with (C,)ss = max (W+)
(7)
The absence of a lag aggostic edge can be directly enforced by zeroing the corresponding entries across all lags, described in Section 2.2 Therefore, the focus of this work lies in how properly formulate and enforce the presence of lag-agnoo lag-agnostic edges during optimization.
3.2 PROCESS-INERS SAICESS-INEQQIVVLLENC IN CHARACTERIZING LAG-AGNOSTIC EDGE PRESENCE
We consider the problem of modeling lag-agnostic edge presence using a smooth, non-negative constraint function. Let (p(W));; 0 represent the presence penalty for edge (i,j), such that (p(W)); 0 indicates satisfaction of the lag-agnostic presence prior. With this setup, we extend the objective in Equation (5) include soft constraint penalties follows:
min L(X; Wo.L) + Xp Σ (Cp p(W)) subject to h(Wo) = 0, Wo.1
Here, the prior constraint is relaxed into the objective via a weighted penalty, allowing for greater stability in the presence of the inherent ambiguity associated with lag-agnostic edge semantics.
Maximum-Basse Based Formulation and Process-Inequivalencee A natural and direct approach for encoding lag-agnostic edge presence is to penalize the maximum magnitude across all lags, in line with the definition in Equation
(pmax (W)) ReLU (6- (6-mas) max (W+)
Although this formulation logically equivalent to the lag-agnostic prior at convergence, is not process-equivalent. That is, it does not preserve the prior intention throughout optimization.
In particular, directly applying a penalty to max, (W,) introduces a bias toward whichever lag happens to dominate early in training. leaving the rest lags unconsidered, illustrated below:
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Proposition 1. Let (i.j) be a lag-agnostic edge specified be present, and assume the data- fitting loss satisfies V(W, Jayl C 2 0 for all T during optimization (push all (W,)s toward zere D. Suppose
WT + To. ò > (WWD)ss > [(W at initialization. Let Way be the solution of the structure
learning problem defined in Equation {8) using the maximum-based prior loss (Equation) 9 Then,
for sufficiently large Ag, the optimum satisfies: (Wat")ss) Diyl 2 5 and VT + To, (W
This scenario illustrates typical trade-off between data fit and adherence to lag-agnostic prior, where the data-fitting loss does not favor including the prior-specified edge (i,j) at any lag. In this case, the maximum-based formulation selects the lag whose edge has the largest initial absolute weight and disregards other lag options. Once the prior constraint is satisfied by this single lag, the optimizer stops exploring alternative lags, thus incorrectly treating the lag agnostic prior as lag-specific. We refer to this behavior as process inequivalence.
Definition (Process Equivalence).
lag-agnostic loss formulation is process process-equivalent if respects the prior's "lag toug-unknown" semantics throughout the entire optimization process. It must not introduce explicit bias toward any specific lag, but must instead evaluate all candidate lags holistically to select the one(s) most compatible with the data.
The early commitment of the maximum-based formulation breaks this core idea of lag-agnostic priors, that no preference exists among candidate lags, and stops the model from correctly identifying the true lag. To address this issue, the process-equivalenn formulation is needed to avoid such bias during optimization.
3.3
FORMULATIONS
The failure of the maximum-based formulation provides an insight that it is necessary to simulta- neously impact on edge (i, at every lag to avoid bias in specific lags, thus ensuring the process- equivalence to the lag-agnostic prior. Following this insight, we introduce masked formula- tion and logic-dual formulation for the presence lag-agnostic edges.
Binary-Masked Formulation A natural idea is to apply uniform constraints all lagged versions of a given edge. This leads to the following loss function for a lag-agnostic edge (i,j):
(phin (W)) maxx (W+) <6)
Σ
ReLU (6 I(W ()),,
(10)
where T(.) is the element-wise indicator function. The binary mask activates the loss only if all corresponding lag-specific edges below the threshold 5. When active, the penalty equally encourages all lag-specific edges to grow.
This design ensures that the constraint behaves in accordance with the intended semanties:
Proposition 2. A lag-agnostic edge (i.j) is present in WoLL if and only if (pbnn (Wo.L))ss = 0. where Pbin()) defined by Equation (10).
More importantly, this formulation also maintains consistency during optimization, as shown below: Proposition 3. Let (i be lag-agnostic edge with known ordering of conffict degrees with respect to the data data-fitting loss:
0 ((W,)) Vi(W.))ef) <....
VIOw
where Tı corresponds to the lag most aligned with data fit. Suppose all (Wr)ij are initialized with do < 8. Then, optimizing Equation with the prior penalty Pbin will result in edge (i. appearing only at lag T1.
This result demonstrates that (-i selects the lag-specific edge most compatible with the data, at least under an ideally consistent ordering of loss gradients and identical initialization. In this sense, the maseed formulation seeks the optimal structure under the lag-agnostic presence constraint, maintaining both consequence and process equivalence with the original prior. This ensures the optimization remains equivalent to the intended causal semantics throughout training.
This assumption ensures analyze the non-trivial scenario where the data alone is insufficient to satisfy the prior constraint, thus forcing the lag-agnostic loss to actively influence the optimization path.
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Logic-Dual Formulation We now introduce second process- eerievleeeuvvaeentt formulation by deriving the edge presence constraint as the logical dual of the edge absence under lag- aiggasoostic priors:
(a = 0, where a(W) = Σ WW1
Here, |Wily 0 serves as the condition for the absence of lag-specific edge, and the summation expresses an "AND" logic: all corresponding lag-specific edges must satisfy this absence condition.
In contrast, the presence of lag-agnostic edge follows "OR" logic: the edge is considered present at least one lag-specific edge is active. We capture this behavior using a product-based formulation:
(Po((W)) II (6-U(6-((W)))
(11)
where R (1y))yl) evaluates to zero if any lag-specific edge exceeds the threshold 8. The product thus encodes a logical "OR": the constraint is satisfied (i.e., the penalty vanishes) when any one of the edges is sufficiently strong.
Proposition 4. lag-agnostic edge (i,j) present Wo.L if and only if (Por(Wo:))); where (-) is defined in Equation 11
This formulation, like the binary-masked version, simultaneously considers all lag-specific edges during optimization and remains process-equivalet to the original prior (cf. Proposition 3
Compared to the binary-masked formulation, the logic-dual formulation is fully continuous, avoiding the discontinuities introduced by the indicator function. This results in smoother optimization dynamics. However, the product-based formulation suffers from scale sensitivity: as the number of candidate lags increases, the magnitude of the loss can decrease rapidly, even when the constraint is violated, leading to weaker optimization signals.
To mitigate this, we normalize the loss magnitude as follows:
(Por (Por(W)))s = II ReLU (6- I(W))))).
(12)
where each ReLU term is scaled by å to standardize its contribution. This normalization ensures that, when (W-)is 0, the loss yields a consistent penalty regardless of the number of lags, preserving balanced optimization across varying temporal horizons.
3.4 DATA-DRIVEN INITIALIZATION FOR STABLE OPTIMIZATION
The lag-agnostic edge presence constraint introduces additional non-convexity into the structure learning objective, as stated in the following result.
Proposition 5. Let minw XXW be convex, linear VAR optimization problem. Then, the augmented problem
min L(W; + C, p(W)
(13)
W
is non-convex for set of aag-agnostic edge-presence constraints Cp E (0, 1]xxd and our proposed prior loss p(W).
This added non-convexity can destabilize optimization and lead to convergence to suboptimal solutions. The following toy example illustrates this effect: Example (Multiple Local Optima under Lag- Agnostic Constraint). Consider time series with two variables, X, and X2, over three time steps and maximum lag L = 2, with:
= 0, 4). X = [4, 2, 1).
Assume lag-agnostic prior indicating that X₂ + Xi. We fit a linear model predicting Xa,1 3,1
Xx1 (Wo),, X 2 + (W1),,1223 (W22,1X1122

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and optimize the following loss with binary-masked lag agnostic constrain
min L(W) + 100 (max |W7|2,1 < W
Σ Rel.U(1 (W+l2.1),
(14)
C(W = (4 (1--(W6)) (Wo),, - 4(W2)2,1) |W]],.
This setup yields at least three local optima:
(i) 2,2 [1, 0, 0.72), (ii) W22,1 = [0, 1, 0.47], (iii W 10, 0, 11,
with corresponding losses 1.73, 1.48, 1.0, respectively, Although all satisfy the lag lag-agnostic constraint, only (iii) is good optima.
This example (see Appendix B for proof, and see Appendix C for more discussions) demonstrates that incorporating lag-agnostic priors transforms the originally convex problem of unconstrained time-series structure learning (without the acyclicity constraint) into non-convex one. As result, different initializations may lead the optimization process to suboptimal solutions such as (i) or (ii), where the lag-agnossin constraint is prematurely satisfied by incorrect lag. This underscores the importance of a principled initialization strategy to promote convergence to favorable optima.
Two-Stage Optimization via using the outcome of the unconstrained data-fitting problem, which guides optimization toward the most data-consistent lag. For Example [ this strategy naturally favors the correct lag, leading the optimization to recover solution (iii) and effectively avoid the suboptimal solutions (i) and (ii). Specifically, we adopt two-stage process:
Initialization A practical solution is to initialize
Stage I: Solve the unconstrained structure learning problem:
Widata- arg min L(X; Wo.L) subject to h(W)) Wot
(15)
Stage 2: Use the result as initialization for the lag-agnostic constrained objective:
min L(X; + Ap Σ (C, o(W)), subject to h(W)) = O, with w(O2 W-data WoL
(16)
This formulation preserves the original task structure, enabling existing optimization and thresholding methods to be applied directly. In our implementation, we use either the binary-masked loss Poin (W) or the logic-dual loss Por (W)) to encode lag-agnostic edge presence constraints.
4 EXPERIMENTS
This section presents the main experimental results and analysis. We begin with experimental setup.
Synthetic Data. We generate synthetic time-series data based on Rrdös-Rényi (ER) random graphs, where each possible edge is included independently with equal probability. To control graph sparsity, we denote settings as ER-k, where is the ratio of the number of edges the number of nodes d. Linear time-series samples are generated using a linear VAR process under two noise settings: Gaussian (Gauss) and Exponential (Exp). The sequence contains T samples (time steps). For incorporating lag- lag-agnostic priors, we randomly select a proportion p% of the true -ag-ognostic edges as prior. The generation nonlinear data is detailed in the specific corresponding experiments.
Real-World Data. We evaluate on the DREAM4 gene regulatory network dataset Maarbach et al. 2009). a standard benchmark for time-series structure learning. It provides gene expression trajectories with known regulatory interactions under various perturbations. As the ground-truth specifies edges but time lags, this dataset well-suited for evaluating models with lag-agnostic structural priors.
This example also applies the logic-dual formulation.

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Metrics. We report structural metrics including Structural Hamming Distance (SHD), True Positive Rate (TPR), False Discovery Rate (FDR), Fl score, and the recovery rate of prior lag-agnostic edges. We also use the area under the ROC curve (AUROC) for summary graph evaluation, and test-set regression loss to assess predictive accuracy when lag-specific ground truth is unavailable.
Setup. Default settings: prior percentage p - 80, loss weight Ap = 0.5, edge threshold 0.1, maximum lag L = 3; others follow the backbone default. Backbones inlcude DYNOTEARS (Pamfil ct al. 2020), LIN (Liu and Kuang 2023), RHINO Gong et al. (2023), and NTS-NOTEARS (Sun et al.
2023). ). Our methods are denoted Backbone& (binary-masker loss) and Backbone* (logic-dual loss). Initialization strategies include "Init 0" (zero), "Init Data" (data-driven), and "Init Random" (random). We also test the maximum-based formulation that is theoretically flawed, denoted as Backbone". Experiments run on an AMD Ryzen 9 7950X (4.5 GHz) CPU and 32 GB RAM.
ER2
ER4
BR2
ER4
ER2
ER4
Score SEnR)
Edge Recovery Rate
Canas
0.
SHD
(Number of Nodes)
(Number Nodes)
(Number Nodos)
Algan
DESOT WAREEAASSROOE UE
DYNDTEARS GORA
Dastaa
DANOTEARS tee Dara)
DENOTEARS Caa Dnia
DOND Qne Deau
Figure 1: Comparison between time-series structure learning with and without lag-agnostic priors.
4.1 EVALUATION OF OVERALL PERFORMANCE
We compare our method, augmented with lag- agnootiti edge presence priors, against the data-only baseline across varying node counts and graph settings. Figure reeports SHD, Fl score, and edge recovery rate on synthetic data with 250 time steps. A more comprehensive evaluation is available in Appendix E We observe that both the logic-dual formulation (DYNOTEARS* with Init Data) and the binary-masked formulation DYNOTEARS& with Init Data) consistently outperform the data-only baseline. Moreover, methods initialized with data-based strategies consistently outperform their pero-initializer counterparts, highlighting the benefit of data-driven initialization in improving optimization stability.
As for the formulation of prior losses, we observe that 1) The flawed maximum-based formulation (DYNOTEARS", Init 0) underperforms the other process-equivalent methods (DYNOTEARS& and DYNOTEARS*, Init 0) in most cases, which is expected due to its inherent process ineeivvllee....ee.ee... 2) DYNOTEARS* consistently outperforms DYNOTEARS and DYNOTEARS& under Init O, suggest- ing that its fully continuous loss formulation leads to more stable convergence without a principled initialization. 3) Even when Init Data slightly masks the flow of the flawed formulation, our process- equivalent methods paired with Init Data still maintain the best performance stably.
4.2 COMPARISON WITH LAG-SPECIFIC PRIOR INTEGRATION
To demonstrate the robustness of our method, we compare it not only against "no prior" baseline but also against a Lag-Specific (I.S) baseline given imperfect information. In this setup, the LS baseline receives correct summary graph priors, but the associated lag is incorrect for 4% of them (denoted LS- presents partial results from experiments using 20-node ER4 graphs, Gaussian noise, and a maximum lag of 5, evaluated across varying percentages of prior knowledge. A more comprehensive evaluation is available in Appendix [
q). Table I
The results in Table I are revealing. While the oracle baseline with perfect lag information (LS-0) achieves the best performance, our agg-sttiitic method performs remarkably close to this upper bound withou requiring any specific lag knowledge. However, LS's performance brittle: With just
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Table Partial comparison results between our method with lag-specific prior integration methods.
5% Priors
10% Priors
30% Priors
50% Priors
Method
SHD1
FIf
SHD1
FI1
SHD
F1
SHD
F11
DYNOTEARS L.S. O, Perfect Lags
13.00:+:3110
0.91
13.00+3.110
0.9100002
13.00=3.110
0.91:00.02
13.00=3.10
0.91+:0.02
10.33.1...3
0.93::0.22
8.00.12.37
0.95:10.02
4.83-1
0.97:10.01
4.83-11.33
0.97
1S-10, 10% Error
11,67433.4
0.92-0002
8.17422.4
0.95-0.02
6.67::2.58
0.96:1:002
6,5042243
0,96++0.02
LS-30, 30% Error
12.50=+3.62
0.92=0.03
9,50+t3.27
0.94=0002
8.83:11335
0.94+00.11
8.33+2.116
0.95+=0.02
1.S-00, 50% Error
12.50:43.62
0.92:10.03
10.67422.73
0.93:00022
11.17:1.47
0.93::0001
11.83=1944
0.92:00011
DYNOTEARS' (Init 0)
12.00.1.2.45
0.92:10.03
10.17::2.51
0.93 0.003
8.50.1..24
0.94.10.02
7.831..63
0.95.10.02
DYNOTEARS' Data)
11.50:::2226
0.92.:0.02
9.33
0.94-+0.02
7.83::2.04
0.94::0002
6.814
0.954:0.11
DYNOTEARS& (Init D)
11.67:1.2.13
0.92100022
9.83113991
0.93
6.67.1..10
0.96.10.02
7.00
0.95100.33
DYNOTEARS& (Jnit Data)
10.83++:.97
0.93=00001
8,50+13.78
0.94
5.83+11977
0.97:10.01
6.17+22.71
0.95
DYNOTEARS* (Init
10.87:12.552
0.93:10.02
8.831:2.55
0.94
6,00:1..8
0.96.100.01
5,93111.22
0.96:10.02
DYNOTEARS" (Init Data)
10,50++2.35
0.93=0002
8,00-:22337
0.95=00.2
5,33+
0.97=00.01
1.50+11776
0.96+00.11
10% incorrect lag information (LSS-10), our lag-agnostic method already exceeds its performance. As the noise in the lag increases (LSS-0, LS-50), the LS method's performance degrades significantly, falling significantly behind our lag-agnostic approach. This highlights the practical value of our method in real-world scenarios where precise temporal information is often noisy or unavailable. Additionally, the pocess-equivalent formulations (& and *) consistently outperform the flawed, process-inequivee formulation (*), confirming that maintaining the intended semantics throughout the optimization process leads to superior and more reliable structural recovery.
4.3
GENERALIZATION TO NON-STATIONARY AND LON-LINEAR BACKBONES
LIN
.10
330
50
RHINO
.5
-10
-30
.50
16
Score
0.70
Scaled SHD
1.0-
Ft Score
0.7
0.65
0.6
0.60
855
0.5
-D
20
20
10
20
5
10
20
Brownian Dim
Brownian Dim
Node Number
Node Number
(a) Results for LIN with various prior amount.
(b) Results for RHINO with various prior amount.
Figure 2: Results of applying our method to LIN and RHINO for non-stationary, nonlinear data.
To demonstrate the versatility of our framework, integrate it with state-of-the-an backbones designed for complex non-stationary and non-linear time-series data: LIN (Liu and Kuang (003) and RHINO (Gong et 2023). We use the logic-dual loss for lag-agnostic priors and the data-driven initialization. Our method with 5% lag-agnostic priors is denoted as LIN-p or RHINO-p here. We adopted experimental setups from their respective papers to ensure a fair comparison: MLsedased nonlinear data; (LIN) 5-node ER graphs (density=0.5). 1 lag, ID and 2D Brownian motion, intervention nodes; (RHINO) (5. 10, -ndee graphs, lags, connection factor=2, spline product noise, intervention history=3. We report results with 5000 samples for LIN and RHINO in Figure 2 with more comprehensive evaluation available in Appendix D The results show that our lag-agnostic prior framework consistently enhances the performance of both backbones, with improvements scaling directly with the amount of available prior knowledge. These findings provide strong evidence that our method serves general and effective plug-in for sophisticated, optimization-bant based discovery methods, confirming its utility for challenging data conditions.
4.4 PERFORMANCE ON REAL- WORLD DATA
We tested our framework on the DREAM4 dataset using various sample sizes (m). We supplied our method and a "random-lag" baseline with 25% and 75% of the true lag-agnostic edges priors. As the lag-specific truth is unknown, we also test static NOTEARS with and without priors. Results of summary graph recovery quality and test-set regression loss are reported in Table

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Table 2: Results of data-only and prior-based methods on DREAM4 dataset.
DREAM4-63
DREAM4-126
DREAM4-189
@ Priors Method
Loss.]
AUROCT
Loss
AUROCT
Loss.
AUROC1
NOTEARS
8.20=0.73
0.54+0.03
8.02=0.92
0,55+0.03
7.89#0.73
0.55+0.03
NOTEARS-Prior
7.57=0.52
0.66#0.02
7.57=0.72
0.67=0.02
7.66+0.61
0.68+0.03
DYNOTEARS
1.81=1.14
0.58=0.03
5.61#1.04
0.62+0.04
4.08+0.60
0.65+0.05
PHOTTEARS-RanomLL...
7.2640.63
0.64+0.03
6.01=0.83
0.68=0.04
4,94+0.61
0.71+0.06
DYNOTEARS' (Init 0)
6.59=0.13
0.70=0.05
5.47=0.12
0.72+0.04
5.42+0.09
0.74+0.05
25%
DYNOTEARS' (Init Data)
7.58=0.11
0.58=0.03
5.43=0.12
0.63+0.03
4.35+0.07
0.66#0.06
DYNOTEARS& (Init 0)
6.6020.57
0.72+0.02
5.79=0.70
0.74=0.03
4,9920.49
0.75+0.05
DYNOTEARS& (Init Data)
7.24+0.91
0.69=0.02
5.1920.88
0.73+0.04
3.80=0.53
0.7420.04
DYNOTEARS* (Init O)
6.6620.54
0.71+0.02
5.82=0.82
0.7420.04
5.0820.49
0.75+0.05
DYNOTEARS* (Init Data)
7.19=1.01
0.69=0.02
5.27=0.92
0.73+0.03
3.85+0.54
0.74+0.05
NOTEARS
8.20=0.73
0.54+0.03
8.02=0.92
0.55+0.03
7.89=0.73
0.55+0.03
NOTEARS+Prior
6.68+0.42
0.89=0.01
6.61#0.53
0.8940.01
6.65#0.42
0.89#0.01
DYNOTEARS
1.
0.58=0.03
5.61=1.04
0.62+0.04
4.08+0.60
0.65+0.05
PHOOTEARS-RandoLL..
7.45+0.98
0.70=0.02
6.20=0.72
0.75+0.03
5,43#0.84
0.7740.05
DYNOTEARS" (Init 0)
7.60=0.1 14
0.89=0.02
5,49=0.14
0.90=0.02
4.41
0.90+0.02
75%
DYNOTEARS" (Init Data)
7.58=0.11 11
0.62=0.04
5.43=0.12
0.67=0.03
4.35+0.06
0.71
DYNOTEARS& (Init 0)
5.24=0.57
0.91=0.01
4.70=0.70
0.92*0.01
4.29:0.64
0.92#0.02
DYNOTEARS& (Init Data)
6.02+0.78
0.90=0.01
4.4620.64
0.91=0.01
3.25+0.33
0.92+0.02
DYNOTEARS* (Init 0)
5.28+0.84
0.90=0.01
4.66=0.74
0.92+0.01
4.27+0.39
0.92+0.02
DYNOTEARS* (Init Data)
6.22+0.72
0.89=0.01
4.8820.85
0.91+0.01
3.40#0.42
0.92+0.02
As shown, our method achieves the best performance across all conditions, outperforming baselines in both predictive accuracy (regression loss) and structure recovery (AUROC). Notably, while priors with imperfect random lags can appear to improve the summary causal graph, they severely degrade the model's s predictive ability (126 and 189 samples). Our lag-agnostic approach, however, consistently improves both, demonstrating its practical value and robustness in the real-world setting. Besides, the process-equivalent formulations (& and #) outperforms the flawed formulation (*) in nearly all cases, which demonstrates the necessity of the correct loss formulation.
4.5 ABLATIONS, SUPPLEMENTARY EVALUATION AND RESULTS
Experiments of comprehensive ablations, evaluation of lag-agnostic edge absence, comprehensive parameter analysis, qualitative results, and supplementary results, are presented in Appendix E
CONCLUSION
This paper introduces novel task: incorporating lag-agnostic structural priors into continuous structure learning from time-series. The goal is to leverage commonly available, coarse-grained prior knowledge to guide the recovery of fine- tie-ggriied, lag-specifie causal structures. We analyze the unique challenges this setting poses, particularly in loss formulation and optimization stability, and propose solutions with both theoretical guarantees and strong empirical performance. Results on synthetic and real-world datasets demonstrate the effectiveness and robustness of our approach. Future work may explore extending lag-agnostic priors beyond individual edges more expressive forms such as causal paths or partial orders, which are widely used in static structure learning.
ACKNOWLEDGEMENTS
This research was supported in part by the National Nature Science Foundation China (No. 62406302, 62137002), in part by the Natural Science Foundation of Anhui Province (No. 2408085QF195), in part by the Fundamental Research Funds for the Central Universities under Grant WK21501 10035, in part by the USTC Research Funds of the Double First-Class Initiative (Grant No. YD91 and in part by the Research Funds of Centre for Leading Medicine and Advanced Technologies of IHM (Grant No. 2025IHM01030).
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STATEMENT ON LLM USAGE
In the preparation of this manuscript. a large language model (LLM), specifically Google's Gemini, was used as a writing assistance tool. The human authors first drafted and completed each section of the paper, establishing all core ideas, analyses, and conclusions. The LLM was then used solely to refine and polish the language of these pre-written drafts to improve clarity, grammar, and readability. The LLM's role was strictly that of an editing assistant; it did not contribute to the research ideation, experimental design, or the generation of any results or substantive content. All authors take full responsibility for the final content of this paper.
A RELATED WORK
A.1 STRUCTURE LEARNING FROM STATIONARY TIME-SERILEE DATA
Structure learning from stationary time-series data, whose fundamental statistical properties don't 't change over time, spans a broad set of methodooogies, which can be mainly grouped into constraint- based, functional causal model (FCM)-based, gradient-bart Granger causality-bar -base,,, and miscella- neous approaches.
Constraint-based Methods. Constraint-hased approaches, such as tsFCI (Entner and Hoyer 2010). PCMCI (Runge et al.) (019b)), and CDANs (Ferdous et al. 2023), rely on conditional independence testing infer temporal dependencies. These methods aim to recover partially directed acyclic graph (PDAG) by exploiting the statistical independencies implied by the causal Markov condition and faithfulness assumptions.
FCM-based Methods. Functional Causal Model (FCM)-based methods, such VarLiNGAM (Hyvärinen et af.] 2010) and TiMINo (Peters et al.) 2014), assume a structural equation model with functional dependencies and noise independence. These methods identifiable under stronger functional assumptions, but may be restrictive when applied to complex nonlinear dynamics.
Gradient-based Methods. Gradient-based approaches reformulate structure learning as differen- tiable optimization problem. DYNOTEARS (Pamfil et al.) (2020) extends the NOTEARS framework (Zheng et al., 20188) to time-series data by jointly learning intra-slice and inter-slice dependencies in linear VAR models. NTS-NOTEARS (Sun al., (2023) further incorporates neural networks and soft supervision for dynamic causal structure discovery.
Granger Causality -based Methods. Granger causality- based methods (Granger 1969 evaluate causality through predictive performance: if the past of variable X, improves prediction X,, then X, is said Granger-cause X, Granger causality assumes strictly inter-slice causality, and typically ignore instantaneous (intra-slice) effects. Modern machine learning methods are actively developing recover granger causality from time-series data. Representative approaches include neural granger causality (Tank et al., (2018), generalized vector autoregression (GVAR) (Marcinkevičs and Vogt 202T), neural additive vector autoregression (NAVAR) (Bussmann et
2021) and amortized causal
discovery (ACD) (Löwe et al. 2022).
Miscellaneous. Other methods, such oCSE, TCDF Nauta et al. 2019). NBCB, and PCTMI, include hybrid or specialized approaches tailored to specific domains. These may leverage mutual information, neural attention, hybrid search strategies and are often evaluated in niche applications.
In summary, constraint- based methods are based on discrete search approaches, while FCM-based methods use analytical approaches to address structure learning from time-series. The gradient-based methods and neural granger causality approaches utilizes continuous optimization methods, which compatible with our introduced task of incorporating agg-anoottc priors. Even method is introduced based on the assumption of both intra-slice and inter-slice causality, it can be directly used for granger causality and benefit the community by leveraging rich lag-agnostic priors.
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A.2 STRUCTURE LEARNING FROM NON-STATIONARY TIME-SERIEE DATA
Learning causal structures from real-world time-series data is often complicated by non-stationarity. where shifts in the underlying dynamics can lead to spurious discoveries if stationary models are used (Song et al.) 2009 Zhang et al.) (017). To address this, significant research effort has focused on methods that can adapt to changing data distributions. One prominent strategy to model the process as sequence of discrete causal regimes. This includes methods that explicitly detect changepoints to segment the data, like CASTOR (Rahmani and Frossard, 2025). as well as those that infer latent interventions to identify different causal models, such as LIN (Liu and Kuang 2023).
An alternative line of work models more complex dynamics or leverages richer data. RHINO (Gong et al., 2023), for instance, captures continuously evolving systems by modeling vistory-dependent noise with deep neural networks. Others, like IDYNO (Gao et al.) 2022), use interventional data from active experiments to robustly identify causal graphs that otherwise unidentifiable from observational data alone.
Investigating the integration of widely available lag-agnostic prior knowledge in such cases is highly valuable but rarely studied. Our work complements these approaches by introducing principled method integrate widely available lag-agnostic priors, and demonstrate its effectiveness these complex backbones.
A.3 USE OF PRIOR KNOWLEDGE IN DIFFERENTIABLE STRUCTURE LEARNING
Prior knowledge has long been recognized as a powerful asset in causal discovery and has been widely integrated into traditional combinatorial structure learning methods (Chen et al.) 2016 Li and Beek 2018; Constantinou
al.] (023). More recently, the differentiable structure learning community (Zheng et al. [018]] Yu et al.) 2019: Zheng et al.) 2020) has begun exploring the incorporation of
various forms of prior knowledge ledge including edge constraints (Chen al.) 2025 Hasan and Ganil
2022), partial orders (Ban et al.] 2024), and ancestral relationships (Wang al. (2024). These
constraint types reflect common structural priors and help connect traditional discrete methods with
modern continuous optimization approaches in machine learning (Zhang et al. 2021).
In dynamic causal discovery, however, the integration of prior knowledge remains far less developed. A few recent efforts, such as NTS-NOTEARS (Sun al. (023), support hard supervision for known lag-specific edges. This study, however, assume that the exact time lag of a causal relationship is known, which is rarely the case in real-world applications. In practice, domain knowledge from fields such as biology, neuroscience, or industrial systems often indicates the presence of a causal relationship between variables, without specifying when the influence occurs (Marbach et al. 2010).
Despite the broad availability of this coarse-grained form of knowledge, no existing methods system- atically support lag-agnostic structural priors, which specify the existence of a causal edge without identifying its exact time lag. This paper addresses that gap by proposing a continuous optimization framework for time-series structure learning that integrates lag-agnostic priors.
A.4 RELATIONSHIP TO STATIC STRUCTURAL PRIORS
Our work focuses on adapting coarse-grained knowledge, commonly available in static (non-temporal) settings, to the dynamic time-series domain. We formalize the relationship between our lag-agnostic edge presence prior and established constraints used in static structure learning (e.g., edge, path, and order constraints) by examining the hierarchy of constraints and the resulting optimization challenges.
Connection:
The Source of Knowledge. The knowledge utilized in our framework-a high-level lag-often
summary graph, such as i + j without a specific lag often originates from the exact same domain experts or L.LM sources used for specifying priors in static causal discovery. Our contribution the first rigorous method to correctly interpret and enforce this common, lag-agnostic knowledge within a temporal, lag-awan lagreware structure learning pipeline.
A Hierarchy of Constraints and Strength. In structural discovery, priors exist in a hierarchy of strength, moving from the most restrictive to the least:
Edge Presence (i + j) Path Existence (i ~m 5) = Partial Order (i x j)
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Our work focuses the strongest constraint, i + (lag-agnostic edge presence), making it the most stringent form of structural prior introduced to the time-series setting to date.
Comparison to Partial Orders < j. A partial order constraint < j) (Ban et al. (0044). is fundamentally forbidden-type constraint: it forbids any path from j to This constraint is gradient-consistenn with the DAG penalty. Both the partial order penalty and the instantaneous DAG constraint enforee acyclicity by pushing structural weights down (toward zero). This alignment results in a relatively stable optimization problem.
Comparison Path/Ancestral Constraints D). Path/ancestral constraints represent an existence-type prior. The fundamental challenge they present is directly analogous to our problem:
Key Similarity (Gradient Confliet): Both path existence priors and our lag-agnostic edge existence prior introduce penalty that must push weights up (away from zero) to satisfy the existence condition. This is in direct conflict with the DAG penalty and regularization, which push weights down. This conflict the primary source of the optimization instability that both our work and recent ancestral constraint methods Ban et al. (025) must solve (cf. Example in Section 3.4).
Key Difference (Nature of Uncertainty): While path constraints involve uncertainty over which set of edges forms the path (e.g.. - k vs. - j), our constraint involves uncertainty over the temporal dimension of a single edge, i.e., determining the correct lag among ((W)) (Wi)y..... (.
In summary, our work introduces the strongest form of existence prior (edge) to the temporal domain. By doing so, we address unique structural uncertainty (over lags) that shares the fundamental optimization difficulty of gradient conflict with modern existence-type static constraints. We posit that future work could build on this foundation by exploring the incorporation of weaker lag-agnostic path or order priors into dynamic structure learning.
A.5 SOURCES AND PRACTICAL AVAILABILITY OF LAG-AGNOSTIC PRIORS
We identify the following sources and factors that contribute to the practical availability and reliability of aggaagnsttic knowledge:
Classical Source: Domain Expertise. The primary and most established source for causal knowl- edge is domain expertise (Amirkhani et al., 2016). This expertise is almost always provided at a high-level, searse-grained, semantic level. For instance, in fields like genetics (relevant to the expert may confirm that "Gene A regulates Gene B" or "Region X influences Region Y," but they rarely specify the exact time lag of this interaction. This pervasive gap between high-level expert knowledge and the low-level temporal detail required by traditional methods (c.g., NTS NOTEARS ) is the central motivation for our lag aggostinstic framework.
DREAM4 experiments ) or neuroscience
Large Language Models. A powerful, emerging source of high-level causal priors is Large Language Models (LLMs). Recent research has focused on developing reliable frameworks to extract summary causal relationships from LLMs. These methods provide inherently lag-annostic and rich priors, bypassing the need for expensive and low-efficiency expert specification. The rapid rise of these data-derived and I.LM-generated priors makes our work, which provides the formal mechanism integrate them into time-series models, both timely and practically significant.
Emerging Source:
The reliability of the LLM--deiveed priors is ensured through both the derivation stage and the integration stage:
Prior Derivation: Research shows that edge priors can be made more reliable by combining their derivation using LLM supervision with data-driven methods.
Our model's 's design ensures robustness even to the inherent imperfections in real-world knowledge. By adopting soft penalty (as discussed in Section 3.2), our framework allows the model to weigh the prior against strong contradictory evidence from the time-series data, making it inherently robust degree of prior error.
Prior Integration:
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B PROOF OF STATEMENTS
Proposition 1. Let (i be a lag-agnostic edge specified to be present, and assume the data-fitting loss satisfies V ((W-)as C 0 for all T during optimization (push all (W;) )ig toward zero )) Suppose WT + To, 5 I( To Digl > ((W=)sgl
initialization. Let Wop be the solution of the structure
learning problem defined in Equation { using the maximum-based prior loss (Equation 9. Then, for sufficiently large Ap. the optimum satisfies: (W7. )sgl< 26 and VT # To. (WT") Kg 8.
Proof. Let Wir (Wr)is and m(W) = max, [orl. The objective is
J(W) L(W) + ( (00))mm(WW))))))))) + pho).
Assume (i) V Ter1 2 0 for every (the data term always contracts each [w,) 1) and (ii) 0 < |wrpl < while |wrol > maxxytro |w:| at initialization. Note that the acyclicity loss consistently push all structural parameters toward Hence, we regard (W)) as part of L here as they both push (Wr)ss toward 0 for the lag-agnostic edge (i.j).
We have m(W) |w!al and, while | < the penalty equals d |wmzol with sub-gradient Dw -A sgn (w.,) and 0w,p= 0 for * To. Because Vw, is bounded the compact set {wo+ S 6J. choose Ap larger than that bound. Then
V (woppl
V Vlrro 1- Xplwral 0,
so a descent step increases |w7,|: for every T + To.
|= Im, 1m,
and descent decreases |wy]. Consequently [wrp grows monotonically and all other magnitudes shrink until the first time t* with [w,, (t')l = 6. At that instant p(W)= 0 and remains zero long as Wryl 2 8; if W.r, ever tries to dip below 8, the same penalty force -Ap sgn(w;.) reactivates and, because Xp still dominates the data term, immediately drives |w,,| back up. Therefore eventually
|w>,| 2 5 > |wr for every T F Tu. so the constraint mW)) 2 5 is satisfied solely through the single coefficient Wro i.e., the edge lag To with the largest initial absolute value.
Proposition 2. A lag-agnostic edge (i.j) is present in Wo.t. if and only if (pbinWao. (Wo.t.) 1).f 0, where (om(() is defined by Equation 10
Proof. Write
m = max OSTSL
(W,))
We first show that presence indicator in (TO) equals 0:
(pbin)s) = 0. Assume the edge is present, i.c. m 2 6. Then the
1(m 8) 0,
hence (Pbin (W));; = 0 (...) 0.
Then we show (Pbin)aj = presence. Suppose (pvin (W)).) 0. By (10) the product of two non-negative factors is zero, so at least one factor is zero.
Case 1: 11(m= 8) 0. Then m 8, hence 7 with (W+) 2 6: the edge is present.
Case 2: Σ ReLU(6 I( )ay 0. Each term of the sum is non-negative; therefore every term
must be 0, which implies (W>); >)yl 2 for some т (otherwise each absolute value would be < 5 and
9This assumption ensures analyze the non-trivial scenario where the data alone is insufficient to satisfy the prior constraint, thus forcing the lag-agnostic loss actively influence the optimization path.
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the indicator in the first factor would be 1, contradicting the product being 0). Hence the edge is again present.
Since presence implies (pbin)is 0 and conversely (pola)y implies presence, have the desired equivalence and complete the proof.
Proposition 3. Let (i be lag-agnostic edge with known ordering of conftict degrees with respect to the data-fitting loss:
0 V IWW, >ylk Vi ( VI...VI VIW--)ol
where TI corresponds the lag most aligned with data fit. Suppose all (Wr)is are initialized with with the prior penalty Phin will result in edge (i, appearing
5n < 8. Then, optimizing Equation only at lag T1.
8)
Proof. We have that the data-fitee tata-fitting loss L(X; o..) satisfies the fixed ordering
0 S In < Drz ... < BrL. = VI(W.)) C.
(17)
Denote Wy = 1(W+)ug (r = O...., . L.) and let
m == max Wy.
Because all w, < 5 at initialisation, I(m 6) 1 and the prior penalty for the pair (i, equals
A Σ (5 wr) (all terms positive).
The negative gradient of the total objective (restricted edge (i.j)) is
C + XpPbin) -r. + Ap (T= 0p......,44
(18)
dw.
Because On < Ir < ... < 9T.. by
17 we have
On + Ap
Dra
Ap >..>> 9YL + Ap-
(19)
Thus Wr, receives the largest positive ascent among all lags, while every w, starts from the same value. Consequently W becomes the unique maximiser m after an arbitrarily small descent step.
As long Inequality (19) remains valid because the ordering (17) is assumed fixed. Hence W,, grows strictly faster than every other w, and reaches 8 first. Let tM be the first iteration where W,, 6. At this moment m = and the indicator in Psin flips to zero, so the penalty vanishes.
8 (so the indicator in Pain stays 1) the update for every lag keeps the form 18
For iterations > t" the objective reduces to alone, whose gradients -O- non-positive (each Ir 2 0 by assumption). Thus v I 0 for every lag. For * T1 we still have (-(t) S w-(() < 5; no mechanism makes them increase. Wry (f) can decrease but remains so the presence condition is fulfilled and the prior penalty never re-activates.
in neighbourhood of t*.
Gradient flow therefore converges a stationary point where
Wr
6,
Wr 0 (7 + 71),
which is exactly the result of proposition We complete the proof.
Propusition 4. A lag agnostic edge (i,j) ) is present Wor. if and only if (por o...))ss 0, where (-) is defined in Equation 17).
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Proof. Define a, (Wy) and write
L II 7=0
ReLU - az).
Por
(+) If Por 0, then the edge is present. Because product of non-negative numbers is zero only when at least one factor is zero, there exists an index I with
ReLU(6 a,,) = 0 - 8 5 - a, 2 6.
Hence ((,,,) )syl 2 the edge is present lag T² (+=) If the edge is present, then Por 0. Assume there exists 7 with art ò. Then
ReLU(6 - a 0,
so the product contains a zero factor and therefore Por = O.
Since each direction holds, the equivalence is proved.
Example (Multiple Local Optima under Lag-A Aggostics Constraint). Consider time series with two variables, X1 and X2, over three time steps and maximum lag L = with:
X = [0, 0, 4), X = [4, 2, 1].
Assume lag-agnostic prior indicating that X2 + X1. We fit a linear model predicting X3,1 as:
Xa1 =Wo),,,), (,
+ (Wi)2,1 (W2),,13 X 1,2,
and optimize the following loss with bisary-masked lag-agnostic constrain 5
min L(W) + 100 maxx IW7 < W
Σ
-1
(14)
C(W) = (4 (4- 2(W1)2,1 4(W2)2,1) +|[
This setup yields at least three local optima:
(ii) W = [0, 1, 0.47], (iil) W 10, O, 1). with corresponding losses L = 1.73, 1.48, 1.0, respectively. Although all satisfy the lag-agnostic constraint, only (ii) is a good optima.
(i) W.221 = [1, 0, 0.72],
Proof. We will show that the three points (with exact values) listed in Example T
23
ii (i (1, 0,
32
(ii) (i)) (0, 15.
(20)
32
(iii) ((i) = (0, 0, 1),
local minimisers of the objective
F(W) (4- (L-())),,111 2(21),,1111666 4(W2)2.1) + |
L(W)
1 sparsity
(21)
+ 100 max W < 1
ReLU(1-|W-le))
T=0
This example also applies the logic-dual formulation.
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Throughout, write w, (W+)2,1 for brevity and consider perturbations w, + Ey with (Eo,E1, E2) arbitrarily small in Euclidean norm.
First note that for every point in 20). max, |,o|| 1, the indicator I(max, |wy| < 1) is zero and the penalty term is inactive.
Let P(e) the perturbed penalty indicator:
P(e) ( max |,, + erl < 1).
For any perturbation sufficiently small, we have two disjoint cases.
Case A: The penalty is triggered (P(e) =1)) Then
1).
F(W +s) 2 100 Σ (1 |,, + +xll)
100,
==0
while F(W()) 1.73, F(W6))) = 1.48, F(W(Hi))) 1. This penalty value (100) is larger than every baseline value,
F(W + - F(((k)) > 0 (k: i, 2i, iii).
Case B: The penalty remains zero (P(x) 0)- This implies max, |w+ + s+l 2 1. Because exactly
one component in each (K)) equals and the others are below 1, we must have e+ 0 for that maximising index f, while |-~ is arbitrarily small for the others.
Consider the case of W = continuous, and only need to prove that for arbitrarily small so and
(1, O, 23
32 we have that &o > 0. this region, F(W) is fully
F(wo + Eo: W1, w2) 2 F(wo, W1, W2):
(22)
F(wo, w1 + E1, w2) 2 F F(wo, W1, W2): F(wo, w1, u2 + E2) 2 F(wo, w1, W2).
(23)
(24)
Let
1
(4 we -2 - 4w2) 8"
We have:
3
F(wo + Eo, W1, w2) F(Wo, W1, W2) (-2t + 1)5o
0,
which proves 22
Besides, we have:
(-41 + 1)81 18 0
F(wo, W1 =1, w2) F(wo, W1, W2)
(-4t 1)E1 -
which proves 23
Finally, we have:
F(Wo, w1, W2 + E2) F(wo, w)) (-8t + 1)ez = 0 0,
which proves
Hence, we have shown that F(W() + +e) 2 F(W(O) case B. Parallel calculations for (()) and (iitt give the same results as well.
Both Case A and Case B show that for sufficiently small non-zero perturbations F(W ((W+))-F(N) F(W) > Thus each reference point W(O, W(o), ((i)) satisfies the definition of local minimiser. n
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C AMBIGUITY-IDUCCH NON-CONVEXITY OF LAG-AGNOSTIC PRIORS
This section discusses why the lag-agnostic priors introduce additional non-convexity to the objective, and illustrates that this is unique challenge for lag-agnostic prior, which is not encountered by lag-specific priors.
In lag-specific structure learning prior singles remains convex (quadratic data-loss é + linear constraint). A lag-agnostic prior, instead, states only that some lagged edge must be present; the optimisation then decides which lag de-activates the penalty. That logical OR introduces pieces that switch on/off at different locations, yielding non-convex surface even in the linear- VAR setting.
weight, e.g. "(W1);y 2 8", and the objective
Now we consider the following example that replace the lag-agnostic constraint in Example lag-specific one.
Consider two variables (X1, X2) observed at three time steps (I. = 2):
X,, = 10,0, X, 24
We learn weights wi, = (,,)2,1 in
X3.1 X3.2 + +w X2,2 + X1.2
using either lag agnostici prior PLA in Example | or lag-specific prior PLS on the presence of edge in lag 1, .e., parameter wn in this formulation. Then consider the corresponding losses:
Fia(w) (4- win 2w1 4wz) + [oo|
+ [wy]
100 100 [m (wr! 1)
Σ
ReLU(L |w+)),
Fis(w)= = (4 wig 2wi 4ww) 2 + |ool |w]] + |w2|| 100 ReLU(1 - -lwil).
Both use edge threshold 5
1.
Next, we consider the convexity the lag-specific objective. Fis, the squared error is a convex quadratic, & is convex, ReLU(1 - (wrl) is convex in wi. Hence Fis is convex in (wo, Wi, W2).
In comparison, the lag-agnostic loss FLA has been proven to be non-convex, which the following three local optima:
= (1,0,0.7)) == (0,0,000 D= (0,0,, 1)..
Because the penalty switches off as soon as any wi reaches 1, each axis direction creates a separate flat basin. Gradient descent starting A, B, or D converges these points, all stationary and satisfying the lag-agnostic prior. Only D minimizes the overall loss, while A, B persist as (suboptimal) local minima, precisely the phenomenon highlighted in Example
In summary, without the acyclicity constraint the lag-specific objective Fis is convex and has unique (lasso) solution. Replacing the single-lag prior with the ambiguous OR-coostraint PLA produces non-conve, surface with multiple disconnected minima. This non-convexity is intrinsic to lag- agnostic priors and absent in earlier lag-specific formulations, motivating careful initialization and the process-equivaless losses proposed in this paper.
D LIMITATIONS AND BROADER IMPACT
Limitations. While our method demonstrates strong performance across synthetic and real-world datasets while remaining identical computational efficiency the used backbone model, several limitations remain. First, the method currently assumes latent confounders or missing variables; its effectiveness under partial observability remains untested. Second, the two-stage optimization (data fit followed by rrioddguiedd refinement) may not always guarantee global optima due residual non-convexity in the loss landscape. Finally, our method assumes the correctness of the provided priors, which may harm the structure learning if with low prior quality.
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Broader Impact. This work contributes improving structure learning in time-series settings where precise temporal annotations are scarce but higher-level causal insights are available, an increasingly common situation in domains such biology, neuroscience, finance, and sensor systems. By enabling the use of coarse coarse-grained prior knowledge, our method supports more practical and accessible causal discovery pipelines. However, with all data-driven causal inference tools, there is a risk of overinterpreting or misapplying recovered structures, especially in high-stakes fields like healthcare policy-making. We emphasize that the results should be validated with domain expertise and, where possible, interventional or experimental studies.
Overall, we hope this work encourages the development of structure learning methods that bridge the gap between real-world prior availability and fine-graines causal inference in dynamic systems.
E COMPLETE EXPERIMENT RESULTS
This section presents a series of supplementary analyses. We begin by evaluating key modules, testing our method on the non-linear NTS NTS-ASSEARS backbone and integration edge absence priors. We then present comprehensive parameter analysis results, followed qualitative results including experiments with an unknown maximum lag, and an analysis varying structural weights. Finally, we provide supplementary results for all experiments.
ER2
BR4
EXI
ER4
ER2
LR2
CHISS
10
10
0.99
3.90
641
64)
0.50
0.35
035
0.56
0.35
0.9
SHD
Fi Score
SHD
Fl Score
Algorithms
Baseline
Algerithms
Bascline
DYNOTEARSA
Uii Random)
DYNOTEARSA
Randon)
Data)
DYNOTEARS*
Random)
(Iii
DYNOTEARS*
(Ink Data)
(a) Results on 250 samples.
(b) Results on 1000 samples.
Figure 3: Comparison between data-based initialization and random initialization.
E ABLATIONS INITIALIZATION STRATEGIES
We assess the effectiveness of data-driven initialization by comparing it with random initialization. Specifically, we generate random Wo.z initializations and evaluate both two-stage optimization using data-driven initialization (Init Data) and direct optimization from these random initialization (Init Random). Results on 30-node graphs with 250 and 1000 samples are shown Figure
For DYNOTEARS&. Init Data consistently generally yields better performance than Init Random, reducing the variability across runs and leading to more stable outcomes. For DYNOTEARS* the difference is less pronounced in the low-sample setting due to its inherently stable, fully continuous loss formulation. However, in the high-sample regime, Init Data clearly outperforms Init Random. These results confirm that data-driven initialization enhances both performance and stability when incorporating lag-agnostic structural priors.
E.2 ABLATIONS ON PROCESS-EQUIVALENS LOss FORMULATION
We evaluate the impact process-equivallnce in loss design by comparing our logic-dual formula- tion (DYNOTEARS") with the maximum-basee formulation, which is denoted as DYNOTEARS" Results are shown in Figure 4 DYNOTEARS* consistently outperforms DYNOTEARS' with the performance gap particularly pronounced under Init 0. This supports analysis that the maximum- based formulation is highly sensitive to initialization, as it tends to commit early to a single lag based on initial parameter values, violating the intent of lag-agnostic priors.
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ER4
ER4
ER4
1.0
25
saning
Ratee
CHISS
sineg
Ealle Recovtty Rale
Fl Score
10
F] Soore
0.4
SID
SHD
30
1.0
1.0
45
20
dxg
In Algoritim
Algorithm
Algorishm
Algurithm
Algorillu
Algoritim
Algorifims
DYNOTEARS
(Ini
DYNOTEARS (Init Den))
Algoriitms
DYNOTEARS (Init
DYNOTEARS (Init Dena)
Daseline
DYNOTEARS'
(lnit
DYNOTEARS (lust Dats)
Baseline
KNS
DYNOTEARS (Init my SNN
DYNOTEARS (Init Dana)
(a) Results on 250 samples.
(b) Results on 1000 samples.
Figure 4: Comparison between process-equivalent and -inequivalent loss formulation.
Table 3: Results on nonlinear data with backbone algorithm NTS-NOTEARS.
Sample Count
200
1000
Model
Method
Fl-scoref
SHD4
Recoveryt
F1-scoref
SHD
Recoveryf
ANM NTSNOTEARS
0,63#0.08
48.00=11.73
0.60#0.15
0.75+0.06
31.3347.76
0.68#0.14
ANM NTS-NNOTERRS*-2
0.65+0.06
47.17=8.45
1.00=0.00
0.76+0.05
31.17+7.03
1.00+0.00
ANM WTS-NOTEARS -40
0.68+0.05
45.17+7.03
1.00=0.00
0.79+0.03
28.33++.93
1.00=0.00
ANM WTSS-NOTEARS--6
0.67:0.07
49.00=10.22
1.00=0.00
0.78+0.06
31.00+8.88
1.00=0.00
ANM WTTS-NOTEARS*...
80
0.70+0.05
46.00=7.c0 64
1.00=0.00
0.81
28.33=+6.83
1.00=0.00
ANM NTS-NOTEARS**110
0.70:0.07
48.17#13.18
1.00=0.00
0.81=0.04
28.67+6.22
1.00=0.00
AIM NTSNOTEARS
0.60#0.04
19.1725.12
0.61#0.14 14
0.85+0.04
20.00+6.29
0.77
AIM WITS-NOTEARS*-22
0,6420.02
47.3324.68
1.00=0.00
0.8720.04
1.00+0.00 17.6745.9
AIM STT-NOTEARS*...
0.6720.02
46.00s6.45
1.00=0.00
0.88+0.03
17.6745.61
1.00+0.00
AIM VYSS-NOTEARR*....
66
0.70=0.05
44.83+10.17
1.00=0.00
0.88+0.05
18.33+8.31
1 1.00=0.00
AIM NTS-NOTEARS*-88
80
0.73+0.04
42.50=8.96 9.
1.00=0.00
0,88+0.04
18. 17:7.25
1. 1.00=0.00
AIM NTS-NOTEARS**
0.75+0.04
39.67=9.24
1.00=0.00
0.89=0.04
16.50=7.29
1.00=0.00
E.3
EVALUATION ON NONLINEAR DATA
This section evaluates our method in nonlinear time-series settings using synthetic data and the NTS-NOTEARS backbone. We first describe the data generation process for two nonlinear structural equation models (SEMs), then present experimental results under varying lag lag-agnostic priors and finally the analysis of the observations.
Synthetic Nonlinear Data We construct two types of nonlinear time-series datasets test the robustness of our approach. In both settings, each variable is generated as a nonlinear function of its lagged parents, plus additive noise. The Additive Noise Model (ANM) generates each sample via
(XW)
where (-)) denotes the element-wins wise sigmoid function and is Gaussian noise. Model (AIM) introduces more complex nonlinearity:
The Additive Index
tanh(XW) + cos(XW) + sin( +
Experimental Setting and Results We apply the logic-dual formulation INTS-NOTEARS' with data-driven initialization (Init Data) under both nonlinear models. For each setting, we vary the percentage of available lag-agnostic edge priors and measure performance using SHD and F1 score on ER4 graph with maximum lags and 20 nodes. The results are reported in Table
Analysis As the proportion of prior edges increases, observe steady improvement in Fl score, indicating more accurate recovery true dependencies. SHD exhibits mild variability across settings but generally remains low, suggesting that performance gains not achieved by overfitting or
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inflating edge counts. These results demonstrate that our method effectively leverages coarse -aaniied structural knowledge to improve structure learning even nonlinear time-series settings, confirming its general upplicability beyond the linear case.
Table 4: Results of DYNOTEARS with absence and presence constraints of lag-agnostic edges.
ER2
250 Samples
1000 Samples
Metric Method
Node=20
Node=30
Node=50
Node=20
Node=30
Node=50
Baseline
12.67=4.03
22.33+2.25
37.67=2.25
10.50=3.02
15.33+2.94
19.67=1.75
DYNOTEARS4Absence
9.83+3.13
16.50=3.08
24.33+2.58
9.83+2.79
14.17=1.83
1..50
DYNOTEARS& (Init 0)
11.17=3.43
18.00=4.20
24.33+6.68
7.17+2.04
6.33+3.27
7.33#1.21
SHD
DYNOTEARS& (Init Data)
9.33+2.42
16.00=3.58
27.33+2.50
4.33+1.03
1.33
5.67+2.42
DYNOTEARS (Init 0)
10.00=3.10
15.00=2.00
24.83#3.54
6,50+2.59
5.33+1.63
7.00=1.10
DYNOTEARS* (Init Data)
9.17=2.14
14.00=3.79
25.50=2.43
5.50+2.07
6.00#1.41
7.33+2.58
Baseline
0.83#0.06
0.80=0.02
0.80#0.01
0.85=0.05
0.85+0.03
0.89=0.01
DYNOTEARS4Ahsence
0.86+0.05
0.85+0.03
0.87=0.02
0.86+0.05
0.87=0.02
0.89#0.01
DYNOTEARS& (Init 0)
0.86#0.05
0.85+0003
0.88+0.03
0.90=0.03
0.94+0.03
0.96+0.01
F
DYNOTEARS& (Init Data)
0.8820.03
0.8720.03
0.8620.01
0.9420.01
0.95+0.02
0.9720.01
DYNOTEARS* (Init 0)
0.87=0,04
0.87+0.02
0.88+0.02
0.91
0.9540.01
0.9640.01
DYNOTEARS* (Init Data)
0.8820.03
0.8820.03
0.87+0.01
0.93+0.03
0.95#0.01
0.9640.01
ER4
250 Samples
1000 Samples
Metric Method
Nodc=20
Node=30
Node=50
Nodc=20
Node=30
Nodc=50
Baseline
17.00=2.68
31.17+2.86
57.0026.90
13.00#3.10
22.6745.39
33.33+4.27
DYNOTEARS+Absence
14.00=3.22
24.00#3.16
40.33+4.68
12.50=3.02
22.33+5.05
32.33+4.63
DYNOTEARS& (Init 0)
17.33=4.76
27.00=6.16
16.83=10.01
6.83+0.98
10.33+2.16
14.83+2.64
SHD
DYNOTEARS& (Init Data)
11.83=3.19
21.83=1.94
40.50#11.66 1.664
6.33+1.86
6.83+2.93
11.50+2.07
DYNOTEARS" (Init 0)
13.50=2.81
22.50:4.09
40.5027.c0 64
6.67+2,16 16
10.17:3.66
14.17=1.33
DYNOTEARS* (Init Data)
11.5024.51
20.00=2.00
38.83=10.78
6.50=1.87
7.33+2.50
13.33+3.44
Baseline
0.88+0.02
0.8620.01
0.8420.02
0.91+0.02
0.9020.03
0.9120.01
DYNOTEARS++bbence
0.91#0.02
0.89=0.02
0.89#0.01
0.91
0.9020.03
0.91
DYNOTEARS& (Init 0)
0.89=0.03
0.8920.03
0.88+0.03
0.9620.01
0.9620.01
0.9620.01
F
DYNOTEARS& (Init Data)
0.92=0.02
0.91=0.01
0.90=0.03
0.9620.01
0.9720.01
0.97+0.01
DYNOTEARS* (Init 0)
0.91=0.02
0.90=0.02
0.90=0.02
0.96+0.01
0.96+0.02
0.96+0.00
DYNOTEARS* (Init Data)
0.93+0.03
0.91#0.01
0.90=0.03
0.96+0.01
0.97=0.01
0.97+0.01
E.4 EVALUATION OF AAG-AGNOTTCC EDGE ABSENCE CONSTRAINTS
This experiment evaluates the impact of incorporating lag-agnostic edge absence constraints structure learning from time-series data. Specifically, we enforce the absence a lag-agnostic edge (i.j) by constraining the corresponding lagged edge weights zero across all lags:
VT E 10. L}, (W+); =
Given binary mask C. E (0, 1]dxd, where (Ca).p denotes the absence of the lag-agnostic edge (1,5), the optimization problem becomes:
min L(X; Wo:L), subject h(W)) 0, - 0. Wo.t
We use DYNOTEARS as the backbone solver and denote the model with lag-agnostic absence constraints DYNOTEARS+Abbenne.
In our setup, we randomly select 80% of absent lag-agnostin edges as priors. We compare three config- urations: (1) DYNOTEARS (data-only baseline), (2) DYNOTEARS+Absenc and (3) DYNOTEARS with 80% lag agnostio edge presence priors. Results under varying node counts, sample sizes, and graph densities (with Gaussian noise) summarized in Table
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Results and Analysis We observe that incorporating lag-agnostic edge absence priors improves structure learning performance over the data-only baseline. However, the degree of improvement is smaller than that achieved by incorporating lag-agnostic edge presence constraints at the same proportion. This difference arises because presence constraints actively guide the model toward recovering missing true edges, enhancing both recall and structural accuracy. In addition, by encour- aging the inclusion of informative edges, presence priors can help remove erroneous edges through the regularization, further improving model performance, In contrast, absence constraints merely restrict spurious connections and do not directly assist in identifying the correct ones. As a result, their influence on structure recovery is more passive and less impactful.
These findings highlight that while absence priors are useful for improving precision and suppressing false positives, presence priors contribute more significantly to overall structure recovery, especially in terms of recall and correct identification of lagged dependencies. This underscores the practical importance of integrating lag-agnostic edge presence information when available.
E.5 EVALUATION UNDER VARYING PRIOR RATES
This section studies how the proportion of available lug-agnostic priors influences the structure learning performance of our method. We systematically vary the prior rate and evaluate its effect on several metrics, including F1 score, Structural Hamming Distance (SHD), recall (TPR), accuracy (I-FDR), and the recovery rate of the prior edges.
ER2
ER4
ER2
ER4
ER2
ER4
0.95
30
SSIRE)
0.90
Edge Recovery Rate
1.0
Gauss
0.85
SHD
10
F1 Score
0.80
0.6
60
1.0
dxp
EXX
0.8
40
30
0.2
0.4
0.6 0.8
1.022
04
0.6 088 1.9
0,5
0.4
0.5
0.8
0.2
0.4
U.S
0. 1.0
0.6
0.4
0.6
0.8
1.0002
0.4
0.5 088 1.0
Prior Rate
Prior Rate
Prior Rate
ER2
ER4
ER2
0.95
0.9
Accuracy GSnee
0.90
Gauss
0.5
Algorithms
Rocall
Baseline
0.50
DYNOTEARS& (Inin 0)
DYNOTEARS& (Init Dana)
DYNOTEARS' (Init 0)
0.8
0.7
Exp
DYNOTEARS* Data)
066
0.7
0.2 0.6
1.002
0.6
1.0
02.
04
0.8
1.0022
0.4 0.6
0.8
Prior Rate
Prior Rate
Figure 5: Results under varying prior rates on 250 samples.
Experimental Setup We use synthetic time-series data generated from ER-2 and ER-4 graphs with 30 nodes, under both Gaussian and Exponential noise settings. For each configuration, we compare DYNOTEARS. DYNOTEARS* (logic-dual), and DYNOTEARS& with two initialization strategies: zero initialization (Init 0) and data-driven initialization (Init Data). The lag-agnostic prior rate varied from low high values, and performance results are summarized in Figure for 250 samples and Figure for 1000 samples.
-makedd, each evaluated
Results and Observations As the prior rate increases, we observe consistent improvements in Fl score, SHD, and recall across all methods and settings. This confirms that incorporating more prior knowledge improves the model's ability to recover the true causal structure. Notably, accuracy (1-FDR) for some Init O methods exhibits slight decline in certain cases, likely due to unstable
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ER2
ER4
ER2
ER4
ER2
ER4
20
SSIRE)
Edge Recovery Rate
Causs
Score
SHD
0.6
1.0
dxq
Exp
0.8
0.8
0.6
0.2 0.4 0.6
0.8
1.00.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
088
1.0.2
0.4
0.6
0.8
1.0
0.4 0.2
0.6
0.8
1.00.2
0.4
0.5 0.5 1.0
Prior Rale
Prior Rate
Prior Rate
ER2
ER4
ER2
ER4
1.0
1,00
0.9
Gauss
0.95
Algoritums
Accuracy
Baseline
Recall
0.7
DYNOTEARS& (Inix O)
1.0
DYNOTHARS& (Init Data)
0.9
DYNOTEARS
0.9
DYNOTEARS (Ini: Data)
0.8
0.7
1.0.22
0.6
1.0
0.2
9. 0.4
Prior Rate
Prior Rate
Figure 6: Results under varying prior rates on 1000 samples.
convergence or early overfitting to incorrect lag selections. In contrast, Init Data methods maintain higher and more stable accuracy, demonstrating the benefit of principled initialization under am- biguous prior constraints. These findings highlight the robustness of our approach and reinforce the importance of both informative priors and strong initialization in achieving reliable structure recovery from time-series data.
E.6 EVALUATION WITH VARYING MAXIMUM LAG NUMBERS
To investigate the influence of the maximum allowable lag our method, we generate synthetic time-series data with true lag-specific structures ranging from L 1 to L 10. We evaluate the methods, DYNOTEARS. DYNOTEARS*, and DYNOTEARS&, with both Init 0 and Init Data strategies, using metries: SHD, Fl score, recall (TPR), accuracy (1-FDR), and edge recovery rate.
Experimental Setup Experiments are conducted on synthetic datasets with 30 nodes and two different sample sizes: 250 and 1000. The results 250 samples are presented in Figure 7 while those on 1000 samples are shown in Figure [8] These experiments assess the robustness of each method under increasing temporal complexity.
Results and Analysis On datasets with 250 samples, performance generally decreases as the maximum lag increases. This expected, as longer lags increase the effective model complexity and introduce more potential for overfitting or misidentification of spurious lagged dependencies, especially under limited data. In contrast, when 1000 samples are available, performance improves as L increases, suggesting that the additional temporal depth can be beneficial when supported by suffi- cient data. Among the methods, DYNOTEARS* (both Init 0 and Init Data) and DYNOTEARS& (Init Data) consistently outperform the data-only baseline across lag values. However, DYNOTEARS& with Init underperforms relative to the baseline when L > We attribute this to the increasing non-convexity of the optimization landscape the number of lag candidates grows, which introduces more local optima. The -on-differentiability of the binary-masker masked loss exacerbates this issue when the model is initialized poorly, leading to unstable optimization. These results underscore the importance of both smooth loss design and principled initialization for scaling lag- agnostic priors to longer temporal horizons.
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ER2
ER4
ER2
ER4
BR2
ER4
Edge Recovery Rate
0.8
0.
Score
SHD
0.7
0.8
dx:
0.8
2345678910
12345 6 910 2 2
123456789100 2177777911222337700
(Maximum Lag)
(Maximum Lag)
Z (Maxinnu Lag)
ER2
ER4
ER2
ER4
GoINS
Algoriihms
Rocall
0.7
Accuracy
0.8
0.7
DYNOTEARSA (Init O)
Baseline
DYNOTEARSK (loi Data)
0.8
DYNOTEARS' (Init Dats)
0.6
06
72345678918 9101 2349678910
2 7191011 9641023346678911
Z (Maximum Lag)
L (Maximum Lag)
Figure 7: Results under varying maximum lags on 250 samples
ER2
ER4
ER2
ER4
ER2
ER4
Edge Recovery Rate
0.9
Gauss
10
0.9
0.8
Score
SHD
dx
0.8
12345 910 2
910
1 6 7 2345 6 8
1.3455 9101 9
(Maximum Lag)
L (Maximum Lag)
L. (Maximum Lag)
ER2
ER4
FR2
ER4
Accuracy SWIRO
Ganss
0.98
Algarishms
Recall
0.5
0.95
DYNOTEARS& (Init
Baseline
0.7
Data)
DYNOTEARS' (Init 0)
6xc
0.8
DYNOTEARS' (Init Data)
0.6
8.6
123456
(Maximum Lag)
(Maximum Lag)
Figure 8: Results under varying maximum lags on 1000 samples.
E.7 EVALUATION WITH VARYING PRIOR Loss WFIGHTS
To analyze the sensitivity of method to the strength of the lag-agnostic prior constraint, conduct experiments by varying the weight A, of the prior loss term from 0.01 to 1.0. We evaluate all five model variants, DYNOTEARS. DYNOTEARS* and DYNOTEARS& with both Init 0 and Init
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Data strategies, using five evaluation metrics: SHD, Fl score, recall (TPR), accuracy (1 (1-FDR), and
edge recovery rate.
ER2
ER4
ER2
ER4
ER2
FR4
0.9
1.0
SSIIE)
SWIE))
Edge Recovery Rate
0.9
Gauss
40
0.8
Score
0.8
SHD
20
0.7
200
1.0
dac
dxg
0.9
Exp
100
50
.01
.1
.5
10
01
10
.01
5 10.01 5
10
10
Prior Loss Weight Ap
Prior Loss Weight Ap
Prior Loss Weight
ER2
ER4
ER2
0.5
0.9
Gauss
0.8
0.8
Algorithms
Accurscy
Baseline
Recall
DYNOTEARS& (Init 0)
0.8
DYNOTEARS& (Init Data)
DYNOTEARS* 0)
0.6
dxg
DYNOTEARS* (Init
0.6
Prior Loss Weight Ap
Prior Loss Weight Ap
Figure 9: Results under varying prior loss weights 250 samples.
ER2
ER4
ER2
ER4
ER2
ER4
1.0
1.0
SSIRC
0.9
Edge Recovery Rate
0.9
Causs
Score
0.8
0.8
SHD
200
1.0
0.8
dxg
Exg
0.4
.01
.1
.5
10
00
10
.01
01
10
.1
10
Prior Loss Weight Ap
Prior Loss Weight Ap
Prior Loss Weight
ER2
ER4
ER2
1.0
0.9
Avcuracy CiEg
0.9
Alguritums
0.8
Baseline
0.8
Rocall
DYNOTEARS& (Init 0)
DYNOTEARS& (Init Data)
DYNOTEARS- (Init 0)
0.8
0.75
DYNOTEARS (nit Data)
0.50
0.6
Prior Weight 2p
Prior Weight Ay.
Figure 10: Results under varying prior loss weights on 1000 samples.
Experimental Setup Experiments conducted synthetic time-series datasets with 30 nodes under two sample size conditions: 250 and 1000 samples. For each value of Ap. we apply the lag-agnostic constraint the structure learning objective and compare performance across methods.
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Results 250 samples reported in Figure 9 and results on 1000 samples reported in Figure 10
Results and Analysis We observe a clear difference in behavior between the two initialization strategies. For methods with Init O, performance tends to improve initially as Xp increases, but then deteriorates when the loss weight becomes too large. This pattern aligns with our theoretical understanding: the influence of the prior grows, the optimization becomes increasingly dominated by the ambiguous lag-agnostic constraint, which can distort the loss surface and lead to premature convergence poor local optima, especially in the absence of guiding initialization.
In contrast, the Init Data variants show a more stable trend. Performance improves with increasing Ap and then plateaus at high level, indicating that data-driven initialization effectively mitigates the non-convexity introduced by the lag-agnostic formulation. These results reinforce the importance of principled initialization, especially when prior constraints strongly enforced.
This experiment confirms that the stability of optimizing the problem with lag- aug-agnostic prior con- straints can be significantly improved with informed initialization, maintaining good performance even with large prior strengths.
E.8 EVALUATION UNDER LARGER MAXIMUM LAGS THAN GROUND TRUTH
This experiment investigates how our method behaves when the model's assumed maximum lag Lmodel is larger than the true maximum lag Limes practice. Specifically, we set Lime = 5 and set the used model with mmadel 10. The underlying graph is an ER-2 network with 30 nodes, and Gaussian noise is used to simulate the time-series data.
may not know the specific maximum lag in
Metrics and Evaluation We report two key evaluation metrics: (1) the data-fitting loss (including regularization) computed using the recovered parameters up to lag & (i.e., lag-0 to lag-k), and (2) the maximum absolute edge weight within each lag-specific matrix WR. which reflects the relative contribution of each lag to the fitted model.
18
Beseline
Beacinn
L> Law
02
L> Lene
18
DYNOTEARS& Gait
DYNOTTARSA (Ini
L>
02
00
F(W,A A)
18
DYNOTEABS&
Ly
max(W,)
02
00
DYNOTEARS Calt
DYNOTFARS'
1>
02
L> Lene
DYNOTEARS' (ini Deal
84
DYNOTEARS' (hat
L> Law
02
L Love
17
L (Maximum Lag)
L (Maximum Lag)
Figure 11: Results under larger maximum lags than ground truth.
Results and Observations Figure I1 presents the results for all five methods. We observe that the data-fitting loss decreases as more lags are included up to the true maximum lag Luve = 5, and then remains relatively stable as ie increases beyond this point. This indicates that the methods correctly prioritize learning causal dependencies within the correct temporal horizon and do not overfit to spurious long-lag dependencies.
In terms of the maximum edge weight per lag, all methods identify significant causal strengths for lags 0 through 5, while edge magnitudes beyond lag 5 drop sharply and remain close to zero. This
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behavior suggests that even when the model is allowed consider more lags than necessary, it can accurately isolate the true temporal dependencies and avoid overestimating long-lag effects.
Among the methods, DYNOTEARS* shows more stable performance than DYNOTEARS&, par- ticularly when > Lime- This is likely due to its fully differentiable loss design, which enables smoother optimization across lagged parameters. Additionally, Init Data consistently outperforms Init for both loss functions, resulting in more stable data and sparser edge recovery in the extra lag dimensions. Specifically, the setter-performing methods exhibit lower fitting loss and maintain near-zero maximum edge weights in higher lag levels, confirming their ability to resist overfitting when the model includes excessive lags.
E.9 VISUALIZATION OF LAG- AAG-SPECIFIC EDGE WEIGHTS
W
W1
W
W
Bascline
0:17
0.73
Prior Mask
Ground Truth
0.43 1..2
40.22
6.0
0.37
0.47
6:50
0.11
6:57
Prior
4.42
0.24
0.0
0.22
X..00 AA11
0.37
0:11
942
0.46
0.25
0.36
6.88
002
DYNOTEARS'
0:97 0.22
0.30
004
Edges recovered
using priors
--05
Missing edges
Figure 12: Visualized results of the lag-specific edge weights.
To qualitatively assess the effectiveness of incorporating aag-aagostic priors, we visualize the recov- ered lag-specific edge weights in representative example. We consider synthetic dataset with nodes and maximum lag of 3. The true lagged structures, the structure learned by the baseline method (DYNOTS, using data only), and the structure learned by lag-agnostic prior-based method (DYNOTEARS*) are visualized and compared. The results are shown in Figure 12
Results and Observations From the visualization, we observe that the model incorporating lag- agnostic priors (DYNOTEARS*) more accurately recovers both the locations and magnitudes of the lag-specific edges compared to the data-only baseline. Many true causal connections that are missed the baseline are correctly recovered under the guidance of lag-agnostic priors. Importantly, the recovered weights not only reflect the correct lags but also closely match the ground-truth edge strengths, demonstrating the precision of the proposed method.
These results visually support our quantitative findings, showing that even coarse gaarse-griiedd prior knowledge can lead more accurate and interpretable recovery of fine-grained, lag-specific temporal structures.
E.10 EVALUATION OF PRIOR ROBUSTNESS
We assess the practical robustness of integrating lag-agnostic priors when the provided knowledge is imperfect or noisy.
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E.10.1 ROBUSTNESS TO IMPERFECT EDGE PRESENCE
We evaluated our method's performance (DYNOTEARS* with Init Data) on a 20-node ER4 graph (Max Lag L = 5, Gaussian noise) while increasing the percentage of incorrect edge presence priors (g). The results demonstrate that our framework is highly robust due to its soft-constraint design.
Table 5: Robustness to Imperfect Lag-Agnostic Edge Presence Priors (20-node ER4, L=5). (V)
denotes outperformance over baseline.
Performance
Correct Priors
Wrong Priors
Method Error Rate (g)
SHD
F1
Num.
Recov. (%)
Num.
Recov. (%)
Baseline
15.0
0.897
0% v
7.2 + 1.6
0.953 + 0.011
57
96.20% + 2.05%
5% /
7.8 1 1.5
0.949
0.010
54
95.99% 1 2.73%
5.56% 1 13.61%
Ours 10%
1 2.6
0.941
0.017
96.41% 1 2.29%
11.11% 8.61%
30% v
12.8 1: 2.2
0.916
0.016
40
97.50% 1 1.58%
17
16.67% 1 4.43%
50%
15.833 = 3.9
0.898 + 0.023
29
98.85% = 1.78%
28
19.05% = 11.66%
The results show our method (v) outperforms the baseline with up to 30 % prior error. This robustness stems from successfully leveraging "good" information (recovering 96-99% correct priors) while actively rejecting "bad" information (c.g., rejecting >> 83% of wrong priors at the 30% error level).
E. 10.2 SENSITIVITY TO INCORRECT EDGE ABSENCE
We also evaluated robustness to imperfect edge absence priors. We found that while correct absence priors are helpful (SHD improves from 15.0 to 13.5), the process is highly sensitive to incorrect (false absence) priors. Practitioners must be highly confident in absence priors, enforcing spurious absence constraint is extremely disruptive and leads to rapid performance degradation (Table 6).
Table 6: Sensitivity to Incorrect Lag-Agnostic Edge Absence Priors (20-node ER4, L=5).
Algorithm
Num. Correct Priors Num. Wrong Priors
SHD |
F1-Score 1
DYNOTEARS (Baseline)
15.0
0.897
DYNoTEARS-Absencee
163
0
13.5 + 0.5
0.908 + 0.004
DYNOTEARS-Absse Abbsenc
155
8
22.2 3.0
0.843 + 0,022
DYNOTEARS. Abbsence
147
16
37.7 17.7:00 6.1
0.729 1 0.042
WYNOTEARS-Absetn
114
49
83.8
0.325 1 0.042
DYNOTEARS-AbEN
81
74
111.7 + 1.9
0.000 + 0.000
E.11 INTEGRATION WITH GROUP SPARSITY BASELINES
The effectiveness of our framework is independent of the base regularization strategy. validate this, we conducted an experiment integrating our lag-agnostic prior into the Group Sparsity (GS) baseline. which is a variant DYNOTEARS that applies sparsity regularization at the group (lag-agnostic) level rather than the individual edge level.
We replaced the standard DYNOTEARS backbone with the GS variant and applied both the 'oracle' lag-specific (LS) priors and our lag-agnostic (DYNOTEARS*) priors top. We tested 20-node ER4 graph (L 5, Gaussian noise).
The results in Table 7 demonstrate the substantial value of integrating our proposed prior.
Observation: Our lag- agnostic prior provides massive structural improvement to the GS baseline, reducing the SHD from 21.19 (data-only) 14.23 (with 50% priors). Furthermore, the performance of our lag agnostic method nearly matches the oracle, lagiitormatiiooiiionn baseline (LS--), demonstrating that method effectively complements existing regularization strategies.
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Table 7: Performance on Group Sparsity (GS) Baseline (20-node ER4, L 5, Gaussian).
5% Priurs
10% Priors
30% Priors
50% Priors
Method
SHD1
Fif
SHD
Fif
SHD4
Fif
SHD1
F11
DYNOTEARS (Group)
21.19 2.55 0,80 + 0.02 21.19 2.55 0,80
0.02
21.19 + 2.55 0.80
= 0.02
21.19
+ 2.55 0,80 + 0.02
IS-0, 0% error (Grup)
19.04 1: 2.71 0,82 1: 0.02 16.98
1: 2,45
0.85
1: 0.01
13.67
1: 1.98 0.88
1: 0.01
13.29
1: 2.15 0.89 1 0.01
LS-10, 10% errer (Group)
20.10 3.110 0.81 +
18.04
+ 2.3
0.44
0.02
16.04
+: 2.22
0.86
0.02
15.33
+ 2.04
0.87
0.01
LS-30, 30% errer (Group)
20,10
0.81
= 0.03
18.56
281 0.83
= 0.02
16.69
+ 2.40
0.85
= 0.02
2.33 0.86
1 0.02
LS-50, 50% emer (Group) 20.10 + 3.05
0.81 + 0.03
18.56
2.76 0.83
0.02
18.48
4: 2.58 0.83
0.02
19.69
+ 2.99 0.82 + 0.03
Ours
19:21 1: 2.66 0,82 1 0.02 17:25
1:
2.18 0.84 1: 0.01 14.29
1 1.95 0.88 0.01 14.23 1 2.05 0.88 1 0.01
Table 8: Comparison of Process- Equivalent Formulations (Softmax vs. Proposed). Data: 20-node
ER4, L = 5, Gaussian Noise.
5% Priors
10% Priors
30% Priors
50% Priors
Method
SHID
F1f
SHD4
F1f
SHD4
F1f
SHDD
DYNOTEARS
13.0041..10
0.91:10022
13.0011.110
0.1110002
13.0013.10
0.91:00022
13.00:1.110
0.91:10.02
LS-0, Perfect Lags
10.33+22.3
0,93+00022
8.00
0.95
4.83
0
4.8
0.97=0.01
LS-10, 10% Error
11.67433.4
0.92:0.02
8.17422.4
0.95:00022
6.67:22.58
0.96::0.02
6,50#22443
0.96:20.02
LS-30, Error
12.50:13..2
0,92:10.03
9,50:1::27
0,44:0002
8.83::133
0.94:10.01
8.33-12.16
0.95:10.02
1.S-50, 50% Firo
12.50:13..2
0,92:10.33
10.67:12.73
0.93-10.02
11.17:1477
0.93-10.01
11.83:1944
0.92:10.01
DYNOTEARS' (Flawed, Init Dara)
11.50 +: 2.26 0.92 4 0.02
2.34
0.94
0.02
7.83 + 2.04
0.94 + 0.02
6.83 + 1.47 0.95 + 0.01
DYNOTEARS- -ofttta (Init Data) 10.83 + 2.88 0.93 + 0.02
4.17
0.95
+ 0.02
6.33 + 1.94 0.96
+ 0.01
5.67
+ 2.16 0.96
+ 0.01
DYNOTEARS& Ours (DYNOTEARS*) (Init Data) 10.50 1. 2.35 0.93 1. 0,02 8.00 1. 2.37
10.83
+:
1.
0.95 0.02
0.95
1. 0.02
5.33
1: 1.86 0.97
1 0.01
5.50
1:
1.76
0.96
1. 0.01
E.12 COMPARISON WITH SOFTMAX- WEIGHTED PRIOR FORMULATION
The continuous optimization approach for structure learning naturally invites comparisons with alternative differentiable surrogates. To fully explore the landscape of process-equivalent formulations, we implemented a new baseline, SONOTEASS-oftmal to use a softmax softmax-weighted sum over lags. This approach can be viewed as weighted variant of our binary- maaked loss (ptin), where the penalty for unsatisfied edges is distributed according to the relative magnitude of each lagged weight.
The penalty function for the softmax-weighted prior is defined as:
E((e I(max [W, < o Σ (softmax, (WW) o ReLU(6- - |Wr1))
Psoftmax =
where the inner softmax, (WW)) is calculated element- across the lag dimension (T = O). for a specific edge (i.j). Like our proposed formulations, this loss is also process-equivalent as addresses all lags simultaneously when the constraint is active.
We compared DYNOTEARS- (Init Data) against our main baselines (Table g appended to the Appendix). As expected, the voftmax-weightee approach performs significantly better than the flawed max-based formulation (DYNOTEARS") achieving results on par with our proposed process-equivalen formulations (DYNOTEARS& and DYNOTEARS*). This further confirms that resolving the issue of process-inequuivalene is the decisive factor for performance gains.
E. EVALUATION OF LAG SCALE SENSITIVITY IN LOGIC- DUAL FORMULATION
The Logic-Dual formulation (pot) fully continuous but it suffers from theoretical scale sensitivity when the maximum lag L is large. Specifically, as the number of factors in the product term increases (i.e., as L grows), the penalty magnitude can decrease, yielding a weaker optimization signal even when the constraint is violated. Our solution to this was the normalized form, Par-
To test the practical impact and effectiveness of this normalization under long temporal horizons, ran experiment
small 10-node, ER-4 graph, varying the maximum lag L from 5 up to 50. We compare the performance of the data-only baseline (DYNOTEARS) against our normalized Logic-Dual method (Pag).
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Table 9: Performance Stability of Normalized Logic-Duse Dual Loss (P(z) under increasing Maximum Lag
L. Data: 10-node ER4.
5
L = 10
L = 20
Method
SHD
FIf
SHD
FIf
SHD4
F11
Baseline
8.00 1.41
0.89
0.02
4.00
0.89
0.95
0.01
6.00 +
0.92 1 0.02
Ours (Por)
2.83 0.75
0.96
0.01
1.00 1 0.63
0.99 1 0.01
1.17 1 0.75
0.99 1 0.01
30
40
L
Method
SHD
FIf
SHD
FIf
SHD
FI1
Baseline
9.00 + 1.64
0.88
0.03
7.00
+ 1.34
0.90
+ 0.02
8.00 + 1.58
0.89 1 0.03
Ours (Por)
4.50
0.96
0.94
0.01
3.50
0.84
0.95 + 0.01
3.00 + 0.63
0.96 + 0.01
The results confirm the stability provided by the normalization. The product-based prior loss continues to provide strong, consistent performance gains over the baseline across the entire range, from L = 5 up to L 50. This indicates that our normalization effectively mitigates the theoretical scale sensitivity issue, ensuring the method's practical utility for analyzing data with large maximum lags.
E.14 SCALABILITY AND COMPUTATIONAL EFFICIENCY
We performed both theoretical and empirical analysis address the scalability and practical runtime of integrating our lag-agnostic prior, especially for large-scale graphs (d 2 100).
Theoretical Complexity. The per-iteration complexity for calculating our proposed prior loss (both Phin and Por) is O (. This cost is highly efficient and negligible compared to the standard VAR-based data-fit loss, which has a complexity of OTLLd) where T is the number of time steps. Therefore, the prior loss itself does not introduce a computational bottleneck to the per-iteration cost.
Empirical Scaling. We tested the overall runtime on large synthetic graphs (ER4, Gaussian noise, L. = 3, 30% priors) with 100 and 200 nodes. The results below compare the runtime of the data-only baseline against our method (DYNOTEARS*).
Table 10: Scalability and Runtime Comparison Large Graphs (d 100). Our method (Ours*) achieves large performance gains despite increased optimization complexity.
Nodes
Algorithm
SHD F1-Score t Runtime (s)
Baseline (DYNOTEARS)
9S
0.815
46.43
100
Ours (DYNOTEARS")
65
0.894
237,89
Baseline (DYNOTEARS) Ours (DYNOTEARS*)
0.778
216.08
200
112
0.863
1364.46
Observation: Our method scales well, achieving significant structural gains (SHD reduced by ze 35%) on large graphs. The observed 5x 6x runtime increase is not from the per-iteration loss calculation, but rather from the optimization process. The combined non-convexity introduced by the DAG constraint and our prior requires the optimizer take significantly more iterations to converge quality solution.
E. ABLATION ON INITIALIZATION PRE-TRAINING
We compared new, simpler data-based initialization strategy in this experiment: Init VAR (New): Pre-training a simple VAR model by optimizing only the data-fit loss L(X; Wo.L), omitting the DAG constraint h(W)).
The experiment was conducted on a 20-node, ER4 graph (L 5, Gaussian noise), testing both our proposed formulations (DYNOTEARS& and DYNOTEARS*)) Results are reported in Table 11
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Table 11: Ablation Study on Initialization Strategies (20-node ER4, 5).
Loss Function
Initialization Strategy
SHD
F1-Score
Accuracy
Recall
Baseline
(No Prior)
20.0
0.859
0.984
0.762
Init O
7.50
0,951
0.985
0.921
DYNOTEARS& (Binary Masked)
Init VAR
5.50
0.964
1.000
0.931
Init Data (Ours)
5.17
0.967
1.000
0.935
Init Init VAR
6.67
0,957
0.987
0,929
DYNOTEARS* (Logic Dual)
6.17
0.960
1.000
0,923
Init Data (Ours)
5.50
0.964
1.000
0.931
The results confirm our design intuition. Init VAR consistently outperforms Init O, demonstrating that starting from a state with good data-fit (even without the DAG constraint) is superior to a cold start. Crucially, the Init Data strategy consistently the best performer across both loss formulations (DYNOTEARS& and DYNOTEARS*). This superiority is attributed to its ability to solve the non-convex DAG constraint first, providing stable, acyelic instantaneous structure (Wo) before introducing the second non-convex prior loss. This effectively avoids the initial gradient conflict and promotes convergence a better local optimum.
E.16 SUPPLEMENTARY RESULTS
This section presents supplementary experimental results that extend the findings reported in the main text. These additional results cover broader range of evaluation metrics, graph configurations, and data sample sizes provide a more comprehensive view of method performance. The specific settings for each experiment are detailed the corresponding figure or table captions.
ER2
ER4
ER2
ER4
ER2
ER4
Edge Recovery Rate
Causs
Score
SHD
Exp
d (Number of Nodes)
(Number of Nodes)
d (Number of Nodos)
ER2
ER4
ER4
0.95
Causs
0.90
Algorithms
Rocall
Accuracy
Bascline
0.7
DYNOTEARS& (Tait 0)
DYNOTEARS& (Imii Data)
DYNOTEARS* (Ini 0)
0.8
Exp
DYNOTEARS* (lait Data)
0.6
50.9
40
d (Number of Nodes)
(Number of Nodes)
Figure 13: Supplementary results the overall comparison on 250 samples.
34

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ER2
ER4
ER2
ER4
ER2
ER4
1.0
SNTRED
Edge Recovery Rate
Gauss
19
FI Score
SHD
1.0
Exp
0.8
0.6
20
10
42
10
20
30
50
10
20
30
40
50
20
70
40
52
10
30
30
40
50.00
20
30
50
d (Number of Nodes)
d (Number of Nodes)
(Number of Nodes)
ER2
ER4
ER2
1.0
3.9
Gauss
Algorichms
Accuracy
Baseline
Recall
0.7
DYNOTEARS& (Inii 0)
DYNOTEARS& (Inii Data)
DYNOTEARS (Ini 0)
0.8
0x:
DYNOTEARS
d (Number of Nodes)
(Number Nodes)
Figure 14: Supplementary results of the overall comparison 1000 samples.
Table 12: Supplementary results of the overall comparison with lag-specific prior integration on 200 samples.
GF
NI Method
SIDD
5% Prios
SHID
10S Prios
SIID
30%
Priors
SID
50% Prooe
DYNOTEARS
4000016666
0.8410006
20001666
0.55:0005
4000056.3
0.30-0006
40.00=0665
0.30:40000
1.5.0.
Lags
37.17-66.05
0.61:0005
ER-2 Exp
LS-10, 10% knvo
39.1756.24
0.040005
57.5945.577
0.6240.05
34.1755911
0.66
33.50-5.011
0.67:20.04
1S-30. 30% Feoor
30.17:634
0602005
38.3324411
06120m
358345499
0.642004
37.00-2888
0.63:20044
LS5500 50%
39.176624
0.0020055
58.3344-119
38.0044.000
41.17
0.60:20002
(Log Agnostic)
3783455.6
0004
33.50=6.255
DYNOTEARS
17.3544.54
0.77
173014454
0.7:1:000
17.334434
0.27 0.07
12.33-4..4
0.77.10.77
LS Perfect Lags
1650.14.77
0.78:1006
14.50-14.44
08110007
10.00:13.16
0,88 0.04
9,17=3443
0.89.1..4
HR-2
LS 30, 30% Erur
17.17:14,26
0.77110066
16.30-14.00
0.9.10007
13.001:3:52
0.05
33.53
0.0:100.4
LS 50, 50% Emur
17.17:14.26
0.77100066
16.3311400
0.791:0007
15.3311499
0.01
15.17-3.119
0.81:10003
Agnostic)
16.9043999
0778--0006
156745334
0.00-0000
12.33-5:200
0:06
0.88:00003
DYNOTEERS
36.0.25.54
0.774:033
36.0045554
0.77+000
36,50
0.77:00.0
LS-O, Perfect Lags
3283-16.21
0.79-10.03
29333--337
0.221-0.44
26.00:16.00
0.84
25
0.84:10.03
BR-4
L.S300
LS-10. 10% Feoor
36.01155544
33.8326.37
0.77-2000
07012004
00000660 60
17.1776997
0.50-:0044
0.110004
2883-6662
0.04
31.1 17:176646
0.0100044
0.79:20.03
L.S 50% Emur
36.00:15.48
0.7710003
32
88
0.801:0004
36.33
0.78
0.76:10.04
Dars (1 Agnocic)
13.7217.19
0.7012000
20 33:5511
0.87:000
26. 66.00666666
0.84
27 4.82
0344:000
DYNOTEARS
2300014.17
0351
231001-141
0351000
23.001-4:17
0.65 0.03
23.00
0.510003
Lags
0.02
ER-
Guuss LS-30, 30% Bror
LS 10, 10% Erur
23.17:4.077
22.67:14.84
0.85
0355+000
19.67-15.13
19.33-5.11 16
0.571:0.00
0.57-000
17.6742.88
1..33
0.89
0.89
0.02
18.33244222
16.67:24.59
0.88-20022
0.90-10.03
LS 50%
2317444.07
0.35--003
21-33-5.22
0.86--0.5
20.67-4:3.20
0
23.53-24.46
0.85-40.22
(Lag Agnosti)
35

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Table 13: Supplementary results of the overall comparison with lag-specific prior integration on 1000 samples.
GT
NT Method
SHD
5%% Prions
F17
SHD
10% Priors
FTT
SHD
30% Priors
FT
SHD L
50% Phices
FT1
DYNOTEARS
14.1745.91
0:50::000
14. 14.17:5:91
0.50::0.09
14.1745.911
14.17=5.91
0.80:20.09
LS O, Perfect Lags
12.17:15.88
0.83110.99
11.83-1.122
0.44:0007
8.171:5.91
0.90:00008
7,67=4.27
0.90:10.05
LS-10, 10% Emur
13.00:55.77
0.0:1:009
12:33455113
0.83::0007
12.0026.26
0.84:0009
11.00:5.066
0.86:00077
ER-2 Exp
LS 30% Erur
13.00:15:87
0.82::0:9
14.1745511
0.81::005
13.17455.64
0:03
12.17:5.00
0.80
50%
13.00:1..77
0.821:009
14.174...19
0.11:0008
0.79:10.10
Ours (Lag Agnosie)
12.39:26.09
083:0000
12:0045144
0.4::0007
5.33
0:00
5.83
087:100066
DYNOTEARS
3..50:23.22
08540005
30.50:2..02
0.540005
3.02
0.85=005
10:50= 1.02
0.5:0005
Perfect Lags
9.00+3.10
0.57-0005
7.33+1197
0
2.83-22.22
0.96 0,03
3.50
0.95:40.22
ER-2
Gauss
1-5-1. 10%
I.5300 30% Eroor
10:0022:45
10.0:22.45
0.640004
0.6-0044
R50
088:0005
0.720000
6.17
4.834:994
0.92
0.44
1.33=1221
5.8
094:20022
0.220.03
LS-50, 50% Boor
10:00:2.45
036-+:004
Kx3+299
0.770005
7.50
0:90
8.17=1994
0.89:20.03
Agnostcc)
9.0012.97
08740005
0.801:004
DYNOTEARS
15.0011..0
050:1001
15001150 50
0.0000000
15:001190
0.90 0:01
15:00 1,90
0.90:10001
LS-0, Perfect Lags
1183::264
0.92+-0.22
2.67
0.94
6.50
0
7.00-2.45
0.95-1-.02
ER-
LS 10% Ervor
14.33:12.0
13.50:42.43
0.01
0.90100.22
13.9
31.50:1:.94
0.1100002
0.921-0.0
11.57::6.09
0.04
13.00:22.68
0.92-40.02
LS 50% tnvr
14.35#2880
0.90+-0002
14.33+22999
0.00-0002
17.17+3322
0:02
23753=4476
0.8-2-0003
Ouss (Lag Agnosic)
13.17:42.56
0.00
11:00::3:29
0.93
4..50
0
2.79
0.9:
DYNOTEARS
1300::3.10
0.000002
15.0:3.100
09
0.91200.22
1.500 Perfect Lags
10.33
0.93110.22
2.00
0.5::0022
1.34
0.7
4.3
0.97:1.011
ER-4
Guuss
IS-10, 10% Froor
LS 13-30,300 30% Emur
12.50:13..2
11.6723.44
0.9210003
0.92:0022
6.172264
0.11133277
0.410002
0.550.0
1.63
6.5722.55
0.94:0001
0.96
2.33=2.16
6.50=2.41
0.95:0002
0.96:20022
15-50, 50%
12:50:13.22
0.920003
1067:2277
0
11.174147
0.9210.01
(Lag Agnosti))
10.50:22.35
0.9200022
800
0.5500022
5.3541366
097
2.672.116
0.960000
ER2
ER4
ER2
ER4
ER2
ER4
Gauss
Grauss
20
1.00
0,70
0.85
SHD
Edge Recovery Rate
ER2
ER4
ER2
ER4
Gauss
Gunsss
Algorithms
DYNOTEARSA Random)
0.30
0.85
0.90
0.88
0.90
0,95
0.90
DYNOTEARS& (Ini: Dasa)
DYNOTLARS (nnit Random)
DYNOTEARS (nii Dats)
0.75
0,80
0.85
0.80
0.85
0.90
0.60
0.70
0.30
0.50
Recall
Accuracy
Figure 15: Supplementary results various initialization strategies on 250 samples.
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BR2
ER4
BR2
Ciass
Guuss
Gauss
1.00
0.90
0.95
EXD
0.98
0.80
0.90
1.00
0.50
0.95
SHD
F1 Score
Edge Recovery Rate
BR2
ER4
ER2
ER4
Grnuss
GaNsS
Algorifums
Baseine
DYNOTEARSA Ramke))
0.85
0.90
0.95
0.90
0.95
0.96
0.98
1.00
0.98
0.99
1,00
DYNOTFARSA
DYNOTEARS' (lui Rundom)
DYNOTEARS' (lnh Data)
0.50
0.90
0.90
0.95
0.88
0.90
0.93
0.95
1,00
Recall
Accuracy
Figure 16: Supplementary results of various initialization strategies on 1000 samples.
ER2
ER4
ER2
ER4
ER2
BR4
30
1.0
25
SSHIC
0,9
Edge Regovery Rate
Crauss
20
0.38
SHD
1.5
1.0
FI
.50
0.8
0.9
0.7
0.7
0
Init
Intt
Dota
Int
Dota
Algorithm
Algorithm
Algorithm
ER2
ER4
ER2
ER4
1.00
0.9
0.95
Glnsss
Algorihms
0.90
Baseline
Acouracy
DYNOTEARS"
Recall
0.7
DYNOTEARS& (Init 0)
3.8
DYNOTEARS- (Ini Dut))
DYNOTEARS& (Ini: Data)
0.7
Inat
Init Unita
Enit
Init Data
Init
Lnit De5t
lnst 0
Init Dota
Algorithm
Algorithm
Figure 17: Supplementary results of comparison to maximum-based formulation on 250 samples.
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BR4
ER2
ER4
1.0
Edge Recovery Rute
Gauss
Score
0.8
SHD
30
1.0
25
dxg
0.9
5x9
20
Algorithm
Agorithm
Algorithm
ER2
ER4
ER2
ER4
1.0
1.05
0.9
Accureey SSnee
Ganss
Algorilima
Baseline
DYNOTEARS' (luit D)
Recall
09.95
DYNOTEARS' (Bzit U)
DYNOTEARS
DYNOTEARS& (Init Data)
0.9
DYNOTEARS* (Beit Dana)
0.7
Isk 0
Tnit
Data
Init Date
Inst
Init Tante
Emit
Data
Algorithm
Algorithm
Figure 18: Supplementary results of comparison maximum based formulation on 1000 samples.
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LIN
LIN-0.05
LIN-0.1
LIN-0.3
LIN-0.5
BA, 1000
BA, 5000
CR. 1000
ER, 5000
15.0
16
SHD
12.5
10.0
1.0
0.70
0.95
Fl Score
0.30
0.9
0.7
0.75
0.65
0.80
1.30
Accuracy
1.0
0.75
0.85
0.8
0.9
0.70
0.30
0.8
1.08
1.30
Recall
1.00
0.96
0.95
0.95
0.90
0.90
0.90
2D
10
20
10
20 Brownian Dim
Brownian Dim
Brownian Dim
Brownian
Figure 19: Results on non-stationary and non-linear time-series data with the backbone algorithm LIN. This figure illustrates the performance of the LIN algorithm under various prior rates, evaluated on simulated data with sample sizes of 1000 and 5000. The data was generated using ID and 2D Brownian motion Barabási-Albert (BA) and Erdös- Erdös-Rényi Rényi (ER) graph structures with 5 nodes. The evaluation metrics include Structural Hamming Distance (SHD), F1 Score, Accuracy, and Recall
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Rhino
Rhino-5
Rhino-10
Rhino-30
Rhino-50
ER
Inst.
Lag
Temporal
0.8
PI Score
0.6
0.7
0.7
0.6
0.4
0.6
0.5
0.5
10
20
10
20
5
10
20
Node Number Inst.
Node Number Lag
Node Number Temporal
SF
0.9
0.8
F1 Score
0.6
0.8
0.7
0.7
0.4-
0.6
0.2
0.5
10
20
10
20
5
10
20
Node Number Inst.
Node Number Lag
Node Number Temporal
ER
0.8
0.30
Scaled SHD
1.0
0.25
0.6
0.8
0.4
0.6
0.15
0.2
10
20
5
10
20
10 Node Number Temporal
20
Node Number Inst.
Node Number Lag
0.8
Scaled SHD
1.0
0.25
0.8
0.20
0.4
0.6
0.15
5
20
Node Number
Node Number
Node Number
Figure 20: Results on non non-stationary and non- non-linear time-serics data with the backbone algorithm RHINO. This figure illustrates the causal discovery capabilities of the RHINO algorithm under various prior rates on simulated data generated from Erdös-Rényi (ER) and Scale-Free (SF) networks with 5, 10, and 20 nodes. The simulation involved sample size of 5000, a lag order of 2, noise level of and utilized spline functions to model historical noise. The evaluation specifically assesses the algorithm's 's accuracy in identifying three distinct graph structures: the Instantaneous graph (Inst), which represents causal relationships among variables within the current time point (t); the Lagged graph (Lag), which details causal influences from past time points (t-k) on the current time point (t); and the comprehensive Temporal graph, which combines both instantaneous and lagged relationships. All results are measured using the Scaled Structural Hamming Distance (Scaled SHD) and the F1 Score.
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Table 14: Supplementary results on the DERAM4 dataset.
Sample size
DREAM4-63
DREAM44126
DREAM4-189
Prior Rate
Method
Loss
AUROC
Loss
AUROC
Loss
AUROC
NOTEARS
8.20++.73
0.5420.03
8.02=0.92
0.55#0.03
7.89+0.73
0.55+0.03
NOTEARS+Prior
7.57=0.52
0.66#0.02
7.57=0.72
0.67=0.02
7.66#0.61
0.68+0.03
DYNOTEARS
7.81#1.14
0.58#0.03
5.61+1.04
0.62#0.04
4.08+0.60
0.65#0.05
25%
SANOTEARS-Ranonne
7.26=40.63
0.64=0.03
6.01
0.68#0.04
4.94
0.71
DYNOTEARS& (Init 0)
6.6020.57
0.7220.02
5.7920.70
0.7420.03
4.9920.49
0.75+0.05
DYNOTEARS& (Init Data)
7.24
0.69=0.02
5.1920.88
0.7320.04
3.8020.53
0.7420.04
DYNOTEARS* (Init
6.66+0.54
0.71=0.02
5.82+0.82
0.74=0.04
5.08+0.49
0.75+0.05
DYNOTEARS* (Init Data)
7.19#1.01
0.69=0.02
5.27=0.92
0.73+0.03
3.85+0.54
0.74+0.05
NOTEARS
8,2020.73
0.54+0.03
8.0220.92
0.55+0.03
7.8920.73
0.55+0.03
NOTEARS+Prior
7.13#0.49
0.77=0.01
7.16#0.62
0.78*0.01
7.20+0.50
0.78+0.02
DYNOTEARS
1.81
0.58=0.03
5.61=1.04
0.62+0.04
4.08+0.60
0.65+0.05
50%
SOOOTEARS-Randoll
7.25#0.76
0.66=0.03
6.1699.93
0.71
5.1240.66
0.72+0.06
DYNOTEARS& (Init 0)
5.75+0053
0.81=0.01
5.09=0.59
0.82#0.02
4.56#0.47
0.83+0.02
DYNOTEARS& (Init Data)
6.67+0.82
0.79=0.01
4.9020.58
0.81#0.02
3.5240.33
0.83+0.02
DYNOTEARS* (Init 0)
6.01+0.48
0.80=0.01
5.37=0.61
0.82+0.01
4.54+0.41
0.83+0.03
DYNOTEARS* (Init Data)
6.9640.90
0.79=0.01
5.08+0.83
0.80=0.02
3.65+0.45
0.82+0.02
NOTEARS
8.2020.73
0.5420.03
8.02+0.92
0.55#0.03
7.8910.73
0.55+0.03
NOTEARS+Prior
6.68+0.42
0.89=0.01
6.6120.53
0.89a0.01
6.65+0.42
0.8940.01
DYNOTEARS
7.81+1.14
0.58=0.03
5.61=1.04
0.62#0.04
4.08#0.60
0.65+0.05
75%
SANOTEARSS-andomlaaa
7.45#0.98
0.70=0.02
6.20=0.72
0.75+0.03
5.43+0.84
0.77+0.05
DYNOTEARS& (Init 0)
5.24=0.57
0.91=0.01
4,70=0.70
0.92=0.01
4.29+0.64
0.92+0.02
DYNOTEARS& (Init Data)
6.02+0.78
0.9020.01
4.4620.64
0.91
3.2520.33
0.9240.02
DYNOTEARS* (Init
5.28+0.84
0.9020.01
4.6620.74
0.9240.01
4.2740.39
0,92+0.02
DYNOTEARS* (Init Data)
6.2240.72
0.8920.01
4.8820.85
0,90#0.01
3.40+0.42
0.9240.02
NOTEARS
8.20+0.73
0.54=0.03
8.02+0.92
0.55#0.03
7.89+0.73
0.55+0.03
NOTEARS+Prior
6.27+0.39
0.9920.00
6.32+0.50
1.00=0.00
6.30+0.31
1.00
DYNOTEARS
7.81+1.14
0.58=0.03
5.61+1.04
0.62+0.04
4.08+0.60
0.65+0.05
100%
DYNOTEARSS
7.19=0.79
0.75=0.03
6.1320.92
0.77
5.7340.76
0.79+0.03
DYNOTEARS& (Init 0)
4.91=0.57
1.00=0.00
4.42+0.63
1.00=0.00
3.9740.39
1.00:0.00
DYNOTEARS& (Init Data)
5.6720.58
0.9920.00
4.29+0.53
0,9920.00
3.1620.36
0.9940.00
DYNOTEARS* (Init 0)
5,0720.55
1.00=0.00
4.4220.75
0.9920.00
4.1840.36
0.9940.00
DYNOTEARS* (Init Data)
6.3020.80
0.98=0.00
4.6320.68
0,9920.00
3.3120.38
0.9940.00
41


Reviews:
[{"review_id": "fPUkC2Ga6N", "reviewer": "Reviewer_tkmz", "summary": "This paper proposes a lag-agnostic prior constraint for incorporating prior knowledge into causal discovery algorithms for time series data. The authors highlight the drawback of outcome-equivalent constraints like maximum-based prior and theoretically justify the effectiveness of their process-equivalent prior. Empirically, the paper shows the effectiveness of their method in incorporating the provided prior information.", "questions": "1. Do the authors have intuition for why data driven initialization works?\n2. As noted in the limitations section, the model can go wrong due to incorrect priors. How sensitive is the model to incorrect priors?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is very well-written and easy to read.\n2. The theoretical results are well-motivated and clearly stated. The examples are well-constructed to illustrate the point that the authors are trying to convey.\n3. The empirical results are extensive, and include several ablation studies.\n4. The authors tackle an important problem that is often overlooked. Priors coming from domain-experts are almost always lag-agnostic so it's nice to see a paper that tackles this challenge.", "weaknesses": "1. Although Proposition 1 is illustrative, it is proven under a very strong assumption, i.e. $\\nabla{|(W_\\tau)_{ij}|} \\mathcal{L} \\geq 0$ for all $\\tau$. This is unrealistic, since many edges in real-world applications are, in fact, not forced to 0 by the data. It is unclear whether the principle being illustrated would still hold in such cases.\n\n2. The main experiments in Section 4 do not include the outcome equivalent baseline \"maximum-based formulation\". Although the authors report some experiments in Appendix E.2, the difference between the two methods seems quite small (especially for the init data setting). This is an important baseline that the authors should consider including in Table 1, Figure 1 and Table 2.\n\n3. A minor weakness is that practical sources and reliability of the availability of such lag-agnostic priors could be discussed more concretely, especially in the motivation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4fAmpZIINB", "reviewer": "Reviewer_rVC5", "summary": "The paper addresses structure learning for multivariate time series when practitioners have coarse, lag-agnostic causal priors (edge exists but unknown lag). Authors show a straightforward \"max over lags\" penalty is consequentially equivalent to the desired prior but process inequivalent (induces early bias to a single lag). They propose two process-equivalent penalties that act across all lags: 1) Binary-masked loss $p_\\text{bin}$: activates only when all lag-specific edges are below threshold, then pushes them jointly; 2) Logic-dual/product loss $p_\\text{or}$: product of ReLU terms (OR semantics), with a normalization to reduce scale sensitivity. They also argue lag-agnostic constraints increase non-convexity and propose a two-stage, data-driven initialization.\nExperiments on synthetic VAR graphs (ER-k, Gaussian/Exponential noise), nonlinear/non-stationary backbones (LIN, RHINO), and DREAM4 show consistent gains in SHD/F1/AUROC and lower regression loss vs data-only baselines and vs lag-specific priors with wrong lags.", "questions": "-  Any empirical comparison to LSE or Gumbel-softmax over lags?\n- What happens when $C_p$ contains 20-40% spurious pairs or $C_a$ wrongly forbids true edges?\n- For $p_\\text{or}$, how often do you observe near-zero gradients early? Does the normalization fully fix it as L grows?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Lag-agnostic prior knowledge is common; formalizing it is useful.\n- Simple, plug-in losses applicable to multiple backbones (DYNOTEARS, LIN, RHINO).\n- Initialization story is well-motivated.\n- Broad evaluation", "weaknesses": "- Propositions establish equivalence of penalties but there's no optimization-theoretic guarantee (e.g., convergence to a correct lag under identifiability conditions).\n- The non-convexity example is illustrative but small; more formal landscape analysis would strengthen claims.\n- The product loss can suffer vanishing gradients when many lags are near but below $\\delta$; the normalization helps but may not fully address scale with larger L.\n- How robust are results to noisy/incorrect presence and incorrect absence priors (false positives/negatives in $C_p$, $C_a$)?\n- Baselines focus on NOTEARS-family and a “random-lag” variant. Important time-series causal methods like DYNOTEARS with group-sparsity across lags isn't compared.\n- No comparison to softmax/log-sum-exp surrogates for max (temperature-controlled) which are a natural alternative to address process-inequivalence.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lem6gWzAho", "reviewer": "Reviewer_2t4w", "summary": "This paper addresses the problem of causal structure learning from time-series data when only lag-agnostic prior knowledge is available. The authors propose a continuous optimization framework that integrates such priors into time-series structure learning. The paper first identifies the process-inequivalence issue in naive maximum-based formulations for lag-agnostic priors, which causes bias toward specific lags during optimization (Section 3.2, Proposition 1). To address this, it introduces two process-equivalent formulations, a binary-masked formulation (Eq. 10) and a logic-dual formulation (Eq. 11), that preserve the semantics of lag-agnostic priors throughout optimization (Section 3.3). The authors further analyze the non-convexity induced by lag-agnostic constraints and propose a data-driven initialization strategy (Section 3.4) to mitigate convergence to poor local optima. Finally, through comprehensive experimental validation on synthetic data, non-linear and non-stationary datasets (using LIN and RHINO backbones), and real-world DREAM4 gene regulatory networks (Sections 4.1–4.4), the paper demonstrates that the proposed framework improves causal recovery and stability compared to both data-only and lag-specific prior methods, particularly when temporal information is noisy or incomplete.", "questions": "Complexity and scalability: How does the proposed penalty (especially the product-based logic-dual term) scale with increasing lag length and variable count?\n\nInitialization sensitivity: How sensitive is the method to the choice of unconstrained pre-training in Eq. (14)? Does using different unconstrained learners (e.g., VAR vs. neural backbones) affect performance?\n\nInterpretability of lag selection: Can the authors quantify how often the method identifies the correct lag (when available) rather than merely satisfying the lag-agnostic constraint?\n\nPractical deployment: Are there specific guidelines for tuning λₚ when validation ground truth is unavailable?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Originality: The notion of lag-agnostic structural priors is novel and fills a clear gap between lag-specific causal discovery (e.g., Sun et al., 2023) and coarse-grained prior-based static structure learning (e.g., Zheng et al., 2018). The formal distinction between consequence equivalence and process equivalence is particularly insightful and provides new conceptual clarity (Proposition 1–4, Section 3).\n\nTechnical quality: The theoretical analysis is rigorous. The proofs of process-equivalence (Appendix B) and the illustration of increased non-convexity via Example 1 are convincing. The data-driven initialization strategy is well-motivated and empirically validated.\n\nClarity: The paper is well-structured, with clear notation and well-separated sections. Figures 1–2 and Tables 1–2 (pages 8–9) are clear and they effectively support the claims.\n\nSignificance: The work provides a general, modular mechanism that can be integrated into various differentiable structure learning frameworks (DYNOTEARS, LIN, RHINO). The experiments demonstrate consistent improvement across backbones, suggesting wide applicability.", "weaknesses": "Scalability considerations: The computational complexity of the binary-masked and logic-dual penalties is not analyzed. As both require operations across all lags and node pairs, their efficiency and scaling to large graphs (e.g., d > 100) remain unclear.\n\nInterpretability of lag selection: Although the process-equivalent formulations prevent early bias, the final lag assignments are driven primarily by data fitting. The paper could elaborate more on how reliably the method identifies the true lag rather than simply satisfying priors (discussion in Section 3.3).\n\nAblation clarity: While Appendix E reportedly contains ablations, the main text provides limited discussion of which component (formulation type vs. initialization) contributes most to performance gains.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jxX98nGvD5", "reviewer": "Reviewer_54Cj", "summary": "This paper considers  the integration of coarse-grained lag-agnostic causal priors. \nThe main claim is that lag-agnostic priors can enable the discovery of lag-specific causal links.\nThe main effort of the paper is put on how to integrate such prior during the practical optimization procedure, and it provides both theoretical and empirical analysis.", "questions": "- What is the connection and differences between the Lag-Agnostic Structural Prior and those in the related work? Especially the \"order\" of causal variables, like Partial Orders [1] and Causal Orders [2]. I suggest an additional discussion.\n- Please consider improving the notation about $\\Theta\\_{ij,s}\\:=\\\\{\\\\theta\\\\mid \\|(W\\_s(\\\\theta))\\_{ij}\\| \\\\geq \\\\delta \\\\}$. Readers may expect $\\\\Theta\\_{ij,s}(0)\\:=\\\\{\\\\theta\\\\mid\\|(W\\_s(\\\\theta))\\_{ij}|\\\\geq 0\\\\}$, which does not match the actual definition of $\\Theta_{ij,s}(0)$.\n\n[1] Differentiable Structure Learning with Partial Orders\n\n[2] Causal Order: the Key to Leverage Imperfect Experts in Causal Inference", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- lag-agnostic structural priors is an interesting formulation, and it can be useful in practice\n- The theoretical analysis is clear and rigorous. It reveals the challenges during the optimization and why the proposed  logic-dual Formulation is essential.", "weaknesses": "- In Section 4.2 and Figure 1, it would be more comprehensive if a baseline using equation (9) can be added. Such empirical results would further justify the discussion about process-equivalent approach.\n- The concrete research problem is not sufficiently introduced. For example, the concept of \"process equivalence\" lacks a more formal definition. I suggest a minor adjustment to emphasize this part.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)