Influence without Confounding: Causal Discovery from Temporal Data with Long-term Carry-over Effects
Authors: Fan Li
Zixuan Liu
Yi Zhao
Qi Tan
Jinyang Li
Ke Xu
Shuihai Hu
jingbin zhou
TAN Kun
Venue: ICLR 2026 Poster (2026)
Keywords: Causal Discovery
Reinforcement Learning
QR Decomposition
Long-term Carry-over Effects
Primary area: causal reasoning

Abstract: Learning causal structures from temporal data is fundamental to many practical tasks, such as physical laws discovery and root causes localization.
   Real-world systems often exhibit long-term carry-over effects, where the value of a variable at the current time can be influenced by distant past values of other variables. These effects, due to their large temporal span, are challenging to observe or model. Existing methods typically consider finite lag orders, which may lead to confounding from early historical data. Moreover, incorporating historical information often results in computational scalability issues. 
   In this paper, we establish a theoretical framework for causal discovery in complex temporal scenarios where observational data exhibit long-term carry-over effect, and propose LEVER, a theoretically guaranteed novel causal discovery method for incomplete temporal data. Specifically, based on the \textit{Limited-history Causal Identifiability Theorem}, we refine the variable values at each time step with data at a few preceding steps to mitigate long-term historical influences. Furthermore, we establish a theoretical connection between QR decomposition and causal discovery, and design an efficient reinforcement learning process to determine the optimal variable ordering. Finally, we recover the causal structure from the R matrix.
   We evaluate LEVER on both synthetic and real-world datasets. In static cases, LEVER reduces SHD by 17.29\%-40.00\% and improves the F1-score by 5.30\%-8.79\% compared to the best baseline. In temporal cases, it achieves a 64\% reduction in SHD and a 45\% improvement in F1-score. Additionally, LEVER demonstrates significantly higher precision on real-world data compared to baseline methods.

Published as a conference paper at ICLR 2026
INFLUENCE WITHOUT CONFOUNDING: CAUSAL DIS- COVERY FROM TEMPORAL DATA WITH LONG-TERM CARRY-OVER EFFECTS
Fan Li
Zixuan Liu Yi 2*
Ta
Jinyang Li*
Ke Xu
Shuihai Hu! Jinghin Zhou!
Kun Tan
'Tsinghua University, Beijing, China
Beijing Institute of Technology, Beijing, China
*Shenzhen University, Shenzhen, China
HHuawei Technologies Co., Ltd., China
Zhongguuncuu Laboratory, Beijing, China
ABSTRACT
Learning causal structures from temporal data is fundamental to many practical tasks, such physical laws discovery and root causes localization. Real-world systems often exhibit long-term carry-over effects, where the value of a variable at the current time can be influenced by distant past values of other variables. These effects, due to their large temporal span, are challenging to observe or model. Ex- isting methods typically consider finite lag orders, which may lead to confounding from early historical data. Moreover, incorporating historical information often results computational scalability issues. this paper, we establish a theoretical framework for causal discovery in complex temporal scenarios where observational data exhibit long-term carry-over effects, and propose LEVER, a theoretically guar- anteed novel causal discovery method for incomplete temporal data. Specifically, based on the Limited-history Causal Identifiability Theorem, we refine the variable values each time step with data at few preceding steps to mitigate long-tern historical influences. Furthermore, we establish a theoretical connection between QR decomposition and causal discovery, and design an efficient reinforcement learning process to determine the optimal variable ordering. Finally, we recover the causal structure from the R matrix. We evaluate LEVER on both synthetic and real-world datasets. In static cases, LEVER reduces SHD by 7.29%-40.00%% and improves the F1-score by 5.30%-8.799 compared the best baseline. temporal cases, it achieves a 64% reduction in SHD and a 45% improvement in F1-score. Additionally, LEVER demonstrates significantly higher precision on real-world data compared to baseline methods.
INTRODUCTION
Temporal causal discovery aims to recover the causal structure from time-series data. By revealing mechanistic rather than merely correlational dependencies, it provides principled basis for reliable prediction (Kuang et al. 2018 Shen et al. 2018 Kuang et al. 2020 Peters et al. 2016), counterfactual
reasoning (Swaminathan & Joachims 2015; Wu & Wang. 2018 London & Sandler 2019). and
root-cause identification (Meng et al. 2020 Ikram et al.) 2022 Lin el al. 2024)..
major challenge temporal causal discovery is disentangling instantaneous influences from lagged ones across successive time steps. Many real-world time series exhibit lagged influences that persist far beyond few time steps, giving rise long-term carry-over effects. From the perspective of historical dependency, the value of a variable at certain time depends not only on the current values of its parent variables, but also on their historical traces. Such systems are ubiquitous both industrial production and daily life. For example, when an advertisement is launched, it stimulates consumers' purchase intentions, thereby driving up product sales in the following period. In this system, the current product sales are not determined solely by the advertising intensity at a single moment, but rather by the accumulated effect of advertising over a past period of time.
'Equal contribution
Corresponding authors

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Published as a conference paper at ICLR 2026
Our goal is to recover the causal structure, i.e., directed acyclic graph (DAG) that codes the cause-and- efffeeect relationships, from observational time-series data. Due to the prolonged duration of the carry-over effects, the influence of early historical data is often
difficult to capture with limited observational data or model. Existing methods typically
consider finite lag orders (Runge al., [0099] Rungel
2020 Castri et al. 2023 Debeire
cct al. 2024; SSagiiro et al.) [020 Sun et al., [02
Pamfil et al. 2020
Granger 1969).
However, neglecting long-term carry-over effects may lead to historical values confounding causal discovery. Figure illustrates a scenario of causal discovery using a model with maximum lag order of 2, where observations of vari- ables at different time steps are represented as temporal nodes. The historical node (2 3) simultaneously in- fluences [1{t) [t] and X3 [t]t;; however, as it is not included in the model, spurious causal relationship may be inferred between X and [t (Powell 2018). Apart from the long-term carry-over effects, the super-exponential num- ber of valid causal structures, reaching 4.2 x 101s for graph with 10 nodes, poses an inherent challenge for causal discovery methods. Prior approaches that tempo- rally unfold variables with a fixed lag order (Runge al., 2019; Runge, 2020; Debeire et al. 2024) further exacer- bate the search complexity.
X[t 3]
x(t-2
X|t
X
X1
X2
X3
Unobserrved /Unmodeled
Observed Modeled
Figure 1: Historical confounding effect
In this paper, we establish a theoretical framework for causal discovery in temporal scenarios with long-term carry-over effects, and propose LEVER, novel reinforcement learning (RL)-based causal discovery method grounded in this foundation. Specifically, we introduce scoring function that guarantees causal identifiability
static settings. To address historical confounding temporal data, further establish the Homnited-Histon Causal Identifiability Theorem, showing that refining observations with limited history suffices for maintaining causal identifiability. Guided by the above theory, we propose LEVER, which can discover the causal skeleton from incomplete observational temporal data. First, LEVER refines observations using a limited history to mitigate long-term confounding effects. It then employs RL framework to determine the optimal variable ordering, using the upper-triangular R matrix from a QR decomposition as compact and efficient state representation. Finally, the causal structure is recovered directly from the optimal R matrix.
We evaluate LEVER on both synthetic and real-world datasets. In static cases, LEVER reduces structural Hamming distance (SHD) by 17.29%-40.00%% and improves the Fl-score by 5.30% 8. 8.79% compared to the best baseline. It achieves superior performance while requiring significantly less runtime and memory compared to baseline methods using the RL framework, demonstrating the effectiveness of using the R matrix as the state. In temporal cases, LEVER achieves a 64% reduction SHD and a 45% improvement in Fl-score. Compared to baseline methods for handling historical information, LEVER's 's historical refinement approach effectively addresses long-term carry- over effects without incurring significant additional overhead. Additionally, LEVER demonstrates significantly higher precision real-world data compared to baselines.
Our contributions are summarized below:
We extend static causal identifiability temporal scenarios with long-term carry-over effects, and establish theoretical connection between causal discovery and QR decomposition.
We propose causal discovery method for incomplete temporal data, which can quickly and effectively recover causal structure in the presence of long-term earry-over effects.
We validate the effectiveness of our method on both synthetic and real-world dataset, demonstrating that our method outperforms the baselines in both accuracy and resource efficiency.
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RELATED WORK
The PC algorithm (Sprtes et al. 2000) uses conditional independence tests to construct causal graphs but struggles with high-dimensional data due combinatorial complexity. GES (Chickering
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2002) employs score- basee optimization for efficient DAG learning. NOTEARS (Zheng al. 2018 reformulates causal discovery as continuous optimization with an acyclicity constraint, enhancing scalability. SCORE (Rolland et al., 2022a)) leverages the properties of the scoring function to achieve linear-time causal structure recovery, but this efficiency comes at the cost of shifting the computational burden estimating the scoring function itself. RL--based approaches formulate causal discovery as a Markov decision process and use reinforcement learning to explore the combinatorial space of DAGs. RL (Zhu et al. 2019) generates full adjacency matrix each decision step, which can lead to high computational cost and convergence challenges. CORL (Wang et al. (2021) reduces the
action space by reformulating static causal discovery as finding an optimal topological ordering, but
its neural-network- -based state encoder may introduce substantial computational overhead.
For temporal cases, although existing works on causal discovery have offered rigorous temporal modeling (Stein et al.) 2025: Mastakouri et al.) 2021), , they typically assume that direct causal dependencies span only a finite duration. Granger causality (Granger) 1969) tests whether the past values of one variable help improve the prediction of another variable with given maximum time lag.
PCMCI (Runge et al. 2019) and its variants (Runge] 2020 Debeire et al. 2024 Saggioro et al., 2020 Castri et al. (023) extend PC to temporal settings by unfolding variables over fixed lags and employ momentary conditional independence tests to identify lagged causal relationships, yet exacerbate combinatorial complexity. NTS-NOTEARS (Sun et al. (021a) and DYNOTEARS (Pamfil al..
2020) introduce structural vector autoregressive model assumption with given model order, and solve an optimization problem with an acyclicity constraint. SyPI (Mastakouri al.) 2021) allows multi-lag dependency from historical data and provides a constrained- causal feature selection method determine the causes of an observed target variable. While these methods can effectively capture short-term lagged relationships, they do not consider the potential confounding effects arising from long-term carry-over influences.
There are also several works that address causal structure under subsampling or continuous time kernels. |Pis et al. 2015) and Abavisani al. (2022) aim to learn the causal structure of a system with direct causal dependencies at timescale Ts. given the causal graph at slower timescale ry. [Bellot (ct al. (202I) assume that causal interactions occur between differential time steps and seeks to learn the causal structure of continuous- stochastic process from discretely sampled observations.
Under the assumptions of most existing work, distant historical values may influence the present only through observed intermediate time points. After conditioning these mediators, the remote past is conditionally independent of the present and cannot act as a confounder for contemporaneous relations. In this work, we drop this assumption and allow distant history to exert direct causal effects on present variables. Such latent long-range causes can induce spurious dependencies among observed variables and make causal structure recovery more challenging.
3
THEOREM GUARANTEE
3.1 NOTATION AND SYSTEM MODEL.
Let X[t) Fd denote the vector of observations (i.c., the values of all d variables) at discrete time step t E N. We assume X(t) is generated by time-invariant system with temporal carry-over effects,
XII = Σ ([--r]9(7) + e(l).
where g(T) E IRdx is the impulse response matrix at lag t, and c(t) Rd is structural noise with i.i.d. entries.
Assumption 1 (Time-invariant skeleton). For every r 2 0, the binary support supplg(7):== Pi F 0} is identical. That is, all g(T) share the same directed acyelic skeleton g. Assumption 2 (Stationary lag dependency). Every non-zero entry of g(7) depends only the lag not the absolute time index I.
embody temporal ordering and causal invariance, which are standard in many existing works Gong et al., 2015 2017 Malinsky & Spirtes 2018 Liu al. 2023).
Assumptions I 2
Objective. Given the observed time series ( T + +T]. our goal is to recover the causal skeleton g (i.e., the summary causal graph (Peters al. 2013 Gong et al. 2024
Liu et al. 2023)). In the
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following, first introduce scoring function and demonstrate that minimizing this recovers the true causal topological order in a static setting, thereafter establish the static identifiability of the causal skeleton. We then extend the theoretical framework to partially observed temporal data, thereby aligning the results with our ultimate objective.
3.2 SCORING TOPOLOGICAL ORDERS
To identify the true causal topological order from all possible candidates, ideal scoring function should attain its optimum at the true order. Following this principle, we introduce a scoring function based on the ordinary least squares (OLS) error.
Formally, let Y E Im consist of m (m > d) iidd observations of d random variables for all t O. Let
(X1,..., that follow Equation under static setting, where g(7) II be the set of all possible permutations of the d variables, and define K as the complete DAG whose skeleton is complete graph and topological order is given by π E
Lemma 1. For set of d variables, there exists a bijection between the set of all possible topological orders and the set of complete DAGs.
Definition (Scoring function). The score of topological order a given observations Y is defined as the total OLS error of fitting each variable j with its former variables in π (denoted as pa" (i)):
Score( Y) = E1r - Y pa (() (KT)
(2)
where (K*) TR (J) x1 are the OLS coesficients.
Theorem Assume that Y has full column rank. Let L E: Rmmxdd be the data matrix whose columns are permuied according to the given topological order æ. Let Y* QR be QR decomposition of y with RRdxd upper-triange trian-trianguaa and Vi, Rs > O. Then the score of a can be expressed as
d
Score( Y)
Σ R2
Theorem 2 (Causal topological order identifiability). Let nt" be topological order the true causal skeleton. For sufficiently large sample size m, the expected score of the true topological order n almost surely attains the minimum over all possible topological orders x E 11, i.e.
E [SSoree(T;;
min B[Score(T Y)). sell
3.3 CAUSAL SKELETON RECOVERY
The complete DAG derived from the true causal topological order may include superfluous edges. We will demonstrate that the true causal graph can be recovered from the topological order through weight estimation and pruning.
Theorem Let R E Bdxd be the QR decomposition result of " and the index selector ok represent the selection of the first k elements. Let W E Rdxdd be the estimated weight matrix, where W., represents the estimated edge weight from variable to variable j if there exists a directed edge from node i to node in KT; otherwise, Wij = 0. Then for each variable the estimated weights from its parents to itself can be computed from R as:
W (P) =B((K²)= Ro(;-)),(i-1> Ro(j-1)d>
I
(5)
Theorem There exists a threshold such that, when all edges in K** with absolute estimated weights [Wall < 0 are removed, the resulting graph is identical to the true causal gruph.
3.4 HISTORICAL CONFOUNDING IN TEMPORAL DATA
The aforementioned theorems establish causal identifiability in static scenarios. However, due to the cvrry-over effects in temporal data. these theories may not hold for the original temporal data. To address this, we propose a refinement scheme that mitigates the historical confounding effects, thereby extending causal identification theory temporal settings.
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Original
Agent
V
Causal Structure
Train
Q-network
Replay
Evaluste
Quaalues
V3
States &
Action
Ximonaa outtonde
Causal We weights
eiqeed
Add
Data List
topo. list
QR Dec.
QR Dec:
update
R. Matrix
State: St
Refined Data
Environment
State: St+11
Optimal R Matrix
1. Historical Refinement
Variable Reordering via RL
3. Structure Recovery
Figure 2: Overview of LEVER
Theorem 5. Regressing X(] on its complete history (X(t-T))>2> and taking the residuals preserves causal identifiability ie., the true order minimizes
these residuals.
In the real world, one often has access to only incomplete history. We next show that, under a mild assumption, limited history suffices.
Assumption (Linear recurrence of g(T)). There exists hEN and coefficients , kis that g(T) = kig(T 1) + ... + kT.9(T h) Vr 2 h 1. typical example is exponentially decaying effects, g(T) = g(( - 1).
such
Theorem simited-historg Causal Identifiability. Let (t) be the residual obtained by regress- ing X on X[t 1)....... X(t h]. Under Assumption 3 the true topological order minimizes Score(T; Y).
4 LEVER
We propose LEVER, method to learn the underlying causal structure from temporal data exhibiting long-tern carry-over effects. Figure Suuure2]illan sliding window to extract samples from the temporal data and refine the values at the final time step of each window by regressing them their preceding steps. Next, utilize Deep Q-Network (DQN) framework to efficiently determine the optimal variable ordering. We define the action at each decision step as appending the refined data of specific variable QR decomposition on this data list to obtain the R matrix, which represents the state, and compute the change global score as the reward. An episode concludes when all variables are included in the data list. Finally, we derive the causal weights from the optimal R matrix and prune edges with weights near zero to obtain the final causal structure.
DigureZiilustratess illlustrates the three key stages of LEVER. First, apply
data list. We perform
4.1 HISTORICAL REFINEMENT
For observational data X of length T. sample comprises historical observation matrix H (w- x)xd and target observation Rd By sliding the window with step size Ai, we obtain m = T-
sliding window of length w extract samples. Each
samples.
AF
In practice, prior knowledge about the properties of the impulse response kernel g(T) is typically unavailable. Thus, increasing the window size w improves the likelihood of capturing more complete structure of historical information. However, since the sample size m decreases as w increases, excessively large may reduce the sample size, potentially compromising the accuracy of subsequent regression analysis and causal discovery. Therefore, a balance between historical information completeness and sample size should be considered to select appropriate w.

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We perform regression analysis using recent historical observations to refine the target observation.
For each sample. we flatten its historical observation matrix into 1 x ((w - 1) d) vector H' by concatenating the values of the d variables at each time step. We then regress the target observation using the historical observation vector:
arg IOIH B
X H' B.
(6)
where X, denotes the r-th variable of the target state, and B, E (==-1))xx1 represents the r-th column of the regression coefficient matrix B.
Let the predicted value be = H' B. The refined target state is then expressed as:
x x H
Theorem 6 ensures that the true topological order achieves the minimal score on the residual x.
4.2 VARIABLE REORDERING VIA REINFORCEMENT LEARNING
We employ DQN framework to efficiently determine the optimal variable ordering.
Action. At each decision step, we define the action as appending the refined data of a specific variable data list. Since variables cannot be selected repeatedly, the action space the i-th step is A = (1,..,,
\ .d]
with a size (d - i+ 1), where r is the indices of variables current data list.
State. We define the state the R matrix obtained from QR decomposition of the data list the specific decision step. As established in the preceding theoretical analysis, the R matrix is uniquely determined by the data list and encapsulates complete information about the causal weights while being directly correlated with the score. Furthermore, QR decomposition can be performed stably and efficiently using numerical libraries such as NumPy (Van Der Walt et al.) (01T)) with computational complexity of O(md²) (Golub & Van Loan 2013). Thus, the R matrix serves as compact and informative representation of the data list, accelerating the RI. learning process and enhancing performance.
Reward. We compute the immediate reward of an action as the reduction global score. The global score is defined as the sum of the residual sum of squares for variables within and outside the data list. Formally, it is expressed as:
GlobalScore
R2 +
Σ RSS jēT
where RG,s is the i-th diagonal element of the R matrix, and RSS denotes the residual sum of squares (RSS) when regressing x, on the data in the data list.
Q-Network. The Q-network maps the input environment state to the action with the highest expected return. To improve the accuracy of expected return estimation and ensure training stability, we employ Double DQN (DDQN) approach (Van Hasselt et al.] (016). Specifically, one Q-network selects the optimal action, while a separate target Q-network evaluates the expected return for that action, reducing overestimation biases. Its parameters are updated using a temporal difference learning approach, with the update rule:
Q(s, a) + Q(s, a
(r (r+ max d') ((s.a;0)).
where and denote the current state and action, s' is the next state, is the reward, is the learning rate, 7 is the discount factor, represents the Q-network parameters, and 0- denotes the parameters of target network. The target network's 's parameters f periodically aligned with 0 using a soft update mechanism 0 00 + (1 2)0 where E (0, 1) is a hyperparameter controlling the update rate.
During training, we track the episode with the highest cumulative reward and record the corresponding optimal R matrix R

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Table Performance on static datasets
Graph
Metrics
PC
NOTEARS
RL
CORL
SCORE
LEVER
F1
0.4574
0.7308
0.3809
0.5930
0.8179
0.8899 48.99%
G(10,20) 20)
FPR
0.0949
0.0208
0.0580
0.0206
0.0329
0.0123 40.96%
SHD
19.00
8.00
18.33
12.00
6.67
4.00 40.00%
FI 0.5528
0.7898
0.3770
0.5228
0.7918
0.8337
45.30%
G(20, 40)
FPR 0.0395
0.0062
0.0193
0.0072
0.0207
0.0090
SHD
32.00
13.54
34.00
25.85
15.50
11.60
11.60 14.299
Note: Black bold text indicates the best results among the buselines, and red triangle markers show the perceniage improvement of our methwd compared to the best buseline results.
4.3
STRUCTURE RECOVERY
We compute the causal weights A using It according to Equation 5
()) Ro (o(j-)))---1) R
isjsd.
We then prune edges with weights near zero to obtain a sparse causal structure. The impact of thresholding illustrated in Figure
EXPERIMENTS
Implementation. We evaluate LEVER synthetic datasets, encompassing both static and temporal scenarios, as well as real-world dataset, comparing its performance against eight classical and state-of-the-art baselines. All experiments conducted macOS with 16 GB of unified memory. The software environment utilizes Python 3.10.2 for scripting and PyTorch 2.5.1 as the primary deep learning framework. Implementation and configuration details are provided the Appendix.
Metrics. We use Recall, False Positive Rate (FPR), Fl-score and Structural Hamming distance (SHD) evaluate the performance of causal discovery methods from multiple perspectives.
The static causal discovery baselines include PC (Spirtes et al. 2000), NOTEARS (Zheng al. 2018), RL (Zhu et al., 2019), CORL (Wang al. 202 and SCORE (Rolland al. 2022a). The temporal causal discovery baselines include Granger (Granger 1969), PCMCI (Runge et al.
Baselines.
2019), and NTS-NOTEARS (Sun et al., 2021a). We run baselines with multiple hyper-paramee settings, and report the results with best performance. The chosen parrmmerrrettess are presented in the Appendix.
5.1 SYNTHETIC STUDY
5.1.1 STATIC CASES
For static data, we generate the ground truth causal graphs with the Erdos-Renyi model. We denote the graphs with d nodes and m edges as G(d, m). We generate 500 i.i.d. samples for each graph using the structural Equation X, = EEEEP(() X,- Ays + E,, where Pa(i) represents the topological the graph, Aya is the weight of the directed edge from variable j to variable
parents of variable i, and 6; is an independent random variable following a standard normal distribution. To ensure experimental reliability, we generate at least three datasets with different structures for each graph setting and report the average performance.
It's worth noting that although our primary focus is causal discovery on temporal data, static data can be considered a special case of temporal data, enabling comparison with broader range of related work. Moreover, the performance of LEVER on static data validates the effectiveness of our RL module.
Results. Table D shows that LEVER achieves higher F1-scores and lower SHD compared to all baselines, demonstrating superior accuracy in causal structure recovery. Specifically, LEVER improves the Fl-score by 5.30%-8.79% over the bert-pern baseline and reduces SHD by 17. 17.29%--0.00 0.00 relative the lowest SHD among the baselines.
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(a) 10 nodes
(b) 20 nodes
(c) 30 nodes
Figure 3: Visualization of LEVER's causal recovery
LEVER
Granger
NTS-NOTEARS
PCMO
Granger
0.8
0.6
SHD
0.4
0.
G110 2(10,20) G120,40) 4020.40) G/30,60) G(30,60) expomential inear esponential linear exponentiall linear
G(10.20) G(20
G(20,40) GE30,60) G(30,60) inear
exponential
Inear
exponential
exponential linear
Figure 4: Performance on temporal datasets
Additionally, we compare the efficiency of LEVER with other RL-based causal discovery methods, namely RL and CORL. Owing to LEVER's action space of only (dd each step and its efficient representation of raw data using the R matrix instead of a Beural-network-Baee achieves superior efficiency in both time and memory usage. On the graph with 20 nodes, LEVER requires 0.25 seconds per episode, while RL and CORL require 19.45 seconds and 1.71 seconds, respectively. Furthermore, LEVER maintains low peak memory usage of 10.36 MB, compared to 24.40 MB for RL and 64.68 MB for CORL (with input and hidden dimensions both set to 64).
encoder, LEVER
5.1.2 TEMPORAL CASES
For temporal data with long-term carry-over effects, generate the ground truth skeleton and instantaneous influence weights with the same settings as static dataset, and have the influence weights decay over lag time. We consider three types of decaying schemes, exponential, linear and complicated (specified later). We generate the original data with equation for 2000 time steps, and use the data of last 1000 time steps, during which the system has been operating stably. At least three datasets are generated for each setting, and we report the average performance.
Results. Figure 3 visualizes the results of LEVER recovering causal relationships from temporal data with carry-over effects. It shows that LEVER can reconstruct the structure and weights of the ground truth causal graph with high accuracy. Figure 4 compares LEVER with baselines. LEVER achieves a 64%-88% reduction in SHD and 45% 101% improvement in F1 F1-score compared with the best baseline. Notably, NTS-NOTEARS and PCMCI, temporal extensions NOTEARS and PC, incorporate limited lag-order temporal effects by unfolding variables across multiple time steps. This increases the number of nodes in the causal graph, significantly raising the time complexity. Under 10-node graph settings, the runtime of NOTEARS and PC on temporal data increases by 203 and 69 times compared to static data, respectively. In contrast, LEVER, which refines observations using historical data, incurs only an additional 0.0017 seconds of runtime.
5.2 DEEP DIVE
Visualizing Long-term Causal Dependency We generate two groups of datasets with complicated decay types, where g(T) is linearly expressed by 10 and 20 preceding entries (i.e., h = 10, 20). Figure prreennts heatmaps of g(T) with different time lag 10, 20, 50. It can be observed that the actual corry-over effects persist over a much longer duration than the recurrence order Such long-term direct causal dependencies cannot be fully captured by fixed maximum time-lag model
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Score
(a) h
Accuracy
01224
Start
(b) /2 20
Figure 6: RI. Effectiveness
Figure 5: Heatmaps of g(T) with 10/20/50
as assumed in existing works, which may introduce confounding in causal discovery. In contrast, our method uses the h-step historical data as a lever to mitigate long-range historical confounding effects.
Ablation Study. We conduct ablation study to assess the contributions of the RL-based variable reordering module in LEVER. We replace the RL-based variable reordering with static causal discovery baselines, and evaluate the performance of the modified LEVER temporal datasets with both linear and exponential decay effects. Specifically, we consider five variants: LV-PC, LV- NOTEARS, LV-RL, LV-CORL, and LV-SCORE. Figure compares the performance of these variants with the full LEVER model. The ablation results show that, although our RL ordering module is theoretically replaceable by existing algorithms in terms of objective, it achieves higher accuracy than these existing approaches.
Sensitivity to Assumption Violations. We evaluate LEVER several modified temporal datasets where the values of three randomly selected causal dependencies are set to zero, except at lags of 3/5/10/25. The results in Table that the accuracy of our method does decrease significantly even when the data violate the assumption. That is because the assumptions provide the necessary conditions for the theory hold, yet certain scenarios outside these assumptions may also support the theory. Furthermore, operations such as pruning enhance our method's robustness in scenarios that deviate from the stated assumptions. These results support the generalization of our method to realistic cases.
Table 2: Results under assumption violation
Dataset
SHD
Fl
Standard
0.8947
Modified (lag=3)
5
0.8718
Modified (lag=5)
0.8718
Modified (lag=10)
3
0.9231
Modified (lag=25)
6
0.8421
5.3 REAL-WORLD STUDY
The wt wa_ks v/ dataset (Gamella et al. 2025b a) contains temporal data derived from controlled experiments conducted in wind-tunnel chamber. The ground truth graph comprises 16 nodes and 26 edges, representing the inherent dependencies among key variables within the wind-tunnel chamber. These variables include actuator settings (e.g., fan inputs and hatch position, denoted H), sensor

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(a) LEVER
(b) PCMCI
(c) NTS-NOTEARS
Figure 7: Recovered causal structures on real-world datasets.
parameters (c.g., fan loads Lins Lou),, and sensor measurements (c.g., barometric pressures Paws Pup- Pamb, Pmn)
We take the last 1000 observations for stability consideration. Figure presents the ground truth dependency among variables and the recovered causal structures of LEVER and two baselines. Edges in black represent the recovered relations, edges in gray represent the undetected relations and edges in blue represent the relationships incorrectly inferred by the model. Quantitatively, LEVER achieves a SHD of 22 and an Fl-score of 0.5769, outperforming PCMCI (SHD 35, Fl 0.3636), NTS-NOTEARS (SHD= = 37, Fl 0.3019), and Granger causality (SHD = 48, FI = =0.9941), We do not visualize the result of the Granger causality method due to the excessive number false positive edges it generates, which makes the causal graph overly cluttered and difficult to interpret.
LIMITATIONS
Our theory is built upon the linear time-invariant (LTI) assumption, which limits the theoretical guarantees of our approach in nonlinear scenarios. future work, we will extend the LEVER framework to accommodate nonlinear relationships, thereby broadening its scope and applicability. Additionally, the assumption for using a few historical observations to completely climinate long-term historical effects is that g(T) can be linearly represented by g(T 1),-. g(( h). In future work, we will explore more relaxed conditions and quantify the impact the long-term carry-over effects causal discovery when these conditions not fully satisfied.
CONCLUSION
In this work, we propose causal discovery algorithm from temporal data with long-term carry-over effects. Our approach refines observations each time step with data from a few preceding steps to mitigate long-term historical influences. We employ reinforcement learning framework based on QR decomposition to determine the optimal variable ordering and the corresponding matrix, from which recover the causal structure in polynomial time. Experimental results show that our method outperforms baselines on both synthetic and real-world datasets.
8 REPRODUCIBILITY STATEMENT
We have made every effort to ensure that the results presented in this paper are repro- ducible. All code has been made publicly available at atftttps:/gitta TemporalCausalRR.. git to facilitate replication and verification. The real-world dataset is publicly available in (Gamella et al.) 2025b a). The experimental setup, including training steps, model configurations, and hardware details. is described in detail in the paper (see Section 5 and Appendix A.2). We have also provided full description of LEVER to assist others in reproducing our experiments. We believe these measures will enable other researchers to reproduce our work and further advance the field.
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9 ACKNOWLEDGMENTS
This work was supported in part by the National Science Foundation for Distinguished Young Scholars of China (No. 62425201), the National Natural Science Foundation China (No. 62472036), and Beijing Nova Program. Yi Zhao and Ke Xu are the corresponding authors.
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A APPENDIX
A.1 PROOFS FOR THEOREM GUARANTEE
A.1.1 PROOF OF LEMMA I
proof. Let =
(1, (1, d} be the variable set. Let IT denote the set of all possible topological orders V,, and I. denote the set of all complete DAGs on V.
Construct map f II -> IC follows. For topological order be the directed graph on V that has, for every pair i < j. the directed edge म, + π.. Clearly every unordered pair of vertices receives exactly oriented edge, so the underlying skeleton of f()) is
(T Ta) E II, let f(x)
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complete graph. Because all edges point from earlier to later vertices in directed cycle can exist; hence f(T) E K.
Define mapping g K - II such that, for any complete DAG K K. (KK) denotes the topological order the nodes in K. Clearly g(K) E II. In addition, we can prove that any complete DAG K has unique topological order. Take any two distinct vertices u, u. In complete DAG exactly the edges u -> or + u is present. If - v, then in every topological order u must precede v; otherwise must precede u. Thus the relative order of and fixed. Since this holds for every unordered pair, the full order of V is uniquely determined.
of
Finally, verify that and are mutual inverses:
For any и E II, the graph f(r) was constructed by orienting each pair from earlier to later in The unique topological order of this graph itself, hence ((f(z))
For any K K. its unique topological order the orientation in matches the orientation in f(T). Hence (()(K) = K.
K) dictates that for every pair {u, v)
Therefore, and inverse bijections, completing the proof.
A.1.2 PROOF THEBRREM II
proof. Since performing QR decomposition on the reordered data matrix y equivalent ap- plying Gram-Schmidt orthogonalization with order и (II, TT), the j-th diagonal element of the R matrix represents the norm the orthogonal projection of Y onto the complement of
span T (,((-1)) Let S = span (Y ( and S represent the complement of S. Then,
R projs (Y,)) Y3 Y 1) Bay (k* ||²
||xx,.> Y (x, Br, (K
Therefore, by definition, have
Score(T: y) y)= Σ
R²
j=l
A.1.3 PROOF OF THEBRREM 2
proof. Let B E Rddd be the true weight matrix, where Bry represents the true edge weight from variable to variable if there exists a directed edge from node to node j in K otherwise, Bij By definition, we have = Y + e, where Rmxd the structural noise with i.i.d. entries. Since K" is acyclic, det(1 B) 1.
Define T (1 -1 and Σ o TT T. Then we have
det = 2d det(TT) det(T) = 2d
Let (T) = Var(Y=a n4 Y m ...,, Y nh ) and dx(
cx(z) 0. Let Σ LDLT be the LDL decomposition of Σ, then based on the properties of Schur complement, we have D = diag(c((T),"* ca(TT)). Therefore,
II (t(m)) det
2d
For correct order n², equations (x 0², 27,44m 1, Ee 8z(T)) d holds for every E (I,... .d. Clearly,
detS,
(p(
2d =
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For incorrect order n', since not all of (6x(1))) are equal 1, follows from the AM-GM inequality that 1x=10x(TT)) > d.
By definition, the score of topologiaal order x given d independent observations Y can be expressed Scorem (T;; Y) Σ RSS Then,
m m-k+1
RSS
Scorem (T; Y) o²
(x( (x())..
k=1
According to the properties of Bartlett decomposition,
RSS (T)
1 m
+ Xm k+1
Cs. (m k
where xm-x+1
k+1 is chi-squared random variable with squares of m k + independent standard normal variables.
1 degrees of freedom, i.e., the sum of
For each fixed k, the following equation holds for sufficiently large
m- Ma-k+ k+1
k + Xm k+1
K
k,
a.s.
BIZ
k.1
=1,
where Zz,s 3.i.d. NO0 1). Furthermore, since m-
we have
1 Scorem (T; Y)
a.s.
8u (T).
m
10
Therefore, for correct order nt, ( Scorem (1'; m-000
ad.. For any incorrect order n',
m=+00
Ed Ek=1 8& (TT) > 02d.
m
As result, the differential Am Scorem (T';
Scorem (T; satisfies:
0.3. C(TT') = o² Σ (xT'))
-d ) 0.
Thus.
Am c(T') + o(m) a.s.
A.1.4 PROOF OF THEOREM 3
proof. Since (K*)) are the OLS coefficients of fitting Y.j on Pa (i) it can be presented as
kr(K*) (Y. (1) Y. (1) - Taa (1) Y.
Let " QQ R, then Y.,,, = =Q. R.g.- Therefore,
W pa ()) = (K*)
= =(Q RRo()-1)) (Q. R (Q R. (Q. R.y) = -R QR R2,00-110 TRR,
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Since columns in Q are normalized and mutually orthogonal, we have QTQ I. In addition, R is upper-triangular Then,
W G)a = (R. -1) (()-1)) R. -00--11 - Rag
= (Ro(s- 1), Rot Rall-11)) 1)) Ra(--1),0()-)
Ro Ro(j-1)), Ro(j-1)
Ro1j-11)o(j- (R ( RRo
R, 1) 1)
Rolj- 1).(-1) Ro(j-1) 1ooj-))a--)Roo(--111
A.1.5 PROOF OF THEOREM
proof. Let x² the topological order of the true causal graph. Let {pa" (j)) be parents set of variable j in the complete DAG K*" (consisting with the notation in Definition and (pa" (i) be the parents set of variable j the true graph. Then for all j E [1,...,, dd, (pa" (f) S (pa (f))-
Denote (y(K*) as the real coefficients, then 3, are unbiased estimators of 3 ) i.e. B[3, (K" } (yK)), 33 (K*') _ 35(K**). Therefore, for any positive real number 8, there exists an arbitrarily small w (0.8/2) and an my such that, for all m > mp,
Pr Vk. 13,(K**) )a B,(K* )al < w) c.
Moreover, for any < 1, such mo exists.
Denote A > 0 as the minimum absolute value over all non-zero values in 3y(K*)) Let 8 = A. then when Vk.jj, (3) - 3, (K*') w happens, all zero elements in 3
(K(K') ) (k Q2) satisfy (K" )al < w, and all non-zero elements in 3,(K* )x (A: E 22) satisfy 13,(K")) )al E (13, K )al w. [3, )al w).
Since < 6/2, have Akı $2..ss A2 E Q2 13,(" < 13 )k=l-
Thus, there exists threshold such that, when all edges in K with absolute estimated weights smaller than 0 are removed, the resulting graph is identical to the true causal graph.
A.1.6 PROOF OF THEOREM 5
Proof. By definition, X(t] is generated by the structural equation:
X = Σ Ex1t - (lgT-) cc(),
where g(T) Rddd is the impulse response matrix at lag т, and c(t) Rd is the structural noise vector with i.i.d. entries, following distribution /O0 a²). The causal graph G is defined as the support pattern of the matrix 9(7).
Rewriting the equation above yields:
X|x
Σ x(1-7)0(F)) X(1,(0) e((),
T=1
[g(1)
g(2)
X]I[(I (()) [X 11, X1t . [|0]
c(t).
(10)
[g())
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Thus, obtain:
[ = [X[ - 1), X - - c()((T -0(0)
where B collects the lagged coefficient matrices.
Since (L ((0) -1 is independent of the regressors [X .l..... X[]]], the residual of regressing X [t] onto its past values is:
r[t] = <tt) - -(t))
which can be transformed to the standard expression of the structural equation, i.e.,
r[t] = r(t]g(0) =(t)
Then Theorem 2 and Theorem can guarantee its identifiability.
A.1.7 PROOF THEOREM 6
Proof. Let S X h1 X + 1]. ..... X 1]] the most recent historical observations, and let U [[], ...., X[t h 1]] be the earlier historical observations.
Define the dependency weight vectors:
g(t)
g()) g(h
g(t - 1)
1)
gg(h 1)
g(L)
where dy and Ds represent the dependency weights of U and S, respectively.
By Equation 10
[[]] is expressed as a linear combination of historical observations and structural
noise:
x UT - 9(0)) 5 ((0) -
Define:
W = (1-9(0))-1
so that:
X = U W ShsW + c(t)
the earlier observations U are unavailable or not modeled, regressing X[t] only on S, the residual
is:
r|t]+ = (1 - (
II ([--Ps)(U W + 50sW + c(t)W)
= - P + - Ps)$0 + Ps)e((W,
where Pss S) is the orthogonal projection matrix, and is the identity matrix. Since:
(1 - P S1
residual simplifies to:
r[t] = (T P W + (I Ps)c(t)W.
(11)
Define:
Tg(i + 1-1)
r,
gli 1)
By Assumption 3 for all 5 2 h + 1,
g(T) kig(7 1) + kng(T h))
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implying:
TA = h-- 1-
KATo.
and by definition, U Io-
Since:
X = Σ
X rlg(T) (t),
T=0
we have:
X = Σ X x + w.
Given (I (I-Ps)S Ps)S
0, it follows that:
(I Ps)[X((t h1, h1, - h 11..., X(t - 1] 0,
yielding:
(T- ((-
(12)
(1 Ps) - 1e0-X---+111 -
(13)
(I- Ps) 1 =
(14)
Unfolding X[t h| in Equation 12
- Ps) Σ X - h h- cet hi) hn) W
Since U = [X) X(1)
X(t - h - 1|]. this becomes:
(1 - Ps=Ps)(UTo+ - (()- =0
As W is invertible, obtain:
(T - (U = O.
Similarly, unfolding X(t - h + 1] in Equation 13
(I- (UT, X|( M19(1) e(l - 1)) 0.
Using Equation 12 (T - Xt) h) = 0, this simplifies to:
(I - (UT + c(t h+ 1)) = 0.
Generalizing for i I....,h:
(L-Ps) + e(t h i)) = 0.
(15)
Substitute Equation 15 into Equation
Since Du I'm and I'm = Σne Kig hn- we have:
( Ps) = PS)UTT(1 -(0))= (1- (U-Ps)U
Σ
kig Kof W.
Using Equation 15 (1 - PP)UUn- = - -I[-)s) (3))((-00 - 4), So:
(I- P = -(1 Ps) Σ kne(t g)W.
Thus, Equation becomes:
r[t]+ -(1 -11- Ps) Σ
kige(t g)W + (1- -Ps)s((t).
9=l
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Rewrite:
h
kne(t - - 1 +c(0)
(1 = -(1 - PP))
Define:
elt] (I Ps) Σ kac(t 9) + c(t)
Then:
r|o} - r|d] ( g(0) elt]-
Compute the expected squared norm of elt]:
T
B [el|ett]]]] [|]
Σ kee(t - 9) +c(t)) (I Ps) Ps)
Σ koelt 9) + t(t))) +))((t-)+(((t))
Since - Ps is symmetric and idempotent, and e(t), e(t - h) are independent, we have:
|| = (I-)) (m (m-
E [c(t -1t² Ps)c(t -]]
1
(16)
where La E Raddd is the identity matrix, is the number of samples, and hdd is the number of columns S.
Equation 16 indicates that all columns of e(t] have the same variance. Thus, the proof method from Theorem can be applied to show that the true causal graph achieves the minimal score on r[t]+. L
A.2 EXPERIMENT DETAILS
A.2.1 LEVER IMPLEMENTATION
Q-Network Structure and Training Hyper-parameters. The Q-Network model consists of normalized multi-layer perceptron with two hidden layers and ReI.U activations. The input first processed by Normalization layer, then passed through two linear layers (with ReLU activation between them) followed by dropout. The final output layer maps the hidden representation back to the original input dimension. The mask is applied the output ensure that variables already in the data list are not selected again.
The values and description of hyper- peee-paaameters used in LEVER implementa- Table 3 Our full implementation is publicly available at tion are presented wwteenn///suuiiitteeeaaaallallla TemporalCausallI--BAA http://aanonmmmmuuuAoooooseeeeee Aopen.scieence///uumii
In addition, we evaluate all values in the output weight matrix as potential thresholds and select the one that yields the minimum SHD. The selected thresholds are presented in Table
A.2.2 BASELINES IMPLEMENTATION
We implement PC, GES, NOTEARS, RI, and CORL. using the geastle ggaa library and PCMCI using the tigramite tg library. Granger causality tests are performed with the grangercausalityterr
function from the statsmodels package. For SCORE and NTS- the original code provided by the authors Rolland et al. (2022b); Sun al.) (2021b). We run baselines with multiple hyper- parameer settings, and report the results with best performance.
NOTEARS,
For the PC algorithm, we set the confidence level to a = 0.05 and use the Fisher-Z Z test as the conditional independence test.
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Table 3: Q-Network hyper-parameters
Parameter Name
Value
Description
window
Size of the sliding window used for processing sequential data.
sliding_stop step
Step size for advancing the sliding window.
buffer_size size
10000
Capacity of the replay buffer for storing experience tuples.
minimal_ste _size
Minimum number of experiences in the replay buffer re- 200
quired to start updating the Q-Network.
mizeh_size
64
Number of experience samples drawn from the replay buffer for each Q-Network update.
Ir
0.002
Learning rate for optimizing the Q-Network during train- ing.
hidden_din
16
Number of units in the hidden layers of the Q-Network.
gamma
0.9
Discount factor determining the weight of future rewards in the Q-Network.
target_update
2
Frequency at which the target network's weights are up- dated match the online Q-Network's weights.
epsilon_start
Initial exploration rate for Boltzmann exploration strategy.
epsilon_end
0.01
Final exploration rate for Boltzmann exploration strategy.
Table 4: LEVER selected thresholds
node ct
decay_ _tppe
threshold
10
exp
0.1398
10
linear
0.1720
20
exp
0.1279
20
linear
0,4627
30
exp
0.1372
30
linear
0.1223
For NOTEARS, we set the regularization parameter A 0.1 and the maximum number of iterations to 1000. For each graph, evaluate all values in the output matrix potential thresholds and select the one that minimizes the Structural Hamming Distance (SHD). The selected thresholds are summarized in 5
Table 5: NOTEARS selected thresholds
node_ct
version
threshold
node_ec et
version
threshold
10
vl
0.0105
20
v6
0.0668
10
v2
0.2268
20
v7
0.1565
10
v3
0.0019
20
v8
0.1
20
vl
0.0374
20
v9
0.0370
20
v2
0.0748
20
v10
0.0370
20
v3
0.0484
20
v11
0.1265
20
v4
0.1032
20
v12
0,0703
20
v5
0.0684
For RL, we set the number of training epochs to nb_ see_ppocchh-- 500, the encoder input dimension to imput poppaddimeesion 64, the encoder hidden dimension hidden dim 64, the decoder hidden
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dimension to decoder monder_hidden dim 16, and the number of attention heads in the self-attention transformer to num = 16.
For CORL, configure the batch size satch_size 128. the encoder input dimension as imput dim 128, the embedding dimension as embee dim 128, and the maximum number of iterations iteration 100.
For SCORE, we select the values of the regularization terms GGG and TH, as well as the pruning threshold cutof. f. for each graph by minimizing the SHD. The selected values reported in Table 6
Table SCORE selected hyper-parameter
node cct
version
eta G
eta H
cutoff
node ct
version
eta
eta
cutoff
10
v1
0.05
0,0005
0.05
20
v6
0.001
0.1
0.01
10
v2
0.001
0.001
0.01
20
v7
0.001
0.0005
0.01
10
v3
0.001
0.0001
0.01
20
v8
0.01
0.1
0.01
20
vl
0.001
0.1
0.01
20
v9
0.005
0.1
0.01
20
V2
0.005
0.002
0.01
20
v10
0.005
0.1
0.01
20
v3
0.001
0.001
0.01
20
v11
0.001
0.001
0.01
20
V4
0.001
0.1
0.01
20
v12
0.01
0.001
0.01
20
V5
0.001
0.002
0.01
For PCMCI, Partial Correlation and set the maximum time lag r_mar: 4 to ensure the window size is consistent with our method, and set the confidence level to a 0.01.
For Granger,
the maximum time lag to marlag 4 for the same reason, and significance threshold of p_thrrehh = 0.01.
For NTS-NOTEARS, the regularization coefficients \ 0.005 and X 0.01, and select the threshold that minimizes the SHD. The selected values are reported in Table
Table 7: NTS-NOTEARS selected thresholds
node ct
decay type
threshold
10
exp
0.1940
10
linear
0.1668
20
exp
0.1902
20
linear
0.1315
30
exp
0.1826
30
linear
0.1362
A.2.3 EXPLANATION OF PERFORMANCE UNDER STATIC SCENARIOS
Table I we observe that although the considered problems of LEVER and baselines are essentially the same, LEVER outperforms these baselines in static causal discovery tasks. This may be because, although both our method and the baselines require more data to achieve better performance, in practice the performance of our method is less sensitive sample size than that of the baselines. As result, our approach performs better when the available data are limited. To verify this, we conducted additional experiments in the static setting using 100, 200, 300, and 400 samples (the results reported in the main paper use 500 samples), As shown in Figure g our method has a more pronounced advantage when the sample size small.
A.2.4 IMPACT OF WINDOW SIZE.
We further investigate the impact of window size on algorithm accuracy relative to the theoretical minimum window size + which ensures complete elimination of historical effects. We evaluate LEVER's Receiver Operating Characteristic (ROC) curves under various window sizes on the datasets with complicated decay types where 9(7) is linearly expressed by 10 and 20 preceding
21

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Published as a conference paper at ICLR 2026
LEVER
PC
MOTEARS
RL
COOL
500TE
LEVER
PC
NOTEARS
RL
CO.. 50382
0.6
0.4
0.2
0.96
0.94
Accuracy
Accuracy
0.92
0.90
0.85
200
Samples
Samples
(a) G(10,20)
(b) G(20,40)
Figure 8: Performance comparison under different sample sizes in static scenarios.
entries. To isolate the effect of sample size, we use 500 samples for training under each window size configuration. As shown in Figure when the window size is insufficient, the algorithm exhibits suboptimal performance. Increasing the window size effectively improves accuracy. However, when window size exceeds h + 1, further increases yield less improvement in algorithm performance.
1.0
1.0
0.8
TPR
0.6
TPR
0.6
0.4
window=17
0.4
window=30
window=11
window=21
0.2
window=12
0.0
0.0
0.00
0.05
0.10
0.15
0.20 FPR
0.25 0.30
0.35
0.00
0.05
0.10
0.20
0.25 0.30 0.35
FPR
(a) h = 10
(b) h 20
Figure 9: ROC curves
B LLM USAGE
Large Language Models (LLMs) were used aid in the writing and polishing of the manuscript. Specifically, we used an LLM to assist in refining the language, improving readability, and ensuring clarity in various sections of the paper.
important to note that the L.LM was not involved in the ideation, research methodology, or experimental design. All research concepts, ideas, and analyses were developed and conducted by the authors. The contributions of the LLM were soiely focused on improving the linguistic quality of the paper, with no involvement in the scientific content or data analysis.
22


Reviews:
[{"review_id": "CP1IWz9yAq", "reviewer": "Reviewer_eY1p", "summary": "The paper studies causal-structure recovery from time series with long-term carry-over effects. It (1) proves that a score based on OLS/QR identifies the true topological order in the static case, (2) shows that regressing out a limited recent history suffices under a linear-recurrence assumption to remove historical confounding, (3) uses the R matrix from QR as an RL-friendly compact state and trains a DQN to recover a topological ordering, and (4) recovers edges from the optimal R and prunes small weights. Theory, algorithm, and strong empirical gains on synthetic and a wind-tunnel dataset are presented.", "questions": "1. How sensitive is LEVER when Assumption 3 is only approximately true? Please include an experiment where g(τ) is not exactly linear-recurrent.\n\n2. The RL state uses the R matrix. How does numerical stability affect performance when columns are nearly collinear? Do you need column normalization or regularization in practice?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "• Clean, provable identifiability result linking QR diagonal terms and the order score. Theorems 1–4 give solid, interpretable mechanics for order scoring and weight recovery. \n\n• Practical mechanism to mitigate long-range confounding via residualization on limited history and a clear sufficient condition (linear recurrence) under which limited history suffices. The Limited-history Causal Identifiability theorem is useful. \n\n• Engineering innovation: using the R matrix as an RL state is simple, computationally efficient, and justified by the theory. This yields large runtime and memory savings versus prior RL encoders. \n\n• Strong empirical performance. The method outperforms several static and temporal baselines in F1 and SHD and shows large improvements in temporal scenarios. Results are reported on multiple synthetic regimes and a real dataset. \n\n• Reproducibility: code and dataset links and implementation details are noted in the appendix.", "weaknesses": "1. The related-work section treats prior temporal methods as finite-lag. But there is a body of work that models dependencies without imposing a strict finite lag (rate-agnostic or effectively infinite-memory representations). The paper should explicitly compare and position itself relative to those methods (both conceptually and empirically when possible). Examples the authors did not cite (add and discuss): \n- Rate-Agnostic (Causal) Structure Learning — Sergey Plis et al. — 2015. \n- Generalized Rate-Agnostic Causal Estimation via Constraints (GRACE-C) — M. Abavisani et al. 2023\n- Consistency of Mechanistic Causal Discovery in Continuous Time (Neural ODEs) — A Bellot et al. 2022\n, plus other works that address causal structure under subsampling or continuous-time kernels. Including these clarifies novelty and assumptions. \n\n2. The Limited-history theorem depends on Assumption 3 (linear recurrence of g(τ)). The paper acknowledges this, but lacks an empirical sensitivity analysis showing how performance degrades when the linear-recurrence assumption is violated or only approximately holds. A stronger empirical section on robustness is needed.\n\n3. Figure 3 is hard to parse and compare to ground truth. The visualization layout, legends, and the mapping between recovered and true edges are not easy to read.\n\n4. The RL ordering module is novel here, but the paper would benefit from (a) an ablation that replaces RL ordering with a simple heuristic (e.g., greedy order by marginal variance or SCORE ordering) to quantify the RL contribution and (b) more baselines that explicitly handle long memory when available", "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": "0X1JVEepmz", "reviewer": "Reviewer_1PFq", "summary": "This work introduce a method called \"LEVER\", which uses a (QR-decomposition)-based, \"static\" score and RL to infer a complete topological (causal) order; based on i.i.d. samples from time series. The theory part mainly aims at \"static skeleton recovery\", assuming i.i.d. samples, and then consider \"historical refinement\" via regression of variables on their past values. The baselines of the empirical experiment consider \"static discovery methods\" and embed them into the LEVER framework (of Figure 2), and a couple of temporal baselines (not a comprehensive list, missing main recent ones). \n\nOverall: The only \"temporal\" modeling of the theory part comes in the historical refinement framework, unlike other recent studies, e.g., [Mastakouri, Schölkopf, Janzing ICML 21]; [Sun,Schulte,Guiliang,Pascal,Poupart AISTATs 23]; [Liu, Sun, Hu, Wang, NeurIPS23]; and [Stein,Shadaydeh,Blunk,Penzel,Denzler ICLR 25]. Also, temporal baselines are limited. I see the contribution is more limited to static skeleton discovery, unlike what the current presentation of the paper suggests. Please see more details in the Weaknesses section.", "questions": "I don't think reference [Liu et al., NeurIPS 2023] assumes either \"time-invariant skeleton\" nor \"stationary lag-dependency\" as claimed by authors below Assumptions 1 & 2, no?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The QR-based score function for static skelton learning, as formalized and used in LEVER, and supported by theory in Sections 3.1-3.3, is novel, up to the best of my knowledge.", "weaknesses": "===The theory part===\n\n- Abstract & intro give the impression that the proposed method solves for a \"full\" causal discovery problem, including recovering directions; but objective, theory, and experiments sections mainly solve for \"causal skeleton (adjacency matrix)  recovery\"; while rely on limited connections between the two.  Lemma 1 discusses a bijection between the sets of \"exact\" topological orders & \"complete\" DAGs, i.e., those with all edges directed consistently with a given full ordering. This follows from classic results in graph theory but it does not certainly address the complexity of the \"full\" causal discovery problem, where DAGs could be still recovered based on partial topological orders. Theorems 3&4 determine edge coefficients based on the same assumptions. \n\n- Sec 3.1-3.3 do not seem to model for time-series data including the proposed score. Sections 3.2 & 3.3 explicitly assume i.i.d. samples as stated in the 2nd para of Sec 3.2 ``Formally, let $Y \\in R^{m\\times  d}$ consists of $m (m > d)$ $\\textbf{i.i.d.}$ observations of $d$ random variables''. This assumption is clear in Thm 1 for \"Full column rank\" of data matrix $Y$, the proposed score in Eq (3), and Thms 3&4.  While such an assumption is common for \"static\" causal discovery, it shouldn't be assumed when modeling \"temporal\" dependence.  Sec 3.4 then introduce historical confounding by regressing $X[t]$ on its complete history $\\{X[t−\\tau]\\}_{\\tau \\geq 1}$ and taking the residuals. This is the only place where temporal modeling is assumed. \n\n- There is no explicit relation between \"the theory proposed by authors\" and \"existing work on the same topic\" is provided. Existing work, which provide more rigorous temporal modeling in their theory include: [Mastakouri, Schölkopf, Janzing ICML 21];[Sun,Schulte,Guiliang,Pascal,Poupart AISTATs 23]; [Liu, Sun, Hu, Wang, NeurIPS23]; [Stein,Shadaydeh,Blunk,Penzel,Denzler ICLR 2025]. Only [Sun,Schulte,Guiliang,Pascal,Poupart AISTATs2023] & [Liu, Sun, Hu, Wang, NeurIPS23] are cited in this submission, the other two are not. \n\n=== Experiments === The only temporal baseline considered is [Sun,Schulte,Guiliang,Pascal,Poupart AISTATs2023].", "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": "c0Dm01wJ8f", "reviewer": "Reviewer_Kj1Y", "summary": "The paper considers the problem of causal discovery from time-series data. Specifically, this authors address the case when the system may exhibit long-term carry-over effects, i.e., $X[t]$ may depend on $X[t-h]$ for large or even infinite $h$. Most existing methods assume that the lag order is finite; hence, applying these methods to the considered setting may lead to spurious edges in the recovered causal graph. The authors first show that, for static data (i.e., when only instantaneous causal effects exist), the correct causal ordering can be identified by performing QR decomposition on the observed data. Furthermore, for temporal data, they show that by assuming that the infinite effect function can be summarized by a linear combination of finite effects, the correct causal ordering can be recovered by first regressing $X[t]$ on limited-history data $(X[t-1],\\cdots, X[t-h])$ and then applying the method for static data to the residuals. The authors develop an identification algorithm based on the theoretical results and evaluate its performance on both synthetic and real datasets.", "questions": "1. Should Equation (1) be $g(\\tau) X[t-\\tau]$, since $g(\\tau)$ is a $d\\times d$ matrix?\n2. In the case of static data, is there a reason why the proposed method outperforms existing approaches (e.g., NOTEARS), especially in the simulation results? My understanding is that the considered problems are essentially the same.\n3.  In Section 4.1, does the optimal selection of $w$ always correspond to the $h$ in Assumption 3? If not, I am not sure whether the results in Theorem 6 can be directly applied to the residual $\\dot{\\mathbf{x}}$ in Equation (7).\n4. Can the DQN framework (i.e., steps 2 and 3 in Figure 2) be replaced by existing causal structure learning algorithms?\n5. Is the existence of time-dependent effects (i.e., whether the data are static or temporal) provided as an input under the different settings in the simulation results?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. Clear explanation of all theoretical results, along with motivating examples and detailed proofs.\n2. The overall presentation is clear and easy to follow.\n3. The authors conduct extensive simulations to demonstrate the performance of the proposed algorithm.", "weaknesses": "The main weakness is that the considered setting substantially overlaps with existing work on causal structure learning. In the case with static data, the problem reduces to causal discovery in linear SCMs with i.i.d. noise distributions. It has already been shown that the model is uniquely identifiable under either Gaussian or non-Gaussian noise distributions, and many existing algorithms can be applied to this setting for causal structure learning. In the case of temporal data, the authors show that, under Assumptions 1, 2 and 3, there exists an $h$ such that the residuals can be written in the form of a linear SCM with equal noise variances (line 901). Further, in the proposed algorithm, it seems that the RL component could also be replaced by existing algorithms (see Q4 below).", "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": "DEioWWqhpR", "reviewer": "Reviewer_EqNz", "summary": "This paper addresses the challenge of causal discovery in time series systems that exhibit long-term carry-over effects, where past influences persist far beyond a limited time lag and can create historical confounding. The authors show that instead of modeling the entire history, it is sufficient to regress out only a small recent window of past observations to obtain refined residual signals that preserve causal identifiability. Building on this result, they propose LEVER, a causal discovery algorithm that uses QR decomposition to represent variable relationships and employs reinforcement learning to efficiently search for the optimal causal ordering. Once the order is determined, causal edges are estimated and pruned to recover the final causal graph. Experiments on both synthetic and real-world datasets demonstrate that LEVER outperforms existing methods in accuracy and efficiency, particularly in scenarios with long-term temporal dependencies.", "questions": "1. How should practitioners choose the window size w in settings where the decay pattern of historical effects is unknown?\n2. Can LEVER be extended to handle nonlinear causal relationships?\n3. How does LEVER perform when some variables have very weak or delayed effects that only appear far beyond the chosen history window?\n4. Is the learned causal ordering stable under subsampling?\n5. Could LEVER be combined with domain knowledge, and how would that be incorporated into the RL reward or state representation?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Addresses an important problem, long-term historical confounding, which is not adequately addressed in existing temporal causal discovery methods.\n2. Instead of requiring full historical data (which is often unavailable, expensive, or noisy), the method works only with a small recent window, which is practically useful.\n3. The method can work in both static and temporal settings.\n4. The motivation and algorithmic pipeline are clearly explained.", "weaknesses": "1. The method has been evaluated only on one real-world dataset. Broader validation would strengthen claims of general applicability.\n2. The method assumes the relationships between variables are linear and time-invariant. But in many real-world systems, relationships are often nonlinear.\n3. The method requires choosing the window size, yet no automatic way to select it is given.", "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)