Coarse-to-Fine Learning of Dynamic Causal Structures¶
Conference: ICLR2026
OpenReview: https://openreview.net/forum?id=ooqnLFagKq
Code: To be confirmed
Area: Causal Discovery / Temporal Causal
Keywords: Dynamic causal structures, Granger causality, acyclicity constraints, linear interpolation, time-varying graphs
TL;DR¶
This paper proposes DyCausal, which utilizes sliding convolutional windows to first capture "coarse-grained" causal structures of time series across multiple time steps. It then refines the causal matrices for each time step using first-order Taylor linear interpolation, coupled with an "always differentiable" acyclicity constraint \(h_\text{norm}\) based on matrix 1-norm scaling. This approach enables the stable and efficient recovery of fully dynamic (both instantaneous and lagged causality vary over time) time-varying causal graphs for the first time, significantly outperforming existing methods on both synthetic and real-world data.
Background & Motivation¶
Background: Discovering causal graphs from time series (temporal causal discovery) is a core task in causal learning. Mainstream approaches are divided into constraint-based (PC/FCI with conditional independence tests) and score-based (vector autoregression + sparse regularization, such as NOTEARS / DYNOTEARS / NTS-NOTEARS). Score-based methods have become dominant recently because they can be integrated with deep models and optimized end-to-end using differentiable acyclicity constraints.
Limitations of Prior Work: Most existing methods assume that causal relationships are static—while the data distribution of a time series may change, the underlying causal structure (which edges exist and their strengths) is assumed to be time-invariant. However, in reality, causal relationships can "emerge, change, or disappear": for instance, a patient's disease trajectory may shift with seasons, or traffic causality during morning and evening peaks can be entirely different. A few recent works (e.g., DyCAST) only allow partial causal components (only instantaneous or only lagged) to vary over time, while still imposing static assumptions on the remaining components, which contradicts the fully dynamic nature of reality.
Key Challenge: To achieve "fully dynamic" causal discovery (where both instantaneous and lagged causality change step-by-step), there are two unavoidable obstacles. The first is efficiency and stability—learning an individual causal graph for every time step leads to a parameter explosion and convergence difficulties; models often fail to converge when the number of nodes is large (e.g., 80 nodes). The second is the acyclicity constraint—causal graphs require the instantaneous part to be a DAG, but existing differentiable constraints (\(h_\text{exp}\), \(h_\text{poly}\), \(h_\text{log}\)) either suffer from numerical explosion/vanishing or are only optimizable within a limited spectral radius range. Crossing these boundaries necessitates restarting training with a lower learning rate, which is slow and unstable.
Key Insight: The authors leverage a key observation—since causality evolves continuously and smoothly over time, identifying coarse-grained states at a few time steps should, in principle, allow for the approximation of the entire time-varying trajectory. This transforms "hard step-by-step learning" into "coarse sampling + interpolation refinement," which is both computationally efficient and stable.
Core Idea: Modeling time-varying causal trajectories in a "coarse-to-fine" two-stage manner—convolutional windows encode coarse-grained states, linear interpolation completes each step, and an always-differentiable norm-scaled acyclicity constraint is applied to recover fully dynamic causal graphs efficiently and stably.
Method¶
Overall Architecture¶
DyCausal aims to solve the following: given \(N\) time series of equal length sharing the same time-varying causal structure \(X=\{X^1_{1:T},\dots,X^N_{1:T}\}\) (with \(d\) variables per time step), it outputs the weighted adjacency causal matrix \(W_t\) for each time step \(t\). This is decomposed into an instantaneous part \(W^\text{ins}_t\) and a lagged part \(W^\text{lag}_t\) (with maximum lag \(\tau\)), where \(W^\text{ins}_t\) must be a DAG.
The overall pipeline consists of three stages: "Coarse Encoding → Fine Interpolation → Reconstruction Optimization": ① A sliding convolutional window of width \(N\) and length \(K\) scans the sequence with stride \(S\). At each position, a CNN encodes the data into a coarse-grained state vector \(Z_t\), which is then "parallel decoded" into a causal matrix—thus only \(\lfloor (T-K)/S\rfloor\) matrices need to be computed; ② First-order Taylor linear interpolation is applied between adjacent coarse matrices to complete the causal matrices for every time step, obtaining a fine-grained time-varying trajectory; ③ An end-to-end optimization is performed using a score-based objective (reconstructing "effect" from "cause" + \(\ell_1\) sparsity + \(h_\text{norm}\) constraint), followed by pruning small elements to obtain the dynamic causal graph \(G\).
%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
A["时间序列 X<br/>N 条 · 长 T · d 变量"] --> B["粗到细编码<br/>滑动卷积窗 + 并行解码<br/>得粗粒度 Wt"]
B --> C["线性插值细化<br/>一阶 Taylor 补全每个时间步"]
C --> D["范数缩放无环约束<br/>hnorm：1-范数缩放 W∘W"]
D --> E["重构优化<br/>因→果重构 + ℓ1 + 中心路径"]
E -->|阈值 δ 剪枝| F["动态因果图 G = {Gt}"]
Key Designs¶
1. Coarse-to-Fine Encoding: Sliding Window + Parallel Decoding to reduce "Step-by-Step Learning" to "Few Steps + Interpolation"
To address the parameter explosion and convergence issues of learning causal graphs for every time step, DyCausal does not learn a graph at every step. Instead, it uses a sliding window of width \(N\), length \(K\), and stride \(S\) to scan the sequence. Each window \(X_{t:t+K}\) is encoded into a state vector \(Z_t=\text{CNN}(X_{t:t+K})\in\mathbb{R}^{dm}\), a coarse-grained representation of the causal structure. \(Z_t\) is then decoded into the causal matrix \(W_t\in\mathbb{R}^{dm\times d\tau}\).
Direct decoding would lead to a parameter complexity of \(\mathcal{O}(d^3m^2\tau)\). Therefore, the authors designed a two-level parallel decoding scheme: \(Z_t\) is first split into \(m\) segments \(\{z_{t,i}\in\mathbb{R}^d\}\) and decoded into \(\{\tilde z_{t,i}\in\mathbb{R}^{d\tau}\}\) using \(m\) parallel MLPs. Then, elements at the same positions are reorganized into \(d\tau\) vectors \(\{u_{t,j}\in\mathbb{R}^m\}\) and decoded into \(\{\tilde u_{t,j}\in\mathbb{R}^{dm}\}\) using \(d\tau\) parallel MLPs, which are concatenated to form \(W_t=[\tilde u_{t,1}|\cdots|\tilde u_{t,d\tau}]\). This reduces the parameters from \(\mathcal{O}(d^3m^2\tau)\) to \(\mathcal{O}(d^2m\tau+d^2m^2\tau)\) and the computational complexity to \(\mathcal{O}(d^2m^2\tau)\). For linear causality, \(m=1\); for nonlinear, \(m>1\) represents the number of hidden units.
2. Fine Interpolation: First-order Taylor to complete every step from coarse states
The causal matrices obtained in the previous step are separated by \(S\) time steps. Based on the assumption that causality evolves continuously and smoothly, the authors posit that if \(S\) is sufficiently small, the change in the causal matrix within the interval \([t, t+S]\) can be approximated as a deterministic linear function using the first-order Taylor formula:
This is equivalent to linear interpolation between two coarse matrices. Consequently, encoding and decoding are not required for every step—calculating only \(\lfloor (T-K)/S\rfloor\) matrices allows for the estimation of matrices for all time steps. This "coarse-to-fine" approach saves computation: even with \(S\ge 2\), the workload is reduced by more than half, while interpolation ensures smooth and stable trajectories. Because both instantaneous and lagged parts are interpolated simultaneously, DyCausal can model fully dynamic causality.
3. Norm-scaled acyclicity constraint \(h_\text{norm}\): Making the constraint "always differentiable and optimizable"
To solve the issues of existing acyclicity constraints (explosion/vanishing or boundary constraints), the authors modified the log-det type constraint. While \(h_\text{log}=-\log\det(\alpha I-W\circ W)+d\log\alpha\) is more efficient than \(h_\text{exp}\) or \(h_\text{poly}\), it is only optimizable in the limited space \(\{W:\rho(W\circ W)<\alpha\}\). Once the spectral radius exceeds \(\alpha\), retraining is required. Using the fact that "matrix norm \(\ge\) spectral radius," the authors scaled \(W\circ W\) by its 1-norm:
Where \(\alpha\) is a constant slightly larger than 1 (e.g., \(1.001\), denoted as \(1^+\)), requiring no additional tuning. Because \(\rho\big(\tfrac{W\circ W}{\|W\circ W\|_1}\big)<1^+\) always holds, the scaled matrix always stays within the optimizable space, eliminating retraining and hyperparameter searching. Key property (Theorem 1): \(h_\text{norm}\ge 0\) and \(=0\) if and only if \(W\) is a DAG—since \(\tfrac{W\circ W}{\|W\circ W\|_1}\) shares the same cycle characteristics as \(W\circ W\). Theorem 2 further proves \(h_\text{norm}\) satisfies three stability criteria: E-stability (no explosion to infinity), V-stability (no rapid vanishing to 0), and D-stability (constraint and gradients are computable almost everywhere). This is particularly important because directly using the spectral radius \(\rho(A)\) yields uncomputable gradients during repeated eigenvalues, an issue \(h_\text{norm}\) avoids.
Loss & Training¶
The observed sequence is rearranged into a "cause" sequence \(X^c=\{X^c_t:=[X_t|\cdots|X_{t-\tau}]\}\) and an "effect" sequence \(Y=\{Y_t:=X_t\}\). The score-based objective requires the causal model to accurately reconstruct effects from causes while satisfying the acyclicity constraint (when \(m=1\), \(\hat Y_{t:t+K}=W_t X^c_{t:t+K}\)):
The problem is solved using the central path method: the reconstruction and sparsity objective is multiplied by a coefficient \(\mu\) and added with \(h_\text{norm}\) as a regularizer. In each iteration, \(\mu\) is decreased (default decay \(\gamma=0.1\), iterations \(r=4\)) before minimization. Finally, a threshold \(\delta\) is used to prune small elements to obtain \(G\). The method is flexible: for nonlinear cases, \(m>1\), \(W_t\) acts as the first layer of the reconstruction network, and \(\tilde W_t\) is obtained by \(\ell_2\) aggregation. for ODE models, it uses incremental reconstruction \(\hat Y_{t:t+K}=\hat Y_{t-1:t-1+K}+\text{MLP}(\sigma(W_tX^c_{t:t+K}))\), and since ODEs do not involve instantaneous causality, \(h_\text{norm}\) is removed and \(r=1\).
Key Experimental Results¶
Main Results¶
Synthetic data utilized ER random graphs for ground truth, with edge weights set as sine/cosine functions to simulate dynamics. Comparisons included DyCAST (partial dynamic), DYNO (DYNOTEARS), and NTS-NO (NTS-NOTEARS, static). Evaluation was conducted at the start, middle, and end time steps for large graphs with \(T=50\) and nodes \(d=\{20, 80\}\):
| Time Step | Method | 20-node F1↑ | 20-node SHD↓ | 80-node F1↑ | 80-node SHD↓ |
|---|---|---|---|---|---|
| \(W_1\) (Start) | DyCausal | 91.43 | 7.40 | 93.85 | 18.20 |
| DYCAST | 29.51 | 45.43 | — (Non-conv) | — | |
| DYNO | 3.52 | 78.40 | 0.34 | 294.90 | |
| \(W_{25}\) (Mid) | DyCausal | 94.38 | 4.10 | 95.20 | 13.40 |
| DYNO | 69.20 | 25.00 | 72.81 | 82.90 | |
| \(W_{50}\) (End) | DyCausal | 90.38 | 8.20 | 92.52 | 22.00 |
| DYCAST | 28.64 | 44.14 | — | — |
Key observation: Static methods like DYNO/NTS-NO fail significantly at the boundaries of the sequence (confusing boundary and middle distributions). DyCAST, due to its ODE-based trajectory, fails to identify causality with drastic weight changes (opposite signs at boundaries) and fails to converge on 80 nodes. Only DyCausal maintains high accuracy across all segments in fully dynamic scenarios.
Real-world data (CausalTime: Traffic, AQI, and Medical), with additional CUTS+ and JRNGC baselines:
| Method | Traffic F1↑ | AQI F1↑ | Medical F1↑ | Traffic Precision↑ |
|---|---|---|---|---|
| DyCausal | 54.81 | 63.16 | 56.86 | 87.41 |
| CUTS+ | 45.21 | 58.20 | 30.23 | 80.10 |
| JRNGC | 52.04 | 55.36 | 52.05 | 68.04 |
| NTS-NO | 46.07 | 40.09 | 34.27 | 67.43 |
Ours ranked first in Precision and F1 across all scenarios. High Precision is particularly valuable in real-world data without verifiable ground truth as it provides more credible edges. Visualization in the Traffic subset clearly shows causality splitting into two distinct modes at \(t=20\) (e.g., \(x_4 \to x_8\) strength is near zero before \(t=20\) and emerges thereafter), proving it captures real dynamic changes that other algorithms discard as noise.
Ablation Study¶
| Configuration | Conclusion |
|---|---|
| w/o Linear Interpolation (Appendix C.2) | Interpolation is essential for identifying dynamic causality; removing it prevents recovery of time-varying trajectories. |
| Acyclicity Contrast: \(h_\text{norm}\) vs \(h_\text{exp}/h_\text{poly}/h_\text{log}/h_\rho\) (Fig. 4) | As \(k\) and \(d\) increase, \(h_\text{exp}/h_\text{poly}\) explode or vanish (E/V unstable); \(h_\text{log}\) degrades rapidly on cyclic graphs as \(d\) increases; \(h_\rho\) is stable but has tiny gradients on cyclic graphs and exponential runtimes. Only \(h_\text{norm}\) is stable and efficient in all settings and penalizes cycles best. |
| \(h_\text{norm}\) vs \(h_\text{log}\) (Appendix C.10) | Confirms that the improvement of \(h_\text{norm}\) is effective and necessary. |
Key Findings¶
- Linear interpolation is the "lifeline" for fully dynamic discovery: Removing it reverts the model to step-by-step learning, which is expensive and hard to converge. "Coarse sampling + interpolation" allows even 80-node graphs to converge stably (where DyCAST fails).
- Boundaries best distinguish method quality: Static or partially dynamic methods collapse at the start and end of sequences due to drastic weight changes (sign inversion). Ours shows its strongest advantage at these boundaries.
- "Always optimizable" acyclicity constraints bring more than just accuracy: They eliminate retraining and hyperparameter search, and also provide significant acceleration in static causal discovery (Appendix C.9).
Highlights & Insights¶
- The "Continuous smoothness \(\Rightarrow\) reconstructible via coarse sampling" observation is clever: It transforms a combinatorial explosion problem of "step-by-step graph learning" into "few key steps + Taylor interpolation." This is a successful application of signal continuity to causal discovery, and by interpolating instantaneous and lagged parts together, it achieves true fully dynamic modeling for the first time.
- Norm-scaling for acyclicity is a reusable trick: A single matrix inequality ("matrix norm \(\ge\) spectral radius") expands the optimizable space of \(h_\text{log}\) to the entire space while keeping cycle characteristics and gradient zeros unchanged. Any differentiable causal discovery work using log-det constraints could adopt this.
- A single framework handles linear, nonlinear, and ODE models: By simply adjusting \(m\) and the reconstruction formula (adding MLP for nonlinear, using incremental reconstruction and removing \(h_\text{norm}\) for ODE), the framework demonstrates strong generalization.
Limitations & Future Work¶
- Strong dependence on the "smooth continuity" assumption: Linear interpolation relies on the premise that causality changes smoothly and \(S\) is small enough. Although the paper claims acceptable accuracy on non-smooth dynamics (Appendix C.12), abrupt jumps are naturally disadvantageous for first-order Taylor approximation.
- Assumption of shared causal structure across all sequences: In reality, different individuals or regions may have different causal mechanisms. This homogeneity assumption limits the scope of application.
- Instantaneous DAG requirement vs. unconstrained lagged parts: This relies on the assumption that "lagged causality naturally does not form cycles over time." If hidden confounders or feedback loops exist, the reliability of recognition might be questioned.
- Potential improvements: Replacing the fixed stride \(S\) with an adaptive one (denser sampling where causality changes rapidly), or using higher-order interpolation/splines to handle non-smooth dynamics.
Related Work & Insights¶
- vs DyCAST (Partial dynamic): DyCAST uses ODEs for trajectories and only allows one part (instantaneous or lagged) to be dynamic. Ours uses "convolutional windows + linear interpolation" to make both parts fully dynamic. DyCAST fails under weight inversions and non-convergence at 80 nodes, whereas Ours is more stable and accurate due to coarse encoding.
- vs DYNOTEARS / NTS-NOTEARS (Static temporal causality): These use differentiable acyclicity constraints for static autoregressive discovery. They are insensitive to changing distributions and collapse at sequence boundaries. Ours inherits the score-based + acyclicity framework but adds coarse-to-fine time-varying modeling and replaces the constraint with \(h_\text{norm}\).
- vs \(h_\text{log}\) (Bello et al. 2022): Ours performs norm-scaling on this basis, changing "limited optimizable space and retraining" into "always differentiable and optimizable" while retaining its beneficial properties and adding E/V/D stability.
- vs \(h_\rho\) (Nazaret et al. 2024): \(h_\rho\) is stable but suffers from vanishing gradients on cyclic graphs and exponential runtime. \(h_\text{norm}\) avoids this D-instability issue via norm-scaling.
Rating¶
- Novelty: ⭐⭐⭐⭐⭐ First to perform fully dynamic discovery; both coarse-to-fine and norm-scaled constraints are novel and self-consistent.
- Experimental Thoroughness: ⭐⭐⭐⭐ Synthetic (multi-node/long-sequence/static-dynamic) + four real datasets + specific acyclicity comparisons; detailed in appendix, but main text tables are sparse.
- Writing Quality: ⭐⭐⭐⭐ Motivation, obstacles, and methods are logically clear; theorems and stability analysis are solid. Symbols are dense, and parallel decoding details require the diagram for full understanding.
- Value: ⭐⭐⭐⭐ Fully dynamic causal discovery is practical for time-varying systems (medical/traffic/finance). Norm-scaled constraints are widely reusable.