Estimating heterogeneous single and interaction treatment effects from observational data under multiple simultaneous treatments is crucial for decision-making. To mitigate estimation variance, previous studies balance representation distributions between every pair of treatment patterns. However, such pairwise balancing scales quadratically with the number of treatment patterns and fails to preserve consistent local proximity structures across patterns, which degrades counterfactual estimation. To address these challenges, we propose the Causal Inference for Heterogeneous Single and Interaction Treatment Effects Network (CIHSI-Net), a deep learning framework built on a novel Barycentric Fused Gromov-Wasserstein Balancing (BFG-WB) objective. BFG-WB aligns the representation distribution of each treatment pattern with a shared Wasserstein barycenter, achieving global alignment while reducing the computational complexity from quadratic to linear, and its Fused Gromov-Wasserstein discrepancy preserves the local proximity structures essential for reliable heterogeneous effect estimation. Simulation studies show that CIHSI-Net consistently outperforms state-of-the-art baselines, and an application to real-world marketing data demonstrates its practical utility in complex multi-treatment scenarios.
Figures & tables
Figure 1: Illustration of neighborhood switching under Wasserstein-based representation balancing. The gray arrows show OT correspondences. Wasserstein minimization aligns the two distributions but can map neighboring points to distant locations, distorting within-distribution neighborhood relations; a structure-preserving match such as j↔j′ would be desirable.
Figure 2: Conceptual illustration of BFG-WB. The gray arrows show a subset of OT correspondences. By evaluating the FGW discrepancy, the alignment is encouraged to preserve within-distribution neighborhood relations.
Figure 3: The architecture of CIHSI-Net. The model consists of three components: the representation learning network ϕ , the task embedding network tw , and the outcome prediction network h . BFG-WB is a regularization term applied to the latent representation ϕ(x) .
ϵCASE
ϵCAIE
Sim.
Method
k=1
k=2
k=3
S={1,2}
S={2,3}
S={1,3}
S={1,2,3}
1
TECE-VAE
1.82 ± 0.09
1.91 ± 0.10
1.80 ± 0.14
3.58 ± 0.11
3.27 ± 0.10
3.30 ± 0.10
6.80 ± 0.18
NCoRE
0.28 ± 0.06
0.30 ± 0.06
0.25 ± 0.05
0.52 ± 0.10
0.33 ± 0.10
0.37 ± 0.10
0.56 ± 0.14
CISI-Net
0.22 ± 0.04
0.24 ± 0.03
0.19 ± 0.06
0.31 ± 0.07
0.11 ± 0.04
0.15 ± 0.08
0.31 ± 0.08
CIHSI-Net
0.19 ± 0.04
0.21 ± 0.04
0.18 ± 0.10
0.28 ± 0.05
0.08 ± 0.05
0.12 ± 0.04
0.24 ± 0.04
2
TECE-VAE
1.60 ± 0.05
1.65 ± 0.09
1.58 ± 0.16
3.01 ± 0.05
2.65 ± 0.08
2.69 ± 0.09
5.50 ± 0.15
Table 1: Comparison of mean estimation errors and standard deviations for ϵCASE and ϵCAIE in two simulation datasets. Here, k∈{1,2,3} indexes individual treatments, and S⊆{1,2,3} (with ∣S∣≥2 ) denotes treatment combinations.
No.
WB
Penalty type
Average ϵCASE
Average ϵCAIE
1
✗
None
0.27 ± 0.06
0.24 ± 0.02
2
✗
W
0.22 ± 0.03
0.22 ± 0.03
3
✗
GW
0.25 ± 0.06
0.24 ± 0.03
4
✗
FGW
0.22 ± 0.02
0.23 ± 0.02
5
✓
W
0.22 ± 0.04
0.21 ± 0.02
6
✓
GW
0.22 ± 0.04
0.22 ± 0.03
Table 2: Ablation study results on simulation dataset 1. Average ϵCASE and Average ϵCAIE denote the means of the reported ϵCASE and ϵCAIE values, respectively. WB indicates whether the Wasserstein barycenter is used. Penalties: Wasserstein (W), Gromov-Wasserstein (GW), Fused GW (FGW), and None.
Figure 4: Estimated CASE (top) and CAIE (bottom) across user groups stratified by pre-promotion service usage.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Hyperparameter
Candidate values
Selected value
BFG-WB coefficient α
{0.1,0.5,1.0,1.5,2.0,5.0}
1.0
FGW trade-off η
{10−5,0.1,0.2,…,0.9,1.0}
0.6
Number of units per FC hidden layer
{20,50,100}
100
Appendix
Table A1: Hyperparameter search ranges and selected values.
Method
K=2
K=3
K=4
K=5
K=6
K=8
CISI-Net
5.98
15.08
31.09
138.54
125.62
1529.83
CIHSI-Net
8.21
9.02
13.92
22.71
23.01
48.32
Appendix
Table A2: Wall-clock training time per epoch (s/epoch) for CISI-Net (pairwise balancing) and CIHSI-Net (BFG-WB), evaluated with K∈{2,3,4,5,6,8} treatments under the same hardware and training configuration. For each K , both methods are trained with the same batch size, while the batch size is adjusted across different values of K . Times are averaged across all observed epochs in 100 trials with early stopping (max 300 epochs, patience = 15).
Figure A1: Average ϵCASE (left) and ϵCAIE (right) under varying numbers of treatments K∈{2,4,6,8} .
ϵCASE
ϵCAIE
No.
WB
Penalty type
k=1
k=2
k=3
S={1,2}
S={2,3}
S={1,3}
S={1,2,3}
1
✗
None
0.28 ± 0.08
0.29 ± 0.04
0.25 ± 0.09
0.36 ± 0.03
0.12 ± 0.04
0.16 ± 0.03
0.31 ± 0.04
2
✗
W
0.22 ± 0.04
0.24 ± 0.03
0.19 ± 0.06
0.31 ± 0.07
0.11 ± 0.04
0.15 ± 0.08
0.31 ± 0.08
3
✗
GW
0.25 ± 0.03
0.28 ± 0.04
0.23 ± 0.10
0.35 ± 0.03
0.12 ± 0.06
0.16 ± 0.04
0.32 ± 0.05
4
✗
FGW
0.23 ± 0.04
0.25 ± 0.03
0.19 ± 0.05
0.33 ± 0.04
0.12 ± 0.06
0.16 ± 0.04
0.31 ± 0.04
5
✓
W
0.22 ± 0.03
0.24 ± 0.03
0.19 ± 0.08
0.30 ± 0.04
0.10 ± 0.05
0.15 ± 0.09
0.28 ± 0.07
Appendix
Table A3: Ablation study result on simulation dataset 1. WB indicates whether the Wasserstein barycenter is used. Penalties: Wasserstein (W), Gromov-Wasserstein (GW), Fused GW (FGW), and None.
Figure A2: Distribution of pairwise Wasserstein (left) and Gromov-Wasserstein distances (right) between representation distributions for all unique treatment pairs. We compare CISI-Net (without balancing), CISI-Net (with pairwise balancing), and CIHSI-Net (ours). Lower values indicate better alignment across treatment patterns.
Figure A3: Sensitivity analysis of hyperparameters α and η on simulation dataset 1. The top row shows ϵCASE(1) - ϵCASE(3) , the middle row shows ϵCAIE({1,2}) - ϵCAIE({1,3}) , and the bottom panel shows ϵCAIE({1,2,3}) .
ϵCASE
ϵCAIE
Method
k=1
k=2
k=3
S={1,2}
S={2,3}
S={1,3}
S={1,2,3}
TECE-VAE
1.23 ± 0.09
1.25 ± 0.06
1.28 ± 0.07
3.22 ± 0.46
1.82 ± 0.08
1.62 ± 0.17
4.00 ± 0.34
NCoRE
0.98 ± 0.16
0.98 ± 0.07
0.78 ± 0.05
2.94 ± 0.48
1.29 ± 0.28
1.37 ± 0.14
5.73 ± 1.20
CISI-Net
0.79 ± 0.01
1.13 ± 0.04
0.96 ± 0.12
1.49 ± 0.14
1.21 ± 0.21
0.96 ± 0.02
1.49 ± 0.11
CIHSI-Net
0.81 ± 0.02
1.02 ± 0.07
0.78 ± 0.07
1.45 ± 0.07
1.09 ± 0.08
0.96 ± 0.04
1.45 ± 0.06
Appendix
Table A4: Semi-synthetic experiment results using Dataset A’s real covariates and observed treatment assignments. Bold values indicate the best performance, and underlined values indicate the second-best performance for each metric.
Figure A4: Sensitivity and uncertainty analysis of the estimated causal effects in the real-world application on Dataset A. The rows correspond to the BFG-WB regularization strength α∈{0.5,1.0,1.5} , and the columns correspond to the FGW trade-off parameter η∈{0.5,0.6,0.7} . In each panel, the top subpanel reports the estimated CASE and the bottom subpanel reports the estimated CAIE across user groups. The shaded regions represent bootstrap confidence intervals.
Figure A5: In dataset B, estimated CASEs for CP 4 and CP 5 (left) and the estimated CAIE for {CP\textsubscript4,CP\textsubscript5} (right) across user groups stratified by pre-promotion service usage. The outcomes are standardized.