The Final-Stage Bottleneck: A Systematic Dissection of the R-Learner for Network Causal Inference
Authors: Sairam S
Sara Girdhar
Shivam Soni
Venue: Accepted by TMLR ()

Abstract: The R-Learner is a powerful, theoretically-grounded framework for estimating heterogeneous treatment effects, prized for its robustness to nuisance model errors. However, its application to network data—where causal heterogeneity may be driven by graph structure—presents critical and underexplored challenges to its core assumption of a well-specified final-stage model. In this paper, we conduct a large-scale, multi-seed empirical study to systematically dissect the R-Learner framework on graphs. Our results suggest that for network-dependent effects, a critical driver of performance is the inductive bias of the final-stage CATE estimator, a factor whose importance can dominate that of the nuisance models.
Our central finding is a systematic quantification of a "representation bottleneck": we demonstrate empirically and through a constructive theoretical example that graph-blind final-stage estimators, being theoretically misspecified, exhibit significant under-performance (MSE > 4.0, p < 0.001 across all settings). Conversely, we show that an R-Learner with a correctly specified, end-to-end graph-aware architecture (the "Graph R-Learner") achieves a significantly lower error. 
Furthermore, we provide a comprehensive analysis of the framework’s properties. We identify a subtle "nuisance bottleneck" and provide a mechanistic explanation for its topology dependence: on hub-dominated graphs, graph-blind nuisance models can partially capture concentrated confounding signals, while on graphs with diffuse structure, a GNN’s explicit aggregation becomes critical. This is supported by our analysis of a "Hub-Periphery Tradeoff," which we connect to the GNN over-squashing phenomenon. Our findings are validated across diverse synthetic and semi-synthetic benchmarks, where the R-Learner framework also significantly outperforms a strong, non-DML GNN T-Learner baseline. We release our code as a comprehensive and reproducible benchmark to facilitate future research on this critical "final-stage bottleneck."

Published in Transactions on Machine Learning Research (04/2026)
The Final-Stage Bottleneck: A Systematic Dissection of the R-Learner for Network Causal Inference
Sairam S
pes pes/uug2Smm22770pesss
Deportment of Computer Science (AIBML)
PES University
Sara Girdhar
pes pussu
Department Computer Science (AIEML)
PES University
Shivam Soni
pesluugBBemm2287@yessu.uuu edu
Department of Computer Science (A18ML)
PES University
Reviewed on OpenReview: oppereveep, net/ forum? QIEOFVSnOP
Abstract
The R-Learner is powerful, theoretically-grounii framework for estimating heteroge- neous trealment effects (HTE), prized for its robustness to nuisance model errors. How- ever, its application to network data-where causal heterogeneity may be driven by graph structure presents critical and underexplored challenges to its core assumption of a well- specified final-stage model. In this paper, we conduct large-scale, multi-sced empirical study to systematically dissect the R-Tearner framework on graphs. Our results suggest that for network-dependen effects, critical driver of performance is the inductive bias of the final-stage CATE estimator, a factor whose importance can dominate that of the muisance models.
Our central finding is a systematic quantification of a "representation bottleneck': we demonstrate empirically and through constructive theoretical example that graph-bling blind final-stage estimators, being theoretically misspecified, exhibit significant underperformance (MSE > 4.0, p 0.001 across all settings). Conversely, we show that an R-Learner with correctly specified, end-to-end graph-aware architecture (the 'Graph R-Learner") achieves significantly lower error.
Furthermore, provide a comprehensive analysis of the framework's properties. We iden- tify subtle 'nuisance bottleneck' and provide mechanistic explanation for its topology- dependence: on hub=dominated graphs, graph-blind nuisance models can partially cap- ture concentrated confounding signals, while on graphs with diffuse structure, GNN's explicit aggregation becomes critical. This supported by our analysis of a 'Hub-Periphery Trade-off," which we connect the GNN over-squa phenomenon. Our findings are validated across diverse synthetic and semi-synthetic benchmarks, where the R-Learner framework also significantly outperforms a strong, non-DML GNN T-Learner baseline. We release our code as comprehensive and reproducible benchmark to facilitate future research on this critical "final-stage bottleneck' here: https: thub. mtall-staggebbttlennek
1 Introduction
The Couble/Debiased Machine Learning (DML) framework, and its R-Learner instantiation, has emerged
cornerstone of modern causal inference (Chernozhukov al. 2018 Nie & Wager 2020). Tts theoreti-

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cal power stems from the principle of Neyman-orthooonnlliyy which constructs objective function where the final causal estimate is robust to first-order errors in the nuisance models for the outcome and treat- ment propensity. The standard and correct narrative surrounding these methods, therefore, emphasizes the importance of using high-quality, flexible machine learning models for these nuisance components.
However, the application of this framework to complex, non-i.i.d. settings like network data introduces new challenges. Specifically, the R-Learner's robustness guarantees rest on critical assumption: that the function class chosen for the final-stage CATE model, r(X), is correctly specified. In this paper, investigate the profound and perhaps under-appreciated consequences of violating this assumption in network settings where the true causal effect is graph-dependent.
Our work suggests a refinement to the conventional DML narrative for this domain. Through a large-scale, multi-seed empirical study, we provide strong evidence that while muisance model quality is important, the primary driver of performance the inductive bias of the final-stage CATE estimator. We find that the choice of the final-stage model can be first-order effect, whose impact can dominate that of the muisance models.
We systematically quantify a catastrophic "representation bottleneck": R-Learners with final-stage estimator that is blind to the graph structure exhibit significant underperformance even when paired with powerful, graph-aware Graph Neural Network (GNN) nuisance models. Conversely, an R-Learner with a correctly specified, graph-aware GNN final stage performs well, even when paired with sub-optimal, graph- blind nuisance models. Our work thus clarifies the modeling challenge: for network data, the most critical architectural choice is ensuring the final-stage estimator can represent the underlying causal mechanism.
To establish this, conduct comprehensive dissection of the R-Learner framework on suite of synthetic and semi-synthetic benchmarks. Our contributions are threefold:
We provide a rigorous, empirical quantification of the final-stage bottleneck, showing with high statistical significance (p 0.001) that misspecified final stage dominant failure mode.
2. We identify and provide mechanistic explanation for novel, topology-dependent "nuisance bot- tleneck," linking it to the GNN over-squashing phenomenon via targeted 'Hub-Periphery' error analysis.
We release our entire experimental framework as reproducible benchmark, including a full suite of ablations and a strong, non-DML GNN T-Learner baseline, to facilitate future research in this emerging domain.
Ultimately, this paper serves as a critical clarification for researchers and practitioners. We provide a clear, emppirically-baceed "hierarchy needs' for network causal inference and demonstrate that success is determined not by the quality of the nuisance models alone, but first and foremost by correctly specifying the function class of the final-stage estimator.
2 Related Work
Our work situated at the intersection of three major lines of research: the statistical framework of Dou- ale/Debiased Machine Learning (DML.,, the algorithmic development of causal meta-learners, and the emerg- ing field of causal inference on graphs.
2.1 Double/Debiased Machine Learning and the R-Learner
A central challenge in modern causal inference is using high-capacily models without introducing bias from their inherent regularization. The DML framework provides general, Aleoretically-grounde solution to this problem (Chernozhukov et al. 2018). To provide intuition for this mechanics, consider the goal of isolating the effect of treatment T on an outcome Y in the presence of high-dimensional confounders X. In observational settings, T and Y are often correlated with X, creating a "spurious' relationship. A naive model might use
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X to predict Y, but regularization in high-dimensional settings often leads to "omitted variable bias" where the model fails to fully control for X (Nie & Wager 2020). addresses this via Neyman-orthogonlliiy which constructs an objective function that is robust (to a first order) to errors in intermediate 'nuisance" models. The process follows a two-stage logic:
1. Nuisance Estimation: We train two separate models: m(X) [| (the outcome muisance) and p(X) = X] (the treatment propensity). These models capture the parts of Y and T that are "explained" by the confounders.
2. Residualization: We compute the residuals: Y Y mx() and T = T p(X). These residuals represent the variation in outcome and treatment that cannot be explained by local or network- based confounders.
3. Causal Estimation: By regressing the outcome residual on the treatment residual, we effectively 'subtract out" the confounding influence, isolating the clean causal signal.
Our work investigates a specific instant in of this framework: the R-Learner (Nie & Wager 2020 The R-Learner minimizes the following "residual-on-resice objective:
f = aaa mint Σ -i=10 -(- (Z:-)(X))7(X.)))
(1)
While DML provides robustness to muisance errors, its theoretical guarantees rest on critical assumption: that the function class chosen for the final-stage Conditional Average Treatment Effect (CATE) model, r(X), is correctly Opecified.Our paper provides the first systematie, empirical dissection of the profound consequences of violating this assumption in the context of network data- failure we term the Represen- tation Bottleneck.
2.2 Causal Meta- and Tree- Based Methods
The R-Learner belongs to a broader family of 'meta-lcarners' for estimating HTE Künzel et al. 2019). which provide general recipes for applying supervised learning models to the task of causal estimation. Alternative strategies include the S-Learner and the T-Learner, the latter which builds separate outcome models for the treated and control populations. Prior to these generic frameworks, tree-based ensembles like Causal Forests were pioncering approach for nan-paraettric CATE estimation (Wager & Athey 2018). In our experiments, we include a strong, GNN-based T-Learner key non-DML. baseline. Our results provide powerful empirical validation of the DML framework's premise, showing that the R-Learner's debiased structure significantly outperforms the more direct T-Learner, even when both use the same powerful GNN architecture.
2.3 Causal Inference on Graphs
Applying causal inference to network data is burgeoning field, as the graph structure roduces complex confounding and potential interference that violate the standard i.i.d. assumption (Aronow & Samiij 2017) The integration of Graph Neural Networks (GNNs), with their powerful inductive bias for relational data Kipf & Welling 2016), is a logical and increasingly active area of research.
The use of GNNs in causality diverse, with applications ranging from uplift modeling (Panagopoulos al. 2024) to instrumental variable estimation. Most concurrently to our work, Khatami et al. (022)) proposed DML-based framework using GNNs to estimate network causal effects, providing crucial, peer-reviewed demonstration of this approach's viability. While this important prior work establishes that an end-to-end GNN pipeline can be effective, the fundamental questions of how and why it works and more importantly, when its components might fail have remained under-characctrrzze.
Our work's contribution therefore not to propose a new mothod, but to conduct systematic disscction of this emerging class of models to produce new scientific understanding. This dissection first reveals the existence and catastrophic impact of the "final-stage bottleneck." " This insight then allows uncover
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deeper, non-obvious mechanism: "Hub-Periphen Trade-off," which link directly to the GNN over- squashing phenomenon, a well-known limitation of message-passing architectures (Alon & Yahav, 2021 Our work thus provides the first mechanistic explanation for the performance of these models on different network topologies, a level analysis not present in prior work.
3 Methodology & The Graph R-Learner
Our work provides a systematic dissection of the R-Learner framework when applied to network data. We now formalize the problem setup and detail the specific R-Learner instantiations used in our comparative analysis.
3.1 Formal Problem Setup
We adopt the potential outcomes framework Rubin 1974h. For each unit (node) in graph G, let (G(1) be the potential outcome if treated (T; 1) and Y;(0) if not (T; =). The observed outcome is Y = -iY(1) + (1 (0). Each node has vector of local features X; (Nie & Wager 2020).
The Conditional Average Treatment Effect (CATE), 7(X, G), is the expected treatment effect conditional on a node's features and its position within the graph NNe & Wager 2020):
T(X9) (1) Y(O)
(2)
Intuitively, captures the heterogeneous response to treatment that varies not only by a node's local at- tributes but also by its structural role neighborhood characteristics within the network (Ma et al. 2022
To identify the CATE from observational data, we make the standard assumption of unconfoundedness. conditional on both local and sufficient graph-based features: {Y(1), (0)} 1 T|X,9 (Aronow & Samii 2017). the context networks, this assumption implics that there are no unobserved variables that simultaneoussy influence both the mreat assignment 7 and the potential outcomes (Y(0), Y(1)), provided we account for the local features and the relevant graph topology (Aronow & Samii 2017 Guo et al.) 2020)) This is stronger requirement than the standard i..dd setting, it necessitates that the estimator effectively captures latent network-based confounders that would otherwise bias the causal estimate (Aronow & Samii 2017
3.2
The R-Learner Framework and the Final-Stagg Bottleneck
The R-Learner Nie & Wager 2020 rooted in Double/Debiased Debiased Machine Learning theory @Cernnzzukov ct al. 2018). soeks a CATE function 7(.) that best explains the relationship between outcome and treatment residuals. This is formalized by the following objective function:
arg arg min E [(Yres Tres T(X
(3)
where the residuals are defined as Yres = y m m(X9)) 9) and Tres T - 9), with m(-) and p(.) being the nuisance models for the conditional outcome and treatment propensity, respectively.
While the orthogonality of this objective provides robustness to errors in the muisance models, its guarantees rest on critical assumption: the function class chosen for the CATE model 7(-) must be flexible enough to approximate the true underlying effect. When the true CATE is a function of the graph structure, but the chosen model for r is graph-blind (i.e., T(X),, a fundamental representation bottleneck occurs. Our work provides the first systematic quantification of the catastrophic impact this bottleneck.
3.3 A 2x2 Grid of R-Learner Instantiations
To systematically dissect the R-Learner framework, we designed 2x2 experimental grid that isolates the impact of graph-awareness both the nuisance and final CATE estimation stages. This allows us to preciscly measure the effects of the "muisance bottleneck" and the "representation bottleneck." " The four resulting R- Learner instantiations summarized in Table 1 For our primary experiments, all GNN components are
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instantiated canonical two-layer Graph Convolutional Network (GCN) Kipf & Welling 2016 otherwise specified in our architectural sensitivity analysis.
unless
Table 1: The 2x2 Ablation Grid R-Learner Instantiations. Our proposed Graph R-Learner is the only fully graph-aware model.
Model Name
Nuisance Models (m,p) Final CATE Model (r)
Baseline
MLP (Graph-Blind)
Linear (Graph-Blind)
Ablation
GNN (Graph-Aware)
Linear (GrahhBllidd)
Hybrid R-Learner
MLP (Graph-Blind)
GNN (Graph-Aware)
Graph R-Learner
GNN (Graph-Aware)
GNN (Graph-Aware)
Baseline Linner) A fully graph-blind R-Learner using a Multi-Layer Perceptron (MLP) for the muisance models and a linear regression on node features for the final CATE estimator (T(X) = X6). This quantifies the performance of a naive application of the R-Learner to graph data.
Ablation LGNN+Linear):: This model incorporates graph structure only in the muisance stage, using GNNs to estimate m(X,9) 9) and g). Its final CATE estimator remains graph-blind. This model is critical for isolating the representation bottleneck.
Hybrid R-Learner (MLP+GNN): This hybrid model uses graph-blind MLP muisance models but employs a graph-aware GNN for the final CATE estimation. This model is designed to isolate the nuisance bottleneck.
Graph R-Learner +GNN+ This is our proposed, end-to-end graph-aware model. uses GNNs for both the nuisance and the final CATE estimation, ensuring its inductive bias correctly specified for all components of the problem.
3.4 External Baseline: The GNN T-Learner
In addition these R-Learner variants, we include a strong, non-DML baseline to validate the utility of the DML framework itself. The GNN T-Learner follows the T-Learner strategy, a standard causal meta- learner (Künzel et al., 2019). It trains single, powerful GNN to predict the outcome Y from both the node features X and the treatment indicator T (Y GNN(X, T)).. The CATE is then estimated by taking the difference between the model's predictions with 7 set to 1 and T set to 0. This baseline allows to test whether the sophisticated, two-stage structure of the R-Learner provides a real advantage over a more direct, single-stage predictive approach.
4 Experimental Design: A Controlled Dissection
To isolate the performance contributions of graph-awaroness at cach stage of the R-Learner pipeline, we conduct a controlled dissection using a series of simulation-basel experiments. simulation-based approach is a scientific necessity for this problem, as is the only way to access the ground-trute trruh CATE and thus directly mcasure the error of our estimators. Our framework designed be challenging and realistic testbed, incorporating the key difficulties of causal inference on graphs.
4.1
Processes (DGP)
For cach simulation run, we generate a graph G (V, E) with 1000 nodes (unless otherwisc specified), each with d 10 independent features X; (O, Ia). To ensure our findings are robust to network structure, conduct experiments across three canonical graph topologies:

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Barabási-Albert (BA): A scale-free model generating graphs with power-law degree distribution, characterized by few high-degree "hubs," mimicking many real-world social networks.
Erdös-Rényi (ER): A random graph model resulting in a more uniform, Poisson-like degree dis- tribution with no inherent hubs.
Stochastic Block Model (SBM): A model that generates graphs with explicit community struc- ture, featuring dense intra-community and sparse communcy connections.
A central challenge in network causal inference is that nodc's treatment and outcome can be influenced by its neighborhood. We explicitly model this network confounding by generating a latent confounder, H, as a non-linear function the graph structure. Specifically, we compute 1-hop and 2-hop neighborhood embeddings for each node using pre-initialized GNNs;
( ReLU(GNN,(XXG)) H () ReLU(GNN,(HH),
(4)
(5)
The treatment. assignment T; is then made dependent both local features X; and the 1-hop neighborhood embedding H violating the standard i.i.d. assumption. The observed outcome Y, is generated as a function of local features, the network confounder, and the true CATE, T3:
Yi = -
(6)
where J((-) is a linear function and E; (O o²) is irreducible noise. This DGP provides robust testbed for evaluating an estimator's ability to deconfound the effect of Tj and accurately recover т₁
4.2 Causal Mechanisms and the Negative Control
A central goal of our study is to test estimator performance under different forms of network-driven causal heterogeneity. We designed several causal mechanisms by varying the functional form of the truc CATE, т. In our main experiments, the CATE is constructed as a function of a node's neighborhood, making graph- blind models theoretically misspecified. Our primary CATE functions are: Simple (1-hop) (T x sin(H())), Higher-Order (r x sin(H(2))) and Interaction (r H()). X).
To rule out the critical confounding explanation that GNN final stage is simply more powerful function approximator, designed negative control experiment (local_x) x). In this setting, the DGP is identical, except the true CATE is function of local features only (T(X) x sin(Xo)). In this world, GNN's graph- aware inductive bias is theoretically unnecessary for the final stage. The theory predicts that the catastrophic representation bottleneck should vanish.
The results, shown in Table 2 confirm this prediction and reveal a deeper insight. The catastrophic gap vanishes, but significant performance gap remains across all models. This is not contradiction, but a powerful, independent validation of the nuisance bottleneck. Even when the CATE function is local, the nuisance components (Y and T) remain heavily confounded by the network. This experiment proves that the representation bottleneck is a real phenomenon tied to the CATE's functional form, and it isolates the standalone importance of using graph-aware models to deconfound the nuisance functions. This finding aligns with the broader literature on leveraging flexible models to control for high-dimensional confounding [arrell
2015 Belloni et al. 2013).
4.3
Semi-Synthetic Experiment on a World Graph
To ensure our findings generalize beyond purely synthetic topologies, we conducted a semi-synthetic ex- periment. This approach provides a crucial bridge between the controlled environment of a full simulation and the complexity of real-world data, allowing us to retain a known ground-truth CATE while testing our methods on a graph with realistic structural properties. For this experiment, we use the well-known Cora citation network (Yang et al. 2016). We take the real-world graph structure and node features X from
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Results for the negative control experiment, where the true CATE function of local features only (r= = f()).. The catastrophic, order-of-maguitude performance gap from the representation bottleneck disappears as predicted. Results are Mean MSE + Std. Dev. over 30 seeds.
Table 2:
Method
Mean MSE + Std. Dev.
Graph-Blind Nuisance Models:
Baseline LMnea+
5.0004 + 1.0247
Hybrid R-Learner (MLP+GNN)
1.4035 + 0.1072
Gruph- Aware Nuisance Models:
Ablation +inNer)n
4.6420 + 0.8303
Graph R-Learner (GNN+GNN)
1.3419 + 0.1117
External Baseline:
GNN T-Learner
2.9344 + 0.4841
the dataset and simulate T and Y on this scaffold using our main network-dependent causal mechanism. Demonstrating that our central findings hold in this more complex sctting provides strong evidence for the practical relevance of the final-stage bottleneck.
5 Results and Analysis
We now present the results from our comprehensive suite of experiments. Our findings are organized to first establish the primary performance bottleneck, then to dissect the more nuanced secondary effects, and finally to confirm the robustness of our conclusions through a series rigorons sensitivity analyses and a semi-synthetic validation. All results are reported as the Mean Squared Error (MSE) of the CATE estimate, averaged over 30 random seeds unless otherwise specified.
5.1 Main Finding: The Universal Representation Bottleneck
Our central hypothesis is that for network-dependent causal elfects, the inductive bias of the final-slage CATE estimator is the most critical modeling choice. We test this in our main experimental setting, using a Barabási-Albert (BA) graph and simple 1-hop CATE function. The results, summarized in Table are definitive.
Table 3: Main results Barabási-Albert graph with network-dependent CATE. The performance gap between graph-blind and graph-aware final-stage models is order of magnitude, demonstrating a catastrophic representation bottleneck. Results are Mean MSE + Std. Dev. over 30 seeds.
Method
Mean MSE St.. Dev.
Graph-Bline Blind Final Stage:
Baseline (MLP+Linear)
4.8235 + 0.9135 4.1659 + 0.8412
Ablation (GnN+Tinear)
External Baseline:
GNN T-Learner
2.3834 + 0.5824
Graph-Aus Aware Final Slage:
Hybrid R-Learner (MLP+GNN)
0.5701 + 0.0856
Graph R-Learner (GNN+GNN) 0.5165 + 0.0527
The evidence for catastrophic representation bottleneck overwhelming. The two models with a graph- blind 'Linear' final stage fail completely, achieving an MSE an order of magnitude higher than those with a graph-aware 'GNN final stage. Notably, the 'Ablation' model, despite using powerful GNN for its nuisance

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End-to-End Graph Awareness Necessary (Main, Result BA Graph)
Mhan Squared Error in CATE Estimation (Lower is Butter)
Baseline (MLP kin)
Ablation KONN (ONN+LUi Lin)
Hybrid & Learnnr (MU*eGGNM
GNN Fleameer (Extemal)
Oraph R-Leamer IDMN (GNN)
Figure 1: Shows the hierarchy of performance over different models.
estimation, still fails. This proves that high-quality nuisance models are insufficient to overcome a funda- mentally misspecified final stage. Furthermore, our proposed Graph R-Learner significantly outperforms strong, non-DML 'GNN T-Learner' baseline (p = 5.43 10-17), validating the robustness of the R-Learner framework itself in this domain. This hierarchy of performance is visualized in Figure [
5.2 Mechanistic Insight: The
Nuisance Bottleneck
Having established the primacy of the final stage, now investigate the secondary 'nuisance bottleneck" by comparing the HHybrid R-Learner (MLP-GNN) model to our full 'Graph R-Learner (GNN+GNN): Table summarizes this comparison across different graph topologies.
Table 4: Analysis of the nuisance bottleneck. The performance gain from using GNN nuisance models is statistically significant graphs with diffuse structure (ER, SBM, Cora) but less pronounced on the hub-dominated BA graph.
Graph Topology
MSE (MLP+GNN)
MSE GGNN+GNN)
P- Valuue
Barabási-Albert, (BA)
0.5701
0.5165
3.68 10
Mndös-Rényi (ER)
0.1127
0.0687
6.49 x 10-5
Stochastic Block Model (SBM)
0.1134
0.0678
4.65 x 10-5
Cora (Semi-Synthetic)
0.3488
0.3352
1.41 x 10-2
The results reveal a clear and nuanced pattern. Using GNNs for the nuisance models provides consistent and statistically significant performance improvement across all tested topologies. However, the magnitude of this effect is topology-dependent The improvement is most pronounced on the uniform-degree ER and SBM graphs. This led us to hypothesize a "Hub-Periphery Trade-off," which we investigate next.
5.3 Direct Evidence: A Targeted Error Analysis the Hub-Perrphery Trade-off
To find a mechanistic explanation for the topology-dependent nuisance bottleneck, we performed targeted error analysis on the hub-dominated Barabási-Albert graph. We partitioned nodes into "hubs" (top 10% degree) and "periphery" (bottom 50% degree) and calculated the Mean Squared CATE Error for each group independently.
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The results, shown in Figure 2 reveal striking and non-obvious performance inversion. On low-degree periphery nodes, the end-to-end Graph R-Learner (GNN+GNN) is substantially superior. In this regime, its GNN-based nuisance models are critical for aggregating the weak, diffuse confounding signals that a graph-blind MLP misses.
Conversely, and more surprisingly, on high-degree hub nodes, the hybrid Hybrid R-Learner (MLP+GNN) model significantly outperforms the fully graph-aware model. We attribute this to the well-known over- squashing phenomenon in GNNs (Alon & Yahav, 2021). The MLP muisance model, by being graph-blind, unintentionally acts powerful regularizer; it protects the hub's unique local signal from being diluted by the noisy aggregation of its massive neighborhood, thereby providing a cleaner, more informative residual to the final GNN stage.
This discovery of a "Hub-Periphery Trade-off" is a critical finding, It suggests that a monolithic, one-size- fits-all GNN architecture sub-optimal for network causal inference and highlights a fundamental tension between aggregating diffuuse information on the periphery and preserving local information on hubs.
Nuisance Bottleneck Analysis on Barebá Graph
Sanity Check (MLP+GNN)
Graph R-Learner (GNN+GNNN)
0.8-
Mean Squared CATE Erro
0.6
0.0
Hubs (Top 10%)
Periphery (Bottom 50%)
Figure 2: The Hub-Periphery Trade-off on the Barabási-Albert graph. The plot shows a clear performance inversion: the end-to-end Graph R-Learner (GNN+GNN) excels on low-degree periphery nodes, while the hybrid Hybrid R-Learner (MLP+GNN) model performs better on high-degree hub nodes.
5.4 Robustness and Sensitivity Analyses
To ensure our findings are robust, we conducted series of sensitivity analyses, presented in Figure
The results confirm the robustness of our conclusions. First, we show that the representation bottleneck persists when replacing GCNs with GATs, proving the finding not architecture-speefff. Second, our analysis of outcome noise shows that the Graph R-Learner's performance degrades far more gracefully than the baselines, demonstrating the power of the DML framework's orthogonalization in isolating causal signal from muisance noise.
Finally, our analysis of sample efficiency (Figure 3(c) reveals that the Graph R- R-Learner's primary strength in the low-data regime (N < 1000) is its stability and robustness to data scarcity. While the baseline and ablation models exhibit significant variance and performance degradation as N decreases, the Graph R-Learner maintains a stable, low MSE. This suggests that its correctly specified inductive bias acts as critical regularizer when observational data is limited, preventing the catastrophic volatility observed in models that rely solely local features.
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Archinettrr Sersivbyyy GOh,
(a) Robustness to GNN Architecture
Attation (CMM+an)) Graph R Loomer (GNN-GWM)
Menth Squaree EXTT F10
Noton Level
(b) Robustness Outcome Noise
Ween Squarded EXTT TIe
Graph Leamee
1080
2000
3000
5000
(c) Sample Efficiency
Figure 3: Sensitivity Analysos. (a) The performance gap persists when replacing GCNs with GATs (Graph Attention Network) (arooor)((Veeiiii (Veličković et al. 2017), demonstrating our findings are not architecture-speciin (b) The Graph R-Learner's error degrades far more gracefully than the ablation model as outcome noise increases, proving its robustness.
(c) The Graph R-Learner exhibits superior stability and robustness to data scarcity. While graph-blind variants become increasingly volatile as N decreases (N < 1000), the Graph R-Learner maintains a consistently low and stable MSE, highlighting its superior sample efficiency.
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5.5 Validation Real- World Graph Structure
Finally, to bridge the gap from synthetic data to real-world complexity, we conducted a semi-synthetic experiment on the Cora citation network. The results, shown in Table confirm that all of our contral findings hold on real-world graph structure. The representation bottleneck remains the dominant effect, and the end-to-end Graph R- R-Learner significantly outperforms all baselines.
Table 5: Results for the semi- synthetic experiment on the real-world Cora citation network. All key find- ings, including the representation bottleneck and the superiority of the end-to-end Graph R-Learner, are confirmed. Results are Mean MSE + Std. Dev. over 30 seeds. Mean MSE Std. Dev.
Method
Graph- Blind Final Stage:
Bascline (MLP+Linear)
2.3150 + 0.1881
Ablation (GNN+Linear)
1.2175 + 0.0665
External Baseline:
GNN T-Learner
2.4012 + 0.3704
Gruph-Awary Final Stage:
Hybrid R-Learner GMIP++NN)
0.3488 + 0.0278 0.3352 + 0.0157
Graph R-Learner (GNN+GNN)
The results on the Cora graph, summarized in Table 5 provide a powerful validation of our central thesis. We observe the same clear hierarchy of performance: the graph-blind final-stage models fail, the T-Learner is a significant but sub-optimal improvement, and the fully graph-aware R-Learner instantiations achieve the lowest error. Crucially, the nuisance bottleneck also statistically significant in this setting (p = 1.41 10-?) ?), confirming that an end-to-end graph-aware pipeline is essential for achieving optimal performance complex, real-world graph structures.
6 Discussion
Our comprehensive suite of experiments has not just evaluated a set of models, but has revealed clear and actionable set of principles for applying the R-Learner framework to network data. We synthesize these findings into 'Hierarchy of Needs" and discuss the deeper implications for both practitioners and rescarchers.
6.1 The "Hierarchy Needs" for Network Causal Inference
Our results suggest a clear, prioritized roadmap for practitioners seeking to estimate network-dependent causal effects. The performance bottlenecks we identified are not equal in magnitude; they form distinct hierarchy of importance.
(Must-Have) A Graph-Aware Final Stage: Our most significant finding is the universal and catastrophic failure of graph-blind final-stage estimators. This 'representation bottleneck" the primary determinant of success. The first and most critical modeling choice use a CATE estimator whose inductive bias (e.g., GNN) represent the underlying graph-dependent causal mechanism. Failing this step renders all other choices irrelevant.
2. (Should-Have) A Robust DML Framework: Our experiments consistently showed that the R- Learner framework significantly outperforms a strong, non-DML GNN T-Learner baseline, particularly on the complex, real-world structure of the Cora graph. This validates the theoretical promise of the DML franework, whose debiasing and residualization procodure provides more robust and efficient path to the causal estimate than direct, single-stage predictive model.
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3. Haave Graph-Aware Nuisance Models: Our analysis the 'muisance bottleneck' revealed real, but secondary, performance gain from using GNNs in the nuisance stage. This gain was statistically significant across most settings, proving its existence. This final layer of end-to-end graph awareness is necessary for achieving optimal, state-of-the-art performance.
This hierarchy provides an unambiguous guide: prioritize the final-stage model, build within an R-Learner framework, and use GNNs for all components achieve the best results.
6.2 A Practical Diagnostic: The
Test
Our findings also yield simple, practical diagnostic for practitioners. To determine if complex, graph- aware model is necessary for their specific problem, we propose the "Two-Model Test":
1. First, fit a simple, graph-blind bascline (e.g., our 'MLP+Lincar' model).
Second, fit a fully graph model (e.g., our 'Graph R-Learner').
A statistically significant and substantially large drop in the out-of-sample CATE estimation error (as demon- strated by our 'POSITIVE" diagnostic results in all network-dependent settings) provides strong evidence for the presence of network-driven causal heterogeneity. This simple test can prevent researchers from incorrectly concluding null effect when in fact their model was merely misspecified for the problem.
6.3 Quantifying the Hub-Periphery Performance Inversion
To provide mechanistic insight into the representation bottleneck, we move beyond qualitative visualizations to a rigorous, quantitative dissection of model performance across the network topology. We investigate the 'performance inversion" hypothesis: that graph-blind muisance models may paradoxically protect signal high-degree nodes by avoiding the over-squahhing effect inherent in GNNs.
As shown in Figure 4 conducted threshold sweep defining 'hubs' from the top 2.5% to the top 25% of nodes by degree centrality. The results, averaged over 30 independent seeds, reveal a robust structural trade-off. At the most extreme hubs (Top 2.5% 12.1%), the Hybrid R-Learner (utilizing MLP nuisances) significantly and consistently outperforms the end-to-end Graph R-Learner.
This provides definitive empirical evidence for the nuisance ovcr-squashing bottleneck. In the Graph R-Learner, the GNN nuisance model aggregates information from massive neighborhoods, which, at central hubs, leads to the exponential dilution of the node's local signal (Alon & Yahav, 2027). Conversely, the graph-blind MLP nuisance model in Hybrid variant acts protective regularizer; by ignoring the noisy neighborhood during the nuisance stage, it preserves cleaner, more informative residual for the final- stage CATE estimator. As the hub defimition broadens toward the periphery (Top 25%), the two models converge, as the over-squashing effect becomes less dominant.
7 Conclusion
In this paper, we conducted the first systematic dissection of the R-Tearner framework for network causal inference. We moved beyond prior work by not just proposing method, but by rigorously quantifying the failure modes and performance contributions of each component in the causal pipeline. Our central finding is the discovery of a catastrophic "final-stage bottleneck," proving that the inductive bias of the final- stage CATE estimator is the primary determinant of success, factor far more critical than the choice of nuisance models. We further provided a mechanistic explanation for a subtle, topology-dependent 'nuisance bottleneck' " by linking it to the GNN over-squashing phenomenon via a robust Hub- Pur-Peerph Trade-off analysis.
Our work provides a clear, empirically-be ers. We validated our findings across comprehensive suite of synthetic, semi-synthetic, and negative control
backeed backed 'hierarchy of needs' that serves as a crucial guide for practition-
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30-Seed Robustness of Hub Performance Inversion
Hybrid R-Leamer (MLP Nuisance) Graph R-Leamer (GNN Nuisance)
3
Squared CATE Error
2
Top 2.5%
Top
Top Top 12.1% Top 15.4% Top 18.6% Top 21.8% Top 25.0%
Hub Definition (Percentile Degree Centrality)
Figure 4: 30-seed robustness sweep of the Hub-Periphery Trade-off. The x-axis represents increasingly strict definitions of "hubs" based on degree centrality quantiles. Error bars indicate one standard deviation across 30 independent experimental replications.
experiments, ensuring that our results are robust to both topological variation and model capacily. Ulti- mately, our results are a definitive statement: for network data, a piecemeal application of graph-awareness is insufficient. Accurate causal inference requires a robust thcoretical framework and an end-to-end modeling pipeline whose inductive biases are correctly specified for the network-driven nature the problem.
Future Work While our study establishes the fundamental hicrarchy for network causal inference, several promising avenues for future research remain. Specifically, investigating the impact of scaling node popu- lations (N) and feature dimensionality (d) could reveal further nuances in the representation bottleneck across deeper architectures. Furthermore, while the Graph R-Learner shows strong performance on citation networks like Cora, future work should extend this systematic dissection to diverse real-world domains such epidemiology or social influence networks to further test the generalizability of the hub-periphery trade-off. Beyond scaling, future architectural studies could further strengthen the "final-stage bottleneck' hypothesis by exactly matching the parameter budgets of ablation models and performing a symmetrical quantification of the degree-regimes where graph-aware nuisances transition from suboptimal (hubs) to optimal (periphery) performance.
Broader Impact Statement
This work aims to improve the reliability of causal inference in domains where network effects prevalent, such as social science, economics, cpidemiology, and biology (Yang et al. 2016). By providing rigorous quantification of the Representation Bottleneck, our research offers concrete guide for practitioners to avoid catastrophic modeling pitsaals where graph-blind est.imaoors fail to represent network-dependent causal heterogeneity.
A critical ethical implication of our findings the risk of falsc-negative conclusions. high-stakes settings-sudh such evaluating public health interventions contact networks analyzing gene perturbations
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in biological systems misspecified final-stage model may fail to detect genuine treatment effect. This could lead to the premature abandonment of effective policies or life-saving treatments.
Potential Mitigations and Cautions:
While end-to-end graph awareness resolves structural bottlenecks, the validity of causal conclusions still rests the fundamental and untestable assumption of uncon- foundedness (Aronow & Samii 2017). Users should exercise caution and integrate domain expertise when applying these methods.
Assumption Awareness:
References
Uri Alon and Eran Yahav. On the bottlencck of graph neural networks and its practical implications, International Conference on Learning Represenlations, 2021. URL htttps://openereeiiet
http:s///pppaeeeeeerr..eeerrrrr wet./forumiidd
1800PhOCVH2
Peter M. Aronow and Cyrus Samii. Estimating average causal effects under general interference, with application to a social network experiment. The Anmals of Applied Stalistics, 11(4):1912-188 1947, 2017. ISSN 19326157. URL http://www jstor org/stablee26362177
Albert-Lááz26 Barabási and Réka Albert. Emergence of scaling in random networks. science, 286(5439): 509 512, 1999.
Alexandre Belloni, Victor Chernozhukov, and Christian Hansen. Inference on treatment effects after selection among high-dimensional controlst. The Review of Economic Studies, 81 (2):608 650, 11 2013. ISSN 0034- 6527. doi: 10.1093 ha/////estu///ee restud/rdt0044. URL https://doi wtt. 10t.1110///essu//rttto.o
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1)CC1-668, 01 2018. ISSN 1368-4221. doi: 10.1111/eccj.1200 12097. URL https:
7/doi. 10. 1111/ectj....e 12097
Max Н. Farrell. Robust inference average treatment effects with possibly more covariates than obscrvations. Journal of Econometrics, 189(1):1-23, 2015. ISSN 0304-4076.
doi: https://doi.
org/10010te/..eeooo..21555. 201..00...0. URL https://ww.scieencddreett www.sacciecedeee....mmmmm htt.://tc.ccccccceeeeaarcrc////e 5030440761500864
Ruocheng Guo, Lu Cheng, Jundong Li, P Richard Hahn, and Huan Liu. A survey of learning causality with data: Problems and methods. ACM Computing Surveys (CSUR), 53(4):1 37, 2020.
Seyedeh Baharan Khatami, Harsh Parikh, Haowei Chen, Sudeepa Roy, and Babak Salimi. Graph machine learning based doubly robust estimator for network causal effects. In Yingzhen Li, Stephan Mandt, Shipra Agrawal, and Emtiyaz Khan (eds.), Proceedings of The 28th International Conference on Artificial Intel- ligence and Statistics, volume 258 of Proceedings of Machine Learning Research, pp. 4366 4374. PMLR, 03-05 May 2025. URL attps::/proceedee
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4156-4165, 2019. doi: 10.1073 10.1073/pnas. 18045.100459116 URL hattps://wwit pnas .arg/org/gg/.o awrg/doiaabs/1ue 1073/pnas. 1804597116
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X Nie and S Wager. Quasi-oracle estimation of heterogeneous treatment effects. Biometrika, 108 319, 09 2020. ISSN 0006-3444. doi: 1110////o/e////ssseeee URL httpp::/dooii asaa076
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A A Constructive Example of the Representation Bottleneck
To formally illustrate the existence and nature of the representation bottleneck, we construct minimal, analytical example. Consider simple 5-node star graph where node C' is the central hub connected to four periphery nodes A, B, D.E. E.
Let each node have a single local feature, defined as:
11=1, =B=1, XD=1, =g=2, Xc = 10
Let the network embedding H; for a node i a function of its neighbors' features (for simplicity, the neighbor's feature if degree is 1, or the average of neighbors' fcatures otherwise). The emboddings are thus:
HA = Hn Hn H = Xo =1=10 10
1
Hc (XA (XA+ XB X X +
+1+++++22 1.25
(9)
This construction creates distinct structural roles: the hub (with H/ 1.25) and the periphery (with H 10). Now, define the true CATE to be non-linear function of this structural embedding, mirroring our main DGP: T, sin(Hf,). This yields gemuinely heterogeneous, network-dependent CATE:
TA TB TD TE sin
(10)
TC = sin
(11)
Now, consider powerful, non-linear but graph-blind final-stage estimator, such MLP, whose fune- tional form is restricted to 9; f(X).. To perfectly recover the true CATEs, this model must learn a function f that satisfies the following constraints based on the observable local features X₁:
f(1) = sin sin(10) (10) (from periphery nodes A, B, D)
f(2) = sin (1) (from periphery node E)
f(10) sin(1.25) (from hub node C)
The model is thus forced to map two different inputs, X 1 and X 2, to the exact same output value, sin(10). More fundamentally, the input to the function, X,, is the wrong information. The true CATE is a function of the structural role (hub periphery), which is captured by H,, not by the local feature X;-
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A graph-aware model, f; gX,, (), trivially learn this function by learning to compute the embedding H; and then approximating sin(.). A graph-blind model, however, is being asked to learn function from variable that has a complex, non-injective relationship with the true causal effect. It is, by construction, thcorctically misspecified. This simple construction proves that even a highly flexible non-lincar model will fail it lacks the correct graph-aware inductive bias, thus remonstratnng the fundamental uature of the representation bottleneck.
B Experimental Configurations
This section contains the full YAML configuration files used to generate the results for each of our main, robustness, semi-synthetic, and control experiments. This is provided for full reproducibility. All experiments were orchestrated by the main script run_ experiments py, which reads one of the following configuration files.
B.1
Configuration: Main Result (BA Graph)
Configuration file: configs/main_bar lasiiii__a__mmmlleeeeeeee simpsee yaml
name: 'Main_ Wesin_Reult____ Bhault_BA_Graph Graph
num_seeds: 30
data_parems:
1000
d: 10
graph_type: 'ba'
cate_type: 'simple_h' real_data_ned nane: mull
noise_level: 0.5
11
12
model_params:
18
nun_layers
14
hidden_dim _imm 32
15
16
braining.params: muisance_epochs: 150
17
18
cate_epochs: 200
19
1r: 0.001
B.2 Configuration: Robustness Check (ER Graph)
Configuration file: configs/robustness ser_graph.yan
name: 'Robustness, ER_ Graph
num_ qum_seeds:
data_ :
graph_type: 'er'
cate_type: 'simple_h' real_data_nane null
10
noise_level: 0.5
11
12 model_params:
13
aun_layers: 2
14
hidden_dim: 32
15
training_params
muisance_ :pochs:: 150
cate_epochs: 200
1r: 0.001
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B.3
Configuration:
Robustness Check (SBM Graph)
Configuration file: confiige/rbbusttesssss. sarpeessssss_nrrhhhp_pep yaml
"Robustness_SBK, Beaeeesss_BX___aap___
nun_seeds: 30
data aata_parens:
10
sypph_type: 'abm'
type: 'eimpl__h'
real_data_nane mull
10
noise_level: 0.5
11
12 model_parama:
nua_layers: hidden_dim: 32
14
training-parame
nuisance_epochs: 150
cate_epochs: 200
1r: 0.001
B.4 Configuration: Robustness Check (Interaction CATE)
Configuration file: http://rruusteesssttteeetetttttt catpaacctioioctteeyaaeeaee cate.yaml
'Robustness, Inaasness_Itteractionon
nun_seeds:
dara_p:rams:
n: 1000 d: 10
graph_typp 'ba'
cate_type: 'interaction' real_data_nane mull
noise_level: 0.5
11
model_params:
layers: hidden_dim: 32
training.params:
17
nuisance_ nuieance_epocher
cate_epochs:
18
B.5 Configuration: Semi-Synthetic Experiment (Cora)
Configuration file: configg//em//syntittiic...a
name: "SemiSyathetse_Cor
num_ seeds: 30
data_paree n: mull d: mull graph_typs: null cate_type: 'simpe__h' real_data_nane: 'cora'
noise_level: 0.5
model_params:
13
nun aun_layers: hidden dim:
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wraining.parami
17 nuisance auisance_epoche: :
cate_epochs:
19
1r: 0.001
B.6
Configuration: Negative Control (Local CATE)
Configuration file: confiigs/controllocaal yam
'Control_Local C
seeds:
pata_params:
1000
10
traph_type:: 'ba'
cate_type: 'local_x'
real_data_pp : mull
noise_level: 0.5
11
12
model_parana nun jua_layers: 2
13
hidden_dim: 32
training.pares
nuisance, spochs: : cate_epochs: 200 1r: 0.001
C Technical Implementation Details
C.1 Parameter Breakdown and Parity
To ensure that performance gains driven by inductive bias rather than model capacity, report the trainable parameter counts for all primary configurations (Table All models utilize hidden dimension of 32 and 2 GNN layers where applicable.
Table 6: Trainable Parameter Counts. The Graph R-Learner achieves better performance with lower parameter budget than the Hybrid variant.
Model Configuration
Component Breakdown
Total Paramcters
Baseline (MLP + Linear)
385 (MLP Nuisance) + 11 (Linear CATE)
396
Ablation (GNN + Linear)
352 (GNN Nuisance) + 11 (Linear CATE)
363
Hybrid R- L-Learner (MLP + GNN)
385 (MLP Nuisance) + 352 (GNN CATE)
737
Graph R-Learner (GNN + GNN)
352 (GNN Nuisance)
+ 352 (GNN CATE)
704
C.2 Statistical Independence of Experimental Seeds
All y-values reported in this study are derived from paired (relational) I-lests comparing model MSEs across N 30 seeds. Each seed represents a full experimental replication: for every iteration, the node features (X), graph adjacency (edge (eeeexindee,,,_ treatment assignments (T), and outcomes (Y) are entirely ro-simulatod Barabási & Albert, 1999). This ensures that our 30 samples are independent and identically distributed i.i.d.) realizations of the data-generating process, satisfying the independence assumptions of the t-test.
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C.3 Multiple Comparison Correction Logs
To validate the "consistent improvement" across topologies, performed Holm-Bonferroni step-down procedure on the raw p-values derived from 30 independent experimental seeds. As shown in Table 7 the performance gain remains significant (a = 0.05) across all tested graphs, including the real-world Cora scaffold.
Table 7: Holm-Bonferroni Correction Logs (30 Seeds).
Topology
Raw p-value
Holm-Adjusted p Significant
SBM
4.64
10-5 5
0.000186
Yes
ER
6.49 10-5
0.000195
Yes
BA
3.68
10-3
0.007370
Yes
Cora
1.44 10-2
0.014446
Yes
C.4 DGP Initialization and Latent Confounding
The latent network confounders (H) are generated using two 'pre-initialized" GNNs (Kipf & Welling 2016).
These GNNs are initialized using standard Xavier (Glorot) normal weights with fixed seed (0). This ensures that the functional mapping of the latent confounding mechanism remains stable and reproducible across all seeds while the specific graph structures and feature values vary.
19


Reviews:
[{"review_id": "1LkrZnXXHz", "reviewer": "Reviewer_neYa", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "R-Learner is a powerful framework for estimating heterogeneous treatment effects but its theoretical guarantees lie on an assumption of a well-specified final stage model. This assumption can be violated in network data where causal heterogeneity may be driven by graph structure. This work systemically dissects R-Learner for network causal inference through experiments on both synthetic and semi-synthetic benchmarks. The results suggest that the performance degradation of R-Learner mainly arises from the inductive bias of the final stage model rather than nuisance model quality. This finding leads to uncovering the Hub-Periphery Trade-off which explains the over-squashing phenomenon in graph neural networks.\n\nStrengths\n- The paper challenges the convention in the literature that nuisance model quality is the most important factor for R-Learner and finds that the inductive bias of the final stage model is the most important one. This finding is critical yet overlooked in the literature and have a significance for both researchers and practitioners.\n- The experiments cover diverse settings including different graph structures, CATE functions, or R-Learner instantiations.\n- Built on the experimental findings, the paper uncovers the Hub-Periphery Trade-off which explains the over-squashing phenomenon observed in graph neural networks.\n- The authors release the code as a reproducible benchmark which can facilitate future research in this domain.\n\nWeakness\n- Overall, the background to understand the paper is not properly addressed and hence the readers who are not familiar with causal inference or graph neural networks are expected to have difficulty in understanding the paper. Specifically, important terminologies are often very briefly described in the paper.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The main claim of the paper is that the performance degradation of R-Learner mainly stems from the misspecified final-stage estimators rather than the nuisance model of bad quality. This claim is supported by rigorous experiments with diverse and convincing setups. Specifically, the authors design a 2x2 experimental grid to isolate the impact of graph-awareness for both nuisance and final-stage models which allow the precise measure of nuisance and final-stage bottlenecks. The experiments are performed on multiple environments across three graph toplogies, three CATE functions, and both synthetic and semi-synthetic datasets. The authors also include a non-DML baseline, different GNN architectures, and negative control experiments to further verify the findings.", "audience_yn": "Yes", "audience_explanation": "The paper covers the emerging field of causal inference, graph neural networks, and debiased machine learning. The audience working in this domain would be interested in knowing the findings of the paper. Also, the authors provide a benchmark to facilitate future research in these domains.", "requested_changes": "(major) \nAs a reader who does not have sufficient knowledge in this domain, I find it quite difficult to understand the introduction and related works sections. The authors are encouraged to describe sufficient background knowledge to understand the paper. For example, this includes the concepts in debiased machine learning (R-Learner, nuisance model), causal inference (unconfoundness, CATE), and graph neural networks (over-squashing).\n\n(major)\nAbbreviations are extensively used throughout the paper but often not properly explained. For example, the paper does not properly explain the meaning of DGP, HTE, GCN, or GAT. The authors are encouraged to provide definitions for these terms.\n\n(minor)\nExperiments on different number of nodes $n$ and number of features $d$ would further strengthen the paper. These experiments can provide a practical implication of whether the effects of the final-stage bottleneck are larger for complex graph structures.\n\n(minor)\nCurrently, the experiments on real-world datasets are only performed on a single dataset (Cora citation network). Experiments on other real-world datasets would strengthen the validity of the results.", "broader_impact_concerns": "The paper is missing a Broader Impact Statement section. The authors are encouraged to write a short section regarding the ethical implications of this work.", "additional_comments": "None.", "extra_scores": {}, "extra_text": {}}, {"review_id": "DfXDAHJ54C", "reviewer": "Reviewer_vJDg", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper investigates the application of R-Learners on network data to study causal relationships in the form of treatment effects. In particular, it addresses a gap in the literature: under-characterisation of how and why the robustness guarantees of R-learners are affected due to violations of the assumption that the final stage CATE estimator is specified correctly / flexible enough to capture the underlying effect, in the context of graph-dependent causal effects (i.e. what happens if GNNs are not used when they should?). It has two key findings (1) “nuisance bottleneck”: in the first stage (nuisance modelling), the critical need of GNNs is especially prominent in diffuse graphs not dominated by hubs, (2) “representation bottleneck”: through experiments that vary how the final stage CATE estimator is constructed (graph-blind vs graph-aware), it reveals that in scenarios where causal heterogeneity may be driven by graph structure, the key driver of model performance is the inductive bias of the final stage CATE estimator. Through their experiments on synthetic data and Cora dataset, the authors also demonstrated how R-Learners outperformed a GNN T-Learner baseline.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Overall, the paper is nicely written with well-motivated experiments conducted to prove their hypothesis, even though it is largely focused on synthetic and standard graph datasets. However, the manuscript could benefit from some additional details to improve clarity.", "audience_yn": "Yes", "audience_explanation": "This would be interesting for researchers in causal ML, particularly those that are working on graph datasets (which is a common industrial use case, especially biomedical).", "requested_changes": "Major\n1. For all 4 models tested, please report the number of trainable parameters.\n2. For all p values reported, what exactly were the statistical tests performed? Since there seems to be quite a number of comparisons, was multiple comparison correction done? \n3. Section 4.1 - ‘pre-initialized GNN’ - how exactly were the weights initialized? \n4. Section 5.3 - It will be great to go beyond just partitioning nodes to top 10% and bottom 50% degree (which are somewhat arbitrary cutoffs). Specifically, one low hanging fruit that could potentially lead to interesting findings would be to see if there’s any threshold you can find where these ‘performance inversion’ evens out / disappear. \n5. Section 5.4, fig 3c - how was the conclusion that ‘advantage is most pronounced in the low-data regime’ arrived at? Advantage implies comparison and fig 3c actually shows the gap between the blue and orange lines narrowing at N<1000, since the drop seems steeper. However, this might be more of an issue of numerical scale / visualisation rather than the actual percentage change in performance (larger data size vs smaller). Nevertheless, this conclusion needs to be revisited and rephrased. \n6. Conclusions from t-SNE plot in fig 4 is not convincing. It might be better to leave this out and focus on improving Section 5.3. \n\nMinor\n- Define abbreviations used at the first instance of using them (e.g. DGP, …)\n- Increase the font size of Fig 3, it’s currently too small. Or reorganise the plots into another layout.", "broader_impact_concerns": "Nothing of note for now.", "additional_comments": "-", "extra_scores": {}, "extra_text": {}}, {"review_id": "tKLzEDoh7L", "reviewer": "Reviewer_8wEX", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a systematic dissection of the R-Learner framework for estimating heterogeneous treatment effects on networks, identifying a critical \"representation bottleneck\" where graph-blind estimators fail catastrophically ($MSE > 4.0$, $p<0.001$) even when paired with powerful graph-aware nuisance models. The authors propose the Graph R-Learner, an end-to-end graph-aware architecture that resolves this bottleneck and achieves significantly lower error. Through a rigorous ablation study and targeted error analysis, the work further uncovers a topology-dependent \"nuisance bottleneck\" linked to the GNN over-squashing phenomenon, demonstrating a \"Hub-Periphery Trade-off\" where hybrid models outperform fully graph-aware ones on high-degree hubs. The contributions are finalized with a \"Hierarchy of Needs\" for practitioners and a reproducible benchmark suite validated on synthetic and semi-synthetic dataset.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The submission supports its claims with relatively strong evidence. The authors quantify a catastrophic \"representation bottleneck\" by demonstrating an order-of-magnitude performance gap ($MSE > 4.0$ vs. $\\approx 0.5$) between \"graph-blind\" and \"graph-aware\" final-stage estimators. They validate that this is a structural failure rather than a function approximation issue using a \"negative control\" experiment, where the gap vanishes when the true CATE is local. Furthermore, the proposed \"Hub-Periphery Trade-off\" is substantiated by a targeted error analysis showing a \"performance inversion\" on hub nodes, which convincingly links the \"nuisance bottleneck\" to the GNN over-squashing phenomenon.", "audience_yn": "Yes", "audience_explanation": "The intersection of Causal Inference and Graph Representation Learning might be quite interesting for the TMLR audience. This paper is particularly relevant because it provides actionable, empirically-backed guidelines (the \"Hierarchy of Needs\") for practitioners applying the popular R-Learner framework to non-i.i.d. data. Additionally, the proposed \"Two-Model Test\" serves as a practical diagnostic tool that methodology-oriented readers will likely find immediately useful.", "requested_changes": "1. The paper currently assumes a high level of familiarity with the DML framework and the specific mechanics of the R-Learner. Terms like \"Neyman-orthogonality\", \"nuisance models,\" and \"residual-on-residual regression\" are introduced rapidly in Section 1 and 2.1.\n\n> Please expand Section 2.1 to include a brief, intuitive primer on DML for readers who are not specialists in causal inference. Specifically, briefly explain why the nuisance stage is separated and what \"debiasing\" intuitively achieves before diving into the specific R-Learner objective function. This is critical for meeting TMLR's criteria of being interesting to the broad audience.\n\n2. The paper relies heavily on \"self-coined\" terms, which can occasionally lead to confusion. For example, \"Final-Stage Bottleneck\" and \"Representation Bottleneck\" appear to be used somewhat interchangeably. Additionally, using the term \"Sanity Check\" as a formal model name in tables and figures (e.g., Table 1, Figure 1) is unconventional and slightly confusing.\n\n> The reviewer recommends renaming the \"Sanity Check\" model to a descriptive technical name (e.g., \"Hybrid R-Learner\") and referring to its role as a sanity check in the text only. Furthermore, please explicitly define the relationship between \"Final-Stage Bottleneck\" and \"Representation Bottleneck\" early on, and stick to consistent usage throughout the text to reduce cognitive load.", "broader_impact_concerns": "N/A", "additional_comments": "The reviewer would like to include a disclaimer stating that they are not familiar with the theoretical aspects of Double/Debiased Machine Learning (DML). Consequently, the theoretical significance of the proposed findings may not have been rigorously assessed.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)