Journey to the Centre of Cluster: Harnessing Interior Nodes for A/B Testing under Network Interference
Authors: Qianyi Chen
Anpeng Wu
Bo Li
Lu Deng
Yong Wang
Venue: ICLR 2026 Poster (2026)
Keywords: causal inference
A/B test
network interference
cluster-level randomization
Primary area: causal reasoning
TL;DR: We develop a novel estimator for A/B test under network interference, which specifically leverages the interior nodes with a regression component for eliminating selection bias and achieving significant reduction in MSE.

Abstract: A/B testing on platforms often faces challenges from network interference, where a unit's outcome depends not only on its own treatment but also on the treatments of its network neighbors. To address this, cluster-level randomization has become standard, enabling the use of network-aware estimators. These estimators typically trim the data to retain only a subset of informative units, achieving low bias under suitable conditions but often suffering from high variance. 
In this paper, we first demonstrate that the interior nodes—units whose neighbors all lie within the same cluster—constitute the vast majority of the post-trimming subpopulation. In light of this, we propose directly averaging over the interior nodes to construct the mean-in-interior (MII) estimator, which circumvents the delicate reweighting required by existing network-aware estimators and substantially reduces variance in classical settings. However, we show that interior nodes are often not representative of the full population, particularly in terms of network-dependent covariates, leading to notable bias. 
We then augment the MII estimator with a counterfactual predictor trained on the entire network, allowing us to adjust for covariate distribution shifts between the interior nodes and full population.
By rearranging the expression, we reveal that our augmented MII estimator embodies an analytical form of the point estimator within prediction-powered inference framework~\citep{angelopoulos2023prediction,angelopoulos2023ppi++}. This insight motivates a semi-supervised lens, wherein interior nodes are treated as labeled data subject to selection bias. Extensive and challenging simulation studies demonstrate the outstanding performance of our augmented MII estimator across various settings.

Published as a conference paper at ICLR 2026
JOURNEY TO THE CENTRE OF CLUSTER: HARNESSING INTERIOR NODES FOR A/B TESTING UNDER NETWORK INTERFERENCE
Qianyi Chen', Anpeng Wu', Bo Li Lu Deng Yong Wang Tsinghua University, 2Zhejiang University, Tencent Inc.
{cqy220mails, 1iboBsen).tsigghuu.... tsinghua. edu. en anpwuězju. edu. en {adamdeng, darwinwang] www.wiiwwgg@tennnnn @tencent.......... com
ABSTRACT
A/B testing on platforms often faces challenges from network interference, where a unit's outcome depends not only on its own treatment but also on the treatments of its network neighbors. To address this, cluster-level randomization has become standard, enabling the use of network-aware estimators. These estimators typi- cally trim the data retain only a subset of informative units, achieving low bias under suitable conditions but often suffering from high variance. In this paper, we first demonstrate that the interior nodes- units whose neighbors all lie within the same constitute the vast majority of the post-trimming subpopulation. In light of this, we propose directly averaging over the interior nodes construct the mean-in-interior (MIl) estimator, which circumvents the delicate reweighting re- quired by existing network-aware estimators and substantially reduces variance classical settings. However, show that interior nodes are often not representa- tive of the full population, particularly in terms of netork--eepeneet covariates, leading notable bias. We then augment the MII estimator with counterfactual predictor trained on the entire network, allowing us to adjust for covariate distri- bution shifts between the interior nodes and full population. By rearranging the expression, we reveal that our augmented MII estimator embodies an analytical form of the point estimator within prediction-pows poweeed inference framework (An- gelopoulos et al., 2023a;b), This insight motivates a semi-supervised lens, wherein interior nodes are treated as labeled data subject to selection bias. Extensive and challenging simulation studies' demonstrate the outstanding performance of our augmented MII estimator across various settings.
INTRODUCTION
A/B testing has long been the gold standard for modern platforms in deciding whether to launch new product features. However, its basic procedures easily fail and lead to misleading conclusions when interference exists-specifically when the classic Stable Unit Treatment Value Assumption (SUTVA) is violated, and unit's potential outcome is influenced by treatments received by adjacent neighbors. Since these influences typically propagate through network topology, such as friendship relations in social networks, we refer to this phenomenon network interference.
In most industrial A/B testing scenarios, the estimand of interest is the global average treatment effect (GATE), defined as the difference between the mean outcomes under global treatment and global control. Under network interference, we position the estimation of GATE as a nontrivial extrapolation task. Rather than relying on graph-agnostic approaches that drastically reduce dimen- sionality to one or two dimensions, e.g., Yu et al. (2022b); Cortez-Rodrighe Rodriguez et al. (2024); Bayati al. (2024), we argue that leveraging the known graph structure much more appealing.
Corresponding author.
'The code is available at https : //gi wwt.tht.o/ccc..... AMT Harness i ig- Interior-No des for- wotwork wetwork--xperrii ments

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Boumdary
100
Inserior
620
400
200
ACoooend Closkes
Dogre
(b)
Figure 1: (a) Histogram of the number of distinct clusters each unit is connected to. (b) Histogram of degree for boundary and interior nodes. The network topology is derived from Facebook net- work (Gui et al., 2015), and clusters are generated through Louvain algorithm (Blondel et al., 2008).
As the most basic approach, the effference-in-meany estimator is graph-agnostie and thus suffers from severe bias in the presence of interference (Karwa & Airoldi, 2018). This issue especially pronounced in early-stage experiments, where the treatment proportion typically falls below 10%. In such settings, treated units often have a high proportion of neighbors in the control group, meaning that the treatment vector in their 1-hop ego network can differ significantly from the global treatment.
To leverage the graph topology, the cornerstone methodology is graph cluster randomization (Hud- gens & Halloran, 2008; Ugander et al., 2013), which ensures that all units within a cluster receive the same treatment. These clusters are typically pre-generated using graph clustering algorithms. Unlike classic unit-level randomization, graph cluster randomization introduces strong correlations among densely connected units. This allows interior nodes to experience an environment that closely approximates global treatment, thereby facilitating GATE estimation.
Moreover, a widely considered regime is the neighborhood interference assumption (NIA), which informally assumes that interference is restricted cluster-level randomization, the Thomisz- (HT) estimator with network exposure indica- tors (Ugander et al., 2013; Ugander & Yin, 2023) is unbiased. This estimator includes only the outcomes of units that satisfy specific exposure conditions e.g. having all neighbors receive the same treatment level. When no ambiguity arises, we refer to simply as the HT estimator and defer its formal definition to Section 2.
unit's I-hop neighbors. Under the NIA and
Under graph cluster randomization, nodes located in the interior of clusters are always clean and thus are included in the HT estimator. However, these interior nodes represent only small fraction of the overall population. In contrast, much larger number of nodes lie along cluster boundaries, yet they are significantly less likely to be selected by the HT estimator. To illustrate this disparity, consider a unit connected to e different clusters, where independent Bernoulli randomization with treatment proportion p is conducted at the cluster level. The probability that this unit is a clean treated node is pe, which can be vanishingly small. To build intuition, Figure la displays the distribution of e in our social network. We observe that interior nodes constitute only 8% of the total population. In contrast, boundary nodes often have large values of c, leading to extreme inverse probability weights of the form (I/p)" in the estimator, resulting in highly inflated variance.
There have been numerous efforts to address the issue of explosive variance in the HT estimator. On one hand, refined randomization schemes have been proposed control the theoretical variance bound of the HT estimator (Ugander & Yin, 2023; Kandiros et al., 2025). However, due to the inherent density of practical social networks, even when sharp theoretical bounds are achieved, the realized variance in practice remains unsatisfactory. On the other hand, alternative estimators have emerged that introduce slight bias in exchange for significant variance reduction, such as the cluster-adaptinc estimator (CAE; Liu et al., 2024). CAE also uses exposure indicators to select clean nodes, but replaces inverse probability weighting with a bilevel averaging of outcomes- within clusters, then across clusters. Nevertheless, we claim that practical clustering algorithms rarely satisfy the strong structural assumptions required by CAE, and that its bilevel averaging step can be further simplified to single average average-leading to substantial variance reduction.
In this paper, we first propose the mean-in mean-in-interior inteeiiorr estimator, which mimics the difference- in-means estimator by assigning much more moderate weights to unit outcomes, in contrast to the exponential
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weights used in the HT estimator- the primary source of its extremely high variance. Our estimator based on the extensive practice of cluster-level randomization and relies solely on the interior nodes of clusters to compute mean outcomes. Intuitively, these interior nodes tend to reside in cleaner environment, closer to global treatment or control conditions. With similar assumptions to Liu al. (2024), specifically that the interior nodes of each cluster representative enough of the whole cluster, we prove the consistency of the MIl estimator.
Furthermore, we observe that the HT, CAE, and MIIl estimators all exclude most boundary nodes; this exclusion inevitably introduces selection bias when there is a systematic discrepancy between interior and boundary units. To address this challenge, we incorporate counterfactual predictor trained on the entire network. Leveraging this predictor, we construct an adjustment term for the MII estimator that mitigates covariate mismatch between interior and boundary units. Moreover, by rearranging terms in the estimator, we provide an alternative interpretation through the lens of semi-supervised learning: interior nodes play the role of labeled data-albedt potential selection boundary nodes contribute representative covariates with partial label information.
Related Works Causal inference under interference has been extensively studied in early works (Sobel, 2006; Rosenbaum, 2007). A foundational methodology in this field is graph clus- ter randomization (Hudgens & Halloran, 2008; Ugander et al., 2013), which allocates treatments at the cluster level and ensures that all units within cluster receive the same treatment. This approach enhances the statistical performance of several classic estimators, making it a common practice. Ad- ditionally, because general interference is often intractable, numerous structural assumptions have been proposed to simplify interference patterns and facilitate analysis.
A significant body of research has centered the partial interference assumption (Hudgens & Hal- loran, 2008; Bhattacharya et al., 2020; Forastiere et al., 2021; Candogan et al., 2023), which posits that the network can be partitioned into disjoint clusters, with interference occurring only within each cluster. While this assumption significantly simplifies the interference structure-allowing cluster-level randomization to produce unbiased estimates- it is somewhat restrictive, particularly in settings such as social networks and online marketplaces.
Beyond this special regime, three main structural assumptions have been developed to restrict the form of interference. The first and most widely studied interference pattern in the literature is neigh- borhood interference (Forastiere al., 2022; Cortez al., 2022: Ugander & Yin, 2023; Liu al., 2024), which assumes that interference is restricted to an individual's 's direct neighbors. Under this assumption, a unit surrounded by neighbors receiving the same treatment level can be considered as being in globally treated or control environment, making it well-suited for GATE estimation. With stronger parametric structure, exposure mapping (Aronow & Samii, 2017; Eckles et al., 2016; Ugander et al., 2013; Baird al., 2018; Baqquez-Barc, 2023) is proposed as summary of the effect of the entire treatment vector on a specific unit through predefined function or representa- tion. Furthermore, the potential outcome model is leveraged to develop more elaborate experimental design (Basse & Airoldi, 2018; Yu et al., 2022a; Harshaw et al., 2023; Chen et al., 2023).
Additionally, there is body of work that leverages graph neural networks (GNNs) as powerful regression models for performing counterfactual prediction on observational data (Leung & Loupos, 2024: Cai al., 2024; Wu et al., 2025). Specifically, the GATE can be viewed as the difference between two counterfactuals at boundary points, i.e., the global treatment versus the global control.
2 BASIC SETTING
2.1 PRELIMINARY
We consider finite population of units interconnected through interference network with known topology. This network is represented as an undirected graph g = (V, E), where the node set V/ corresponds to the unit set [n] = {1, n.,, and the edges described by the adjacency unit as
matrix A. We denote the degree node by deg,, and define the r-hop neighborhood N( (j {(i,j) r), where ji is the shortest path distance between units i and j.
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We adopt the Sayman-Rubin potential outcomes framework (Rubin, 1974). The treatment assign- ment is represented by the vector z = (21, 22,... 2n,) E {0, (0, We define the potential outcome for unit as Y, (z), where Y, may depend the treatment assignments of other units.
The parameter of interest is the global average treatment effect (GATE), defined as:
Y())),
(1)
VE(n]
where 1 and 0 denote the n-dimensional vectors of ones and zeros, representing the global treatment and global control conditions, respectively.
We now introduce our randomization scheme-cluster-levee independent Bernoulli randomization. First, we apply a community detection algorithm to generate clusters, which we take as given in this paper. We denote the resulting clusters C1, C2..... ,, Ck, forming a partition of the node set V.
Second, in each experiment, treatment is assigned at the cluster level, meaning that all units within the same cluster receive the same treatment. The treatment assignments for these clusters are inde- pendently drawn from Bernoulli distributions.
Additionally, for given cluster Ck. the interior is defined as:
Tnt. {i CR, 1) S Ck).
(2)
That is, node belongs to the interior if all of its 1-hop neighbors are also contained within the same cluster. The boundary set is defined as the complement within the cluster: Bnd,, C2 \ Int,- To- gether, the interior and boundary sets partition the cluster. Intuitively, interior nodes are particularly valuable for estimating the GATE Under cluster-level randomization, their 1-hop neighbors share the treatment level, making their local environment closely resemble the global treatment or control.
We then formally introduce the neighborhood interference assumption (NIA), which confines inter- ference effect to the I-hop neighborhood of each unit.
Assumption 2.1 (NIA) We have (2 = (z') ifz, = z holds for all E (i,
This assumption is prevalent in the literature, as it imposes a high-level structural constraint on inter- ference while remaining agnostic to the specific form of the exposure function- is, it does not require a parametric model for how neighboring treatments are aggregated to influence outcomes. Throughout the remainder of this work, we primarily operate under this assumption.
2.2 ALTERNATIVE ESTIMATORS
We now introduce three popular alternative estimators against which we will compare our method. The first the cnfferenccein-meann (DIM) estimator, which pefffernnne-annossie and simply com- putes the difference in mean outcomes between treated and control units.
211
11-2)Y;
ÎDIM
(3)
Eie(n)
Eie(
2;)
The second estimator is the HT estimator with network exposure (Ugander & Yin, 2023).
5x(1)
5z(0) [()
FHT
Y.,
(4)
71 2E
E [[)
where 5; is the exposure indicator: 6; (=o) 1KE jEN.)))))) /// deg, 20, 20). This definition
implies that the HT estimator relies solely the outcomes of clean nodes whose neighbors all receive the same treatment level as the node itself. This exposure condition is commonly re- ferred to full-neighborhood exposure (Ugander al., 2013). Under the NIA, the HT estimator is guaranteed to be unbiased, making it theoretically appealing choice.
Finally, we introduce more nuanced estimator, the CAE (Liu al., 2024). Like the HT estimator discussed earlier, CAE uses the same indicator to select clean nodes. However, instead of applying inverse probability weights, it computes bilevel average over the observed outcomes. As such,
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can be interpreted hybrid of the DIM and HT estimators, To define it, let Eg denote the treatment assigned to cluster Ck. We first compute the within-cluster average:
R,000
Σ sECk
Ejec, 62(2))
(5)
Then, an outer average is taken across clusters, yielding the CAF:
ECAE ΣkE[K] t4yK,,1
ΣNEKKY (1 - YT,D
(6)
EieRKy t ElE(K((1 - t1)
HARNESSING THE INTERIOR NODES
3.1 MEAN-IN-INTERION ESTIMATOR
Now we introduce our MII estimator, which is the difference between the outcomes of treated and control interior nodes. Here, Int UkE[K] Inta denotes the union of the interior sets across clusters.
Eielnt 2iYi
Eielat (1 2,
ÂMIT Ejelnt Zj
Ejemm (1 2z)
As estimator based the difference-in-means approach, the MIl estimator assigns moderate weights to observed outcomes. Empirically, we find that the bias of the CAE estimator is similar to that of the MII estimator, while the MII estimator exhibits substantially lower variance. Furthermore, under assumptions similar to those in Liu et al. (2024), we demonstrate that the MII estimator is consistent, with proof provided in Appendix B.1.
Assumption 3.1 Technical assumptions for the consistency of MIl estimator.
1. The proportion of interior nodes becomes asymptotically uniform across all clusters:
Int,
71
1
ΣNE[K]
Int&
EXE(K] 71
Op
K
(8)
2. The interior nodes provide a sufficiently representative sample of each cluster:
max (1) Yn. (arne(1) =1p)
kE[K]
max
%% Vn. = (1)
kE[K]
Here, Vx(1) and (1) denote the mean outcomes within cluster l and within the interior nodes of cluster respectively. The asymptotics is with regard to the number of clusters, K.
Theorem 3.1 Suppose Assumption 2.1 and 3.1 hold; then the MIl estimator is consistent, i.e.,
fMll - op()).
(10)
We first note that the two parts of Assumption 3.1 specify uniformity across clusters and the rep- resentativeness of interior nodes, respectively. Notably, our requirements are considerably weaker than the corresponding assumptions in Liu et al. (2024)-who for instance, assume mean outcomes within each cluster to be an identical constant. This discrepancy arises partly because our current focus is limited to consistency, whereas their work aims establish Central Limit Theorem (CLTT).
In practice, deriving a CLT and performing subsequent statistical inference under network interfer- ence is intrinsically difficult for global estimands like GATE (Kandiros et al., 2025). With specific experimental design and estimator, even if one accepts various unverifiable technical assumptions, there remain stringent restrictions on network density; for example, the maximum degree of the two-hop connected graph g2 should be o9n (Kandiros et al., 2025).
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In summary, compared to bias analysis, characterizing variance is typically far more complex due to the intricate interdependencies induced by the network. To provide intuition, we illustrate the variance of the MIll estimator using the following potential outcome model:
ESENNaa) 3j deg,
(z) = 32; + h
(11)
Here, 3 is the direct treatment effect, covariate v; is i.idd across all units, and is the interfer- ence function. This model incorporates the classic linear-in-means (Aronow & Samii, 2017) and polynomial interference model (Cortez et al., 2022) as special cases.
An important property of this class of potential outcome models is that each interior node pro- vides an unbiased estimate of the mean outcomes under global treatment or control, regardless of the complexity of h. This suggests simply averaging over all interior nodes with equal weights. Per the Cauchy- Schwarz inequality, equal weighting yields the minimum-variance unbiased esti- mator when only interior nodes are available. principle, adding boundary nodes can further re- duce variance. However, under network interference, doing necessitates delicate weights- ee-g. inverse-propensity weights for network exposure in HT estimator or bilevel cluster-relevant weights in CAE-whie which can actually introduce additional inefficiency.
4 ADJUSTING FOR COVARIATE DISTRIBUTIONAL DISCREPANCY
4.1 MOTIVATION
To begin, we argue that structural assumptions imposed on clusters such as the representativeness of interior nodes or a constant expectation for within-cluster mean outcomes (Liu et al., 2024) are often fragile, precisely because the clustering algorithms used in practice tend to be coarse.
In practice, scalable algorithm for generating clusters is usually heuristic one. Graph clustering for minimizing certain objectives is generally a challenging task, often involving NP-hard com- binatorial optimization problems, such as correlation clustering (Pouget-Abadie et al., 2019) and balanced clustering (Brennan et al., 2022). In practice, social platforms with hundreds of millions of users typically construct millions clusters to meet the immense demand for A/B tests, which can number in the thousands weekly. This large-scale problem necessitates heuristic solutions or fast but heuristic algorithms, rendering more elaborate algorithms, such as causal clustering (Viviano et al., 2025), which involves solving semidefinite programming, inapplicable.
Indeed, on a billion-scale social platform, we observe that certain user engagement metrics-among the primary outcome variables recorded in nearly all experiments- exhibit clear discrepancies be- tween interior and boundary subpopulations. Desensitized empirical evidence of this phenomenon is shown in Figure 2. However, as discussed in our introduction, most estimators without regression component tend to rely heavily on interior nodes, e.g., referring back to Equations (4) and (6).
Boanley
Bazakry
hncric
Imanor
Due Brrhommee
(a)
(b)
Figure 2: Distribution of (a) outcome variable (user engagement) and (b) demographic attribute across interior and boundary units. Data are collected from a billion-scale social platform.

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4.2 BRIDGING THE GAP WITH COUNTERFACTUAN PREDICTOR
Since our MII estimator also employs moderate weighting rather than inverse propensity weighting. is similarly susceptible to the aforementioned selection bias. To address this, we propose training counterfactual predictor over the entire graph. Specifically, we introduce the notion of counterfactual prediction: given treatment vector z, covariate matrix X, and network topology A, we define function f(z. X, A) that predicts outcomes for all units under treatment assignment The function can be flexibly parameterized to fit the data; for example, may be instantiated as a GNN.
Given the observed outcomes (z) under randomization, we train the function f via regression. We then shift the treatment regime to z 1, O, and define the augmented MII estimator as:
FAMII +
Σ
f(1,
1 Σ
x1(1,X,AA)
n jc(m)
$1 iglnt
(12)
1 Σ (1) 80
Σ JE[N]
f(0, A))
iEla
Here, 81 (80) denotes the number of treated (control) units within the interior set. The adjustment term captures the difference between the average predicted outcomes for the full population and those for the treated (control) interior nodes. To illustrate the idea behind the AMII estimator, we consider the following potential outcome model, which includes an additional interaction term between the treatment and a covariate that can be network-dependen
(z))) = (B + omu))2; + (ESENGJA) 3j
(13)
deg,
Here, the v; are covariates that may be i..dd, but we allow u, to be network-dependent and thus non-i.idd.; for example, u, could represent the normalized degree deg, /deg, whose distribution differs between the boundary and interior subpopulations. Additionally, B and a are unknown scalar parameters, and an unknown interference function, which may be non-linear. Given this model and the form of the AMII estimator, we provide the following intuition: regression models with suitably chosen inputs can often capture the interaction term effectively. However, in early-stage experiments (small p). they may struggle to extrapolate the interference function h-precisely the component where the MII estimator demonstrates a comparative advantage.
More broadly, identify the extrapolation of the interference term the central challenge in estimating the GATE under network interference. practice, we typically observe outcomes under low treatment proportions (e.g.. 0%, 5%, 10%), while our goal is to extrapolate to fully treated scenario (p = 100%). A key observation is that for boundary units i, the network exposure term EjEA((,1) ł// deg, tends to concentrate around p. which reflects regime far from global treatment. This highlights that the main extrapolation challenge lies in recovering the interference function h. estimators can
As also shown in Chen & Li (2024), the performance of naive regression- degrade significantly when is nonlinear (e.g., square root, quadratic), in contrast the línear case.
Next, we provide a formal characterization of the intuition of AMII estimator, highlighting the bias reduction achieved by the adjustment term. To manage the complexity arising from network dependencies and the functional form of the regression, we impose parametric assumptions on f.
Assumption 4.1 (Regression Model) Given n samples {uu, Vi, 24, Y.) we assume the trained regression function f has the form:
f(z A,); (3,, &,, + MEAN((90 U,) 15 1(i,
(14)
Here, Bm. ân, denote the estimated coefficients of the linear part, 90 is a learnable transformation
function (e.g., multilayer perceptron), and MEAN denotes the mean pooling of a set of real values.
This functional form mimics partial linear model. We assume the linear part is correctly specified, while the non- non-linen linear part- the interference apply form of one-layer graph convolution.
be misspecified. For this non-linear part,

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Theorem 4.1 Given Assumption 4.1, potential outcome model in Equation (13), and network topol- ogy G. the biases of the MIl estimator and the AMIl estimator are given by:
Bias(farn) = ax(Mnn - 11) Bias(FAMrT) - (Elāâ) - a:)(u 1 (int),
(15)
where jr = Elu,], Piint E[u
Int].
The proof of Theorem 4.1 is provided in Appendix B.2. In summary, we observe that the bias in MII arises from the discrepancy between the interior and the entire population. Furthermore, when the
linear component is correctly specified, the quantity [E[á,,] al is typically much smaller than Jal, indicating a substantial reduction in bias. Additionally, we find that the AMII estimator is harmless adjustment in the absence of distributional differences between the interior and the full population; that is, it does not introduce any additional bias.
Since these results rely on parametric assumptions, we evaluate the performance of the AMII es- timator through a systematic simulation study where structural assumptions are violated in various ways. Despite this, the AMII estimator demonstrates robustness and outstanding performance.
4.3 TALE OF SEMI- SUPERVISION
We then present another interpretation of our method. For simplicity, we focus estimating the mean outcome under global treatment. By rearranging the expression AMII estimator, we obtain:
FAMIL,I
Σ A), n
Σ 2. -Y, X, A))).
(16)
JEIn]
selnt
This embodies a form of the point estimator in PPI (Angelopoulos et al., 2023a;b), and we interpret estimator within that sound framework. Specifically, we treat the outcomes of treated interior nodes as reliable labeled samples, while the outcomes of boundary nodes are used solely for train- ing the predictor f. The goal is estimate the population mean outcome (i.e., the mean label), using a small subset of labeled data (interior units) alongside large pool of unlabeled samples, for which only covariates model-based predictions using a small set of true labels, thereby mitigating estimation eπrоr This perspective aligns with this reorganized form of our estimator.
ideas behind PPI is correct the bias of pure observed. One of the
However, due to the complexity of network interference, the mean of "true labels" still incurs bias. As noted earlier and illustrated in Figure 2, this bias is sourced from the formation of clusters, i.e., there exists substantial distributional discrepancy between interior and boundary units, in contrast the
at-random setting in PPI framework. Hence, our AMII estimator should be first positioned as debiasing using prediction, by which we tackle such selection bias of mean of "true labels". comparison, the core idea of both PPI point estimator and the classic doubly-robust estimator is debiasing the mean of predictions using a few labeled samples.
5
SIMULATION STUDY
5.1 BASIC SETUP
First, since the point of our methodology the reduction of mean squared error (MSE) rather than purely variance reduction, we must demonstrate the effect of bias reduction, especially for the pro- posed AMII estimator. However, due to the complex effect in real platform experiments, c.g., ex- istence of many parallel experiments, temporal effect, launch of other new traits, etc., we have no access to the ground truth of GATE, which hinders the evaluation of bias. Therefore, we follow the convention of existing literature and conduct Monte Carlo simulation with 1,000 repetitions, where randomness arises from treatment allocation (randomization) and exogenous noise in the potential outcomes. We then assess the bias, variance, and MSE based on these repetitions.
Data. We use a Facebook social 2 consisting of 11,586 nodes and 568,309 edges. To generate clusters, we apply the Louvain algorithm and report results using a resolution parameter
Thee network topology is available at http:://eeuutttppsssssooommmmtmt
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y which yields 95 clusters. The resolution primarily affects the number of clusters and has minor effect on the proportion of interior nodes (about 8%). Based on this clustering, we implement independent Bernoulli randomization at the cluster level and evaluate three treatment proportions, E {0.1, 10.1,00. 0.3, 0.5), corresponding to the three-stage experiments conducted prior to the launch.
We focus the estimation of the mean outcome under global treatment, and set Y((0) 0 for simplicity. When this assumption does not hold, a simple aajustment--sutraacnne the baseline level (Yu et al., c0n transform the problem into this setting.
Potential outcome model. We adopt the 2-hop interference framework from Chen & Li (2024) and additionally include two covariates: degree and number of distinct clusters each unit is nected to, denoted by deg, and C₁ respectively. The distributions of these covariates differ between interior and boundary nodes, thereby inducing violation of the representativeness assumption.
1 deg
+ - Oz+ae.
(z) = 3z + Bz + 2 deg
(17)
Here, B is the interference matrix, defined as:
Σ En(D-LA))
B= (11 Ia)
(18)
In this formulation, D is the diagonal degree matrix, and denotes the element-wise product. This model naturally incorporates a linear 2-hop interference, with the intensity characterized by r2. The diagonal element of B is removed to avoid confusion with direct treatment effect.
Moreover, two normalized covariates interact with the treatment vector z. Finally, e ~~ NO(,, represents exogenous Gaussian noise, and a controls the intensity of this noise.
Parameter setting. We set (8,r)) (1, 1), and consider two levels 2-hop interference, "2 E {0, 1}. corresponding to the case that NIA holds and substantial 2-hop interference exists, respectively. Additionally, following Liu et al. (2024), we examine a setting with a relatively low hignal-t-noiis ratio by setting 2. This scenario is particularly relevant in industrial practice, where most new product traits tend to have relatively minor effects on the outcomes interest.
Baselines. We evaluate five methods in total: the Hájek estimator, the cluster-adaptive estimator (CAE), the mean-in meen-in-inteeior (MII) estimator, the counterfactual prediction method using a graph neural network (GNN), and the augmented MII estimator (AMII).
We now provide some additional notes. Since the variance of the HT estimator tends to be explosive in complex scenarios, we adopt its variant-the Hájek estimator (Ugander & Yin, 2023)- ap- plies soff-normalizaiion to the weights, substantially reducing variance at the cost of a small increase in bias. Given that the potential outcome model is fundamentally linear, we use three Chebyshev convolution layers (Defferrard et al., 2016) to construct the GNN, without incorporating any non- linear activation functions. Overall, this not intended be a delicate architecture.
5.2 RESULTS AND DISCUSSION
Due to space constraints, we present only the results using clustering resolution = 5 in the main paper, as shown in Figure 3. The complete experiments iiineenntttiinuuiinnn alternative resolution levels, different covariate specifications, different network topology, and series of ablation studies are reported in Appendix A. Here, we summarize the main conclusions from our results.
First, we observe the outstanding performance of the AMII estimator: it achieves substantially lower bias while maintaining variance comparable that of the three estimators without a regression component. This translates to much lower MSE, even when Notably, the bias reduction becomes even more pronounced in the presence of 2-hop interference (i.e., 12 1). Furthermore, as the coefficient of the interaction term increases from 0.5 to 1, the advantage of AMII becomes even more distinct.
0.1 and GNN perform poorly.
Second, we examine the performance of the GNN estimator. A notable pattern is its strong de- pendence on relatively high treatment proportion, under which substantially more units become informative for estimating outcomes under global treatment.

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Published as a conference paper at ICLR 2026
Bajjkk
CAE
18
MIl
GNN
1.2
AMII
0.0
0.1
0.2
0.3
0.4
0.5
6.1
0.2
0.4
0.5
0.1
022
0.3
0.4
0.5
2.0
1.0
1.6
0.5
1.2
0.4
0.2
0.0
Figure 3: Statistical performance with 72 = (first row) and 72 = 1 (second row).
Next, we investigate the performance of the MIl estimator. It consistently achieves the lowest vari- ance across all settings, while exhibiting similar level of bias as the Hájek estimator and CAE. This bias is expected due to the influence of covariates deg, and q, In the absence of the interaction term and 2-hop interference (see Table 10), we find that Mil is nearly uubiased-simila, to CAE-while still maintaining the lowest variance, making it the best one there. Nonetheless, the AMII estimator also performs well. This demonstrates a form of harmlessness property of the adjustment term.
Moreover, discuss the impact of clustering resolution. The corresponding statistics are provided in Table 1, and results are presented in Tables through 7. The behavior of bias is difficult to analyze since the pattern of between-cluste edges changes during the cluster formation process. Regarding variance, we note that the standard deviation of estimators generally decreases as increases from 2 to 5 (except for the Hájek estimator), as the number of "cluster-level samples" increases. Nonethe- less, there exists an irreducible component of variance, as demonstrated by the fact that estimator variances remain almost unchanged when the number of clusters further doubles (r increases from 5 to 10). This differs substantially from classical regimes and again illustrates the complicated vari- ance structure under network interference.
Last, we discuss an additional ablation study that examines whether training f on the entire graph or only on the boundary nodes yields better performance. The latter is a natural sample-spin splitting idea, but we note that the complex dependency in network data makes it hard to tease out clear story from this choice. The related results are reported in Tables 12 and 13. Empirically, we find that when the treatment proportion is low (c.g. p 0.1) training with the full population is much better. When p increases, the performance of these two approaches becomes similar. The intuition behind this result is that boundary nodes become significantly more informative instance, the expected exposure (1]] = ff grows rapidly for nodes with a high level c. This implies that as p rises, the estimator captures more signal from the local neighborhood structures, effectively leveraging the boundary information that is otherwise sparse in low-exposure regimes.
p increases; for
6 CONCLUSION
In this paper, based on extensive practice of cluster-level randomization, we systematically identify the limitations of existing estimators in GATE estimation and propose the mean-in-interior estimator to eliminate unnecessary reweighting, achieving further variance reduction. Recognizing the heavy reliance on the interior nodes and potential selection bias sourced from clustering, we further propose an adjustment term and enhance its position through a novel semi-supervision perspective. We summarize the adjustment as "debiasing using predictions", which is of independent interest for correcting selection bias. Through series of challenging simulation studies, we demonstrate the remarkable performance of our methodology, especially in the early-stage experiment.
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Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
This research was supported by the National Natural Science Foundation of China (1..2117131).
ETHICS STATEMENT
This work adheres to the ICLR Code of Ethics. Our study does not involve human subjects, person- ally identifiable information, or sensitive data. All datasets used are publicly available and widely adopted in the research community, and we followed recommended practices to ensure fairness, privacy, and reproducibility. The methods we propose are intended solely for academic research and are not designed to cause harm enable misuse. We not aware of any conflicts of interest, sponsorship concerns, or ethical risks associated with this work.
REPRODUCIBILITY STATEMENT
We have made extensive efforts to ensure the reproducibility of our results. All code and scripts nec- essary reproduce the experiments are provided in the supplementary material. Complete proofs of theoretical results are also included in the appendix. Together, these resources are intended to enable independent verification and facilitate future research building upon our work.
USAGE OF LARGE LANGUAGE MODEL (LLM)
The authors employed L.LMs solely to refine wording and correct grammatical emrors in the final manuscript. All ideas, analysis, and conclusions are the authors' own.
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A DETAILED SIMULATION RESULTS
In this section, we provide detailed numerical results. The default setting has been introduced in the main paper, specifically, the clustering resolution is set to y 5, and the potential outcome model follows Equation (17).
A.1 IMPACT OF CLUSTERING RESOLUTION
We first cxamine the influence of the clustering resolution 7, all five methods rely cluster-level randomization. Broadly speaking, y controls the modularity gain required for merging during the hierarchical clustering process. A higher value of corresponds to lower modularity gain, which tends to preserve more small clusters.
We report the basic statistics for three levels of y, including the number of clusters, the proportion of interior nodes, and the proportion of within-cluster edges.
Table 1: The statistics of different clustering schemes.
Y
#clusters
%interior
Swithin-clusters
2
30
8.4
41.6
5 95
7.9
31.7
10
192
7.2
26.6
We then report the results for y E {2, 5, 500 from Table Table 7. As reminder, the default setting reported in the main paper is = 5, which represents a moderate level. The main takeaway here is that clustering resolution has only minor effect on the performance of our methodology, suggesting that the choice of y can be flexibly guided by other considerations.
A.2 IMPACT OF COVARIATE SPECIHCATION
In this subsection, we examine the impact different covariate specifications and the weight of interaction term. Before presenting the detailed results, clarify that the input features of the GNN consist only of node degree deg; and treatment assignment 2,. This implies that, under the default setting, the GNN explicitly includes ingly, we first investigate the following potential outcome model, which removes the normalized degree from the potential outcome model:
of the covariates- dessssddegeeep---- part of its input. Accord-
Y (z) = 3z + Bz ① 1 ac.
(19)
In this model, the GNN no longer has direct access to the covariate that interacts with the treat- ment assignment. The parameter setting remains the same in the main paper, and report the results in Table 8 and Table 9, Compared to Table 3 and Table 6, we find that the impact of this covariate specification is also minor. This suggests that even when the GNN does not explicitly leverage covariates derived solely from network topology, its performance remains largely unaf- fected. Nonetheless, we note that demographic features, which cannot be inferred from the network structure, should be provided as input covariates for regression.
Next, we consider varying the weight the interaction term, which is 0.5 in the default configuration. From the results in Table 3 and Table 6, we observe that the CAE, Hájek, and MIl es- timators all struggle to effectively address covariate distribution discrepancies, even the treatment proportion increases. This suggests that, as the interaction weight increases, the advantage of the AMII estimator becomes more pronounced. Therefore. we omit experiments with higher weights and instead turn to the opposite scenario zero weight-where the interaction term is absent.
The potential outcome model used in this setting is as follows, and the corresponding results are reported in Tables 10 and I1.
Y = 3z + Bz + ac.
(20)
When NIA holds (i.c., r2 = 0), we observe that the CAE, Hájek and MII estimators all achieve near-unbiasedness, consistent with theoretical expectations. In this setting, the MIl estimator attains
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substantially lower variance, making it the best-performing method in this case. In contrast, the GNN estimator does not benefit significantly from the simplicity of this setting it still requires a larger volume of high-quality data (or higher treatment proportion) to perform well. Interestingly, despite the poor performance of the GNN estimator, the AMIl estimator achieves results compara- ble MII. This phenomenon demonstrates form of harmlessness property, further validating the robustness of our methodology.
In the scenario where r2 I, observe substantial bias across all five estimators. Nonetheless, the MII estimator attains the lowest bias and maintains the lowest variance, reinforcing its position as the most effective method in this setting. This observation further validates our choice of assigning moderate and balanced weights to the outcome components, Moreover, AMII continues to achieve performance comparable to MII, even when paired with GNN architecture that not carefully designed.
In conclusion, we find that MII performs best when there is no interaction term and thus no associ- ated discrepancy issue. In addition, AMII demonstrates competitive performance even with simple GNN. Since real-world scenarios almost always involve complex interaction effects, and the clean structure of Equation (20) is idealized, we recommend using AMII in practice whenever it is feasible to train a counterfactual predictor.
A.3 IMPACT OF TRAINING SAMPLE SELECTION
In this subsection, we investigate whether training the counterfactual predictor the full popula- tion or solely on the boundary units yields better performance. This question naturally arises from the idea of sample splitting, although the intricate interdependencies within the data cannot be fully eliminated by this approach. We report the results in Table 12 and 13, and compare them to Ta- ble 3 and Table We observe that training the counterfactual predictor solely on boundary nodes performs worse when p 0.1, but becomes slightly better when intuition is that boundary nodes become more informative as p increases, since the exposure-e.g the proportion of treated neighbors-shine shifts toward the global treatment regime. Nevertheless, the difference between these two approaches remains minor.
0.3, 0.5. The underlying
A.4 IMPACT OF NETWORK TOPOLOGY
Finally, we conduct additional experiment on different network topology² with comparable scale but sparser connecivity-sspeaiiie specifically, its average degree is approximately half that of the network used our earlier experiments. Although the potential outcome model remains the primary focus of our simulation study, we include this experiment for completeness. The detailed results are presented in Tables 14 and 15, alongside comparisons with Tables 3 and
We observe that the performance of the other three estimators which lack regression compo- nent-remaing largely unaffected by changes in the network topology. Additionally, the GNN-based counterfactual prediction estimator demonstrates improved performance this sparser graph, and this enhancement consistently translates to better performance for the AMII estimator as well.
Table 2: Statistical performance (r'2 = 0, = 2)
P
0.3
0.5 Std
Metric
Bias
Std
MSE
Bias
Std
MSE
Bias
MSE
Estimators
Hájek
0.925
0.405
1.020
0.871
0.289
0.842
0.828
0.197
0.725
CAE
0.942
0.455
1.094
0.940
0.237
0.940
0,923
0.159
0.878
MII
0.947
0.302
0.987
0.944
0.130
0.908
0.940
0.092
0.892
GNN
1.796
1.070
4.369
0.632
0.216
0.446
0.429
0.126
0.199
AMIT
0.694
0.465
0.699
0.265
0.169
0.099
0.187
0.100
0.045
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Table 3: Statistical performance (r2 = 0, 5)
P
0.1
0.3
0.5
Metric
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Estimators
Hájek
0.965
0.558
1.243
0,908
0.420
1,001
0.874
0.370
0.901
CAE
0.947
0.342
1.014
0.948
0.193
0.937
0.948
0.136
0.917
MII
0.955
0.242
0.971
0.961
0.130
0.940
0.961
0.092
0.933
GNN
1.497
0.997
3.235
0.650
0.173
0.452
0.430
0.1 14
0.198
AMII
0.597
0.412
0.526
0.249
0.160
0.088
0.165
0.097
0.037
Table Statistical performance (72 0, 10)
P
0.1
0.3
0.5
Metric Estimators
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Hájek
0.954
0.593
1.261
0.929
0.466
1.081
0.890
0.400
0.952
CAE
0.982
0.349
1.086
0.981
0.192
0.999
0.968
0.140
0.956
MII
0.975
0.241
1,010
0,974
0.129
0.965
0.971
0.098
0.952
GNN
1.608
1.009
3,603
0,678
0.169
0.489
0.448
0.110
0.213
AMII
0.642
0.401
0.572
0.242
0.147
0.080
0.153
0.101
0.034
Table 5: Statistical performance (7' = 1, = 2)
P
0.1
0.3
0.5
Metric
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Estimators
Hájek
1.396
0.428
2.133
1.202
0.310
1.541
1.061
0.217
1.173
CAE
1.466
0.459
2.360
1.352
0.243
1.888
1.227
0.168
1.533
MII
1.422
0.329
2.131
1.294
0.149
1.697
1.200
0.103
1.451
GNN
2.247
1.170
6.418
1.101
0.228
1.263
0.770
0.157
0.618
AMII
1.150
0.492
1.563
0.713
0.192
0.545
0.486
0.124
0.251
Table 6: Statistical performance (r2 = 1, 5)
P
0.1
0.3
0.5
Metric Estimators
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Hájek
1.480
0.575
2.520
1.274
0.438
1.814
1.129
0.377
1.416
CAE
1.555
0.345
2.538
1.428
0.195
2.076
1.293
0.138
1.692
MII
1.471
0.274
2.239
1.341
0.154
1.822
1.239
0.100
1.544
GNN
1.917
0.906
4.494
1.142
0.175
1.336
0.792
0.117
0.641
AMII
1.077
0.388
1.311
0.731
0.168
0.563
0.480
0.100
0.241
Table Statistical performance (r2 1, 10)
P
0.1
0.3
0.5
Metric Estimators
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Hájek
1.493
0.605
2.597
1.319
0.476
1.965
1.164
0.408
1.522
CAE
1.628
0.353
2.775
1.486
0.193
2.246
1.334
0.141
1.798
MII
1.514
0.264
2.362
1.379
0.141
1.921
1.271
0.103
1.625
GNN
1.979
0.897
4.722
1.184
0.176
1.433
0.824
0.111
0.691
AMII
1.131
0.395
1.434
0.753
0.150
0.590
0.489
0.100
0.249
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Published as a conference paper at ICLR 2026
Table 8: Statistical performance with one covariate (72 = 0 5)
p
0.1
0.3
0.5
Metric Estimators
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Hájek
0.958
0.557
1.228
0,905
0.416
0.992
0.870
0.366
0.890
CAE
0.935
0.342
0.992
0.936
0.193
0.913
0.933
0.136
0.889
MII
0.949
0.242
0.959
0,957
0.130
0.933
0.958
0.092
0.927
GNN
1.428
0.991
3.023
0.639
0.173
0.439
0.425
0.116
0.195
AMII
0.578
0.394
0.490
0.274
0.160
0.101
0.197
0.100
0.049
Table Statistical performance with one covariate (72 1, 5)
P
0.1
0.3
0.5
Metric Estimators
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Hájek
1.472
0.572
2.493
1,271
0.431
1.800
1.124
0.371
1.401
CAE
1.544
0.345
2.502
1.415
0.195
2.040
1.279
0.138
1.654
MII
1.465
0.270
2.218
1.337
0.151
1.811
1.236
0.099
1.536
GNN
1.857
0,876
4.215
1.139
0.160
1.322
0.785
0.108
0.627
AMII
1.076
0.377
1.301
0.754
0.161
0.595
0.507
0.097
0.267
Table 10: Statistical performance without covariates (r' = 0, = 5)
P
0.1
0.3
0.5
Metric Estimators
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Hájek
0.019
0.558
0,312
0.001
0.415
0.172
0,013
0.369
0.136
CAE
0.023
0.342
0.118
0.016
0.193
0.038
0.008
0.136
0.019
MII
0.011
0.242
0.059
0.003
0.130
0.017
0.002
0.092
0.009
GNN
1.708
0.555
3.224
0.561
0.148
0.337
0.389
0.089
0.159
AMII
0.011
0.264
0.070
0.058
0.177
0.035
0.034
0.118
0.015
Table 11: Statistical performance without covariates (r/2
1, 5)
P
0.1
0.3
0.5
Metric Estimators
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Hájek
0.533
0.572
0.611
0.366
0.427
0.316
0.268
0.371
0.209
CAE
0.586
0.345
0.463
0.463
0.194
0.252
0.338
0.138
0.133
MII
0.504
0.270
0.327
0.377
0.151
0.165
0.275
0.099
0.086
GNN
2.397
0.792
6,373
1.098
0.150
1,227
0.771
0.109
0.607
AMII
0.552
0.284
0.385
0.464
0.176
0.247
0.309
0.120
0.110
Table 12: Statistical performance with regression on boundary (r2 = 0, 5)
P
0.1
0.3
0.5
Metric Estimators
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Hájek
0.965
0.558
1.243
0.908
0.420
1,001
0.874
0.370
0.901
CAE
0.947
0.342
1.014
0.948
0.193
0.937
0.948
0.136
0.917
MII
0.955
0.242
0.971
0.961
0.130
0.940
0.961
0.092
0.933
GNN
2.281
1.030
6.265
0.662
0.167
0.466
0.446
0.118
0.213
AMII
0.740
0.421
0.725
0.144
0.204
0.062
0.092
0.131
0.026
17

---

Published as a conference paper at ICLR 2026
Table 13: Statistical performance with regression on boundary (r2 1, 5)
P
0.1
0.3
0.5
Metric
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Estimators
Hájek
1.480
0.575
2.520
1.274
0.438
1.814
1.129
0.377
1.416
CAE
1.555
0.345
2.538
1.428
0.195
2.076
1.293
0.138
1.692
MII
1.471
0.274
2.239
1.341
0.154
1.822
1.239
0.100
1.544
GNN
2.463
1.211
7.535
1.167
0.190
1.398
0.812
0.117
0.673
AMII
1.080
0.492
1.408
0.625
0.222
0.440
0.386
0.136
0.167
Table 14: Statistical performance on FB-USF (r2 = 0, = 5)
P
0.1
0.3
0.5
Metric
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Estimators
Hájek
0.906
0.646
1.238
0.794
0.508
0,889
0.710
0.435
0.694
CAE
0.954
0.311
1.007
0.925
0.165
0.883
0.896
0.109
0.815
MII
0.944
0.241
0.950
0.943
0.131
0.906
0.943
0.101
0.899
GNN
0.821
0.381
0.819
0.559
0.187
0.347
0.346
0.094
0.128
AMII
0.302
0.256
0.157
0.158
0.138
0.044
0.144
0.101
0.031
Table 15: Statistical performance on FB-USF (r2 1, 5)
P
0.1
0.3
0.5
Metric
Bias
Std
MSE
Bias
Std
MSE
Bias
Std
MSE
Estimators
Hájek
1.432
0.651
2.475
1.188
0.511
1.672
0.989
0.442
1.173
CAE
1.549
0.314
2,498
1.400
0.171
1.988
1.253
0.117
1.583
MII
1.483
0,250
2.261
1.364
0.138
1.881
1.259
0.106
1.597
GNN
1.457
0.366
2.258
1.001
0.142
1.021
0.714
0.094
0.518
AMII
0.879
0.281
0.851
0.633
0.137
0.419
0.504
0.105
0.265
18

---

Published as a conference paper at ICLR 2026
B PROOF
B.1
PROOF OF THEOREM 3.1
First, introduce the basic notations: my Int, denotes the size of the interior set of k-th clus- ter, n& = |C&| denotes the size of k-th cluster. Moreover, we use Inr Fa denote the mean outcome of the interior nodes of the k-th cluster, under global treatment and control, respec- tively. Correspondingly, (1)1), Y& (0) denote the mean outcome of the nodes of the k-th cluster, under global treatment and control, respectively. At last, we use iy as the treatment assignment indicator for the k-th cluster.
Since we employ cluster-level randomization, the treatment assignments of units within the same cluster are highly correlated. Under the neighborhood interference assumption, we begin by rear- ranging our estimator as follows:
fMII= = Eielue 2,Y;
Evelue - (1- 2;
Ejennt 2/
Ejelue 24)
E Ejetnt
Iie 211;(1)
Eielne (1 21)Y;(0)
Ejetne - 2j)
(21)
= EME[K] Exelnta 2:Yi(1) ΣnE(K] Eielnte (1 - 23) ()))
Ejeint 2j
Ejeim (1 2ÿ)
ΣkE[K] lame (
EKE[K] 1x) V (n)
ΣkETKy lgme
ExE[K] )mm
The second equality holds due to NIA (Assumption 2.1) and cluster-level randomization. The third equality transforming the summation over all interior units to that first over interior of k-th cluster, then over clusters. The fourth equality is definition.
By applying Slutsky's lemma, we eliminate the randomness in treatment assignments, yielding:
EkEpK my P&(1) ΣkE(K) m&
EkEjK my Pe(O)
ÍMIT
+ op()).
(22)
ExE[K] my
Assumption 3.1 is adapted from the Assumptions 4.1.1 and 4.1.2 from the original paper of CAE (Liu et al., 2024), while ours are weaker partly because we only pursue consistency. Given Assumption 3.1, we can substitute the component of interior nodes into that of the whole cluster, yielding:
FMII EKEjK] T4 (x. EKE(K] nip (n. + o1)
ΣKE[K] TKk
ΣnEIKi nh
EkejK] nu Yx(1) Ene[K] ny (O( EkE(K] nh
(23)
+ (1())
ExE[K] ni
= =p
B.2 PROOF OF THEOREM 4.1
First, define h(c) E.[h(ee v)]. Recall that the potential outcome model is:
(x(z) (3 amx)2: h( Σ
se/d,g, vz).
(24)
JEN(6.1
Thus, the true GATE is given by:
T 3 + ap + h(1) - h(0).
(25)
Next, with this model, the expectation of MIl estimator is given by:
ÊMIT B axpint h(1) h())
(26)
19

---

Published as a conference paper at ICLR 2026
These together give:
Bias( = a(k - MIna).
(27)
Next, we examine the AMII estimator. We define:
hpp) Rez-Be(()),, [MEAN(19(2)) 15) 1 E (i
(28)
This represents the expectation of the estimate of the interference part given the treatment proportion of randomization equal to p. Notice that the randomness lies both treatment assignment and the covariate Nonetheless, the former diminish when p 0 p 1, since wherein 0, 1 holds, respectively.
An important trait of the functional form of regression function lies in:
1
h(1) E,,
(EAN((((1, lje e (i),
(29)
Int.
sElnt
This holds because swap the MEAN and expectation E,, given the degree deg, fixed. On the other hand, one can easily verify:
h(1) = E.
Σ
MEAN((o(L,)) 1j W 9)
(30)
n iE(m)
Thus, we have:
E[AAMu] = 3 1[m3 (a E(anD)miee + h(1) h(1) + E + E(á,l h(1)
-(n(0) i((0) + R(O))
(31)
B + (a B( B( )) + Bianlu+ h(1) h(0).
This gives the bias of AMII estimator:
Bias(FAMMI) (E[â] - a) | plat).
(32)
20


Reviews:
[{"review_id": "QPvATDzDes", "reviewer": "Reviewer_JmRm", "summary": "The paper addresses the challenge of estimating the Global Average Treatment Effect (GATE) in the presence of network interference, a scenario where a unit’s outcome is influenced not only by its own treatment but also by the treatments of its neighbors. The authors focus on cluster-level randomization and identify limitations in existing estimators and propose new mean-in-interior estimator and augmented mean-in-interior to address the limitations in existing estimators for estimating GATE. Simulation studies are conducted to assess and validate the effectiveness of the proposed estimators and compare with existing estimators across a range of network and treatment settings.", "questions": "1. The nodes within each cluster are partitioned into interior and boundary nodes. In sparse networks, the number of interior nodes may be relatively small, which could adversely impact the performance and stability of the proposed estimators. Could the authors elaborate on how the estimators behave under such conditions, and whether any adjustments are needed to ensure reliable performance in sparse or low-density network settings?\n2. The paper proposes mitigating selection bias in the Mean-in-Interior estimator by training a counterfactual predictor over the entire graph. Could the authors elaborate on the rationale behind this approach?\n3. In the proposed approach, boundary nodes are excluded and only interior nodes are retained for estimating the global average treatment effect under network interference. However, interference is inherently present in GATE since it captures both direct and indirect effects propagated through the network. How do the authors account for or mitigate the influence of network interference on the estimation process, given that boundary nodes—where interference effects may be most pronounced—are omitted from the analysis?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.  The paper addresses the important problem of estimating the average treatment effect under network interference, which has practical significance for causal inference in many real-world applications.\n\n2. The motivation to extend existing estimands is well-articulated, and the comparisons with prior work are thoroughly examined.\n3. Viewing interior nodes as \"biased labeled data\" and approaching the problem from a semi-supervised learning angle is conceptually compelling.\n4. The core idea is clearly explained and easy to follow.", "weaknesses": "1. The proposed estimators rely on strong parametric assumptions, which may limit their robustness and applicability in real-world settings. Moreover, the practical implementation of these estimators could be challenging, particularly in large-scale or heterogeneous networks where model assumptions may not hold or be verifiable.\n2. The impact of network interference on the proposed estimators is not thoroughly addressed. In particular, the paper does not fully analyze how varying levels or structures of interference may affect estimator performance, which raises concerns about their robustness in complex or sparse networks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WFSx2nE6pA", "reviewer": "Reviewer_gJGo", "summary": "The paper proposes a new estimator for the global average treatment effect (also known as the total treatment effect) in the presence of network interference. The work falls under the framework of cluster randomization, where conventional methods (e.g. difference in means or Horvitz-Thompson) suffer from either bias due to inter-cluster interference or large variance due to randomization. Specifically, the authors propose to focus on the interior units whose potential neighbors are all within the same cluster in order to reduce the variance. Furthermore, the bias from sub-sampling is corrected (or partially corrected when mis-specified) by a counterfactual predictor. Under certain assumptions, the bias bound is carefully elaborated, and more simulations are added for model misspecification.", "questions": "1. How can we use the proposed method to do hypothesis testing on GATE? Is there any direct way that we can estimate the variance of the MII estimator along with the asymptotical normality property? \n2. I'm a bit confused about the asymptotic scheme the assumption 3.1 is working on. Are we considering a sequence of networks with an incremental number of clusters? It could be more helpful if the authors can show that the assumption 3.1 is easy to satisfy for common random network models. The $o_p(K^{-1})$ bound for the maximum deviation does not sound trivial to me. My intuition for the bound is around $O_p(K^{-1/2}\\log K)$ for Erdos-Renyi random networks or similar. \n3. The bound in Theorem 3.1 does provide a convergence rate related to either cluster size or number of clusters. It could be refined to provide more insights on designing an experiment.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "- **originality**: the proposed method focusing on interior units is new and novel.  \n- **quality**: the paper is well-written with solid theoretical results and simulations. \n- **clarity**: the paper is clear and well-explained assumptions \n- **significance**: the significance is relatively fair (see weakness part).", "weaknesses": "Although the paper provides a thorough analysis of the bias of the proposed method, the variance analysis part of the MII estimator is missing, which makes it hard to conduct a real test of the causal effects. More detailed questions are left to the questions part.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PS3R2wCUZp", "reviewer": "Reviewer_vtCm", "summary": "This paper proposes two methods for estimating the global average treatment effect (GATE) when the effect of a unit may depend on the treatment assignments of nearby units, with proximity defined by a graph structure over the units.\n\nThe first proposed method, Mean-In-Interior (MII) estimator, averages the outcomes of the interior nodes receiving the same treatment as that of each unit. The authors show that the MII estimator is consistent (Theorem 3.1) under the Neighborhood Interference Assumption (NIA; Assumption 2.1) and the assumption about the interior nodes (Assumption 3.1).\nThe authors also claim that the MII estimator reduces the variance compared to the recently proposed CAE estimator (Liu et al., 2024).\n\nThe second method, called Augmented MII (AMII), corrects the bias of the MII estimator due to the focus on the interior points, using a counterfactual predictor $f$, which predicts the expected outcome given specified treatment assignments of all units (as well as covariates and the graph). Theorem 4.1 presents the bias of this estimator under Assumption 4.1 about the model.\n\nFinally, the paper presents experiments using a synthetic dataset with several specifications of the parameters of the data generating process. The results shown in Figure 2 demonstrate that the MII estimator has lower variance compared to the previous methods, and the AMII reduces the bias leading to reduced MSEs.", "questions": "Major concerns:\n- Does the statement of each theorem include all of its required assumptions? For example, \"treatment is assigned at the cluster level, meaning that all units within the same cluster receive the same treatment\" (l.168): It is not clear if this is an assumption in the theory. Does Theorem 4.1 only needs Assumption 4.1? Please also see the Weaknesses section.\n\n- Is Assumption 4.1 a condition about the true regression function, or the function used in the estimator? Is the function $g$ known to the estimator?\n\n- Could you discuss the contributions in relation to doubly robust/debiased estimators?\n\n- l. 227, \"the MII estimator exhibits substantially lower variance\": Why does the MII have lower variance compared to the CAE?\n\n- I suppose the proposed estimators depends on the clusters through $\\text{Int}_k$. Do the estimators fail if the clusters are not correct?\n\n\nMinor issues:\n- l.466, \"In the absence of the interaction term and 2-hop interference, we find that MII is nearly unbiased\": How did the author draw this observation?\n\n- Is the difference between the distribution in Figure 3 statistically significant?\n\n- Punctuation is missing after the equations.\n\n- l.475, \"increases exponentially\": Isn't it polynomial in $p$?\n\n- l.396, what is the difference between $\\text{deg}_i$ and $c_i$?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- The task of estimation of GATE in the presence of interference is practically relevant and challenging.\n\n- The proposed methods are developed based on interesting ideas.\n\n- Theorems 3.1 and 4.1 provide soundness and strength of the proposed methods.\n\nI set the Contribution score 3: good because the direction of this work looks great.", "weaknesses": "- The proof of Theorem 4.1 (Appendix B.2) uses Eq. (24), which is only introduced \"to illustrate the idea behind the AMII estimator\". (The statement of Theorem 4.1 does not mention this.)\n\n- I would need (at least) some more details about the Eq. (21) in the proof of Theorem 3.1 to verify the proof.\n\n- It is unclear to me what Eq. (13) is illustrating.\n\n- The AMII estimator looks like doubly robust/debiased estimators proposed in the standard treatment effect estimation literature, but there is no reference or discussion on this.\n\n- The statement of Assumption 4.1 is not clear to me.\n\n- The proposed methods are shown to work well on one dataset, which is a little underwhelming.\n\n- The authors claim that the distributions in Figure 3 \"exhibit clear discrepancies\". However, it seems like a subjective observation and the criterion of this claim is not clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BK6D4JYVln", "reviewer": "Reviewer_RLAt", "summary": "This paper addresses the challenge of conducting A/B tests on networks where interference between connected units violates standard causal assumptions. The authors propose the Mean-in-Interior (MII) estimator, which leverages interior nodes lying within the same cluster and *eliminates the inefficiency* introduced by the cluster-relevant weights used in the previous CAE estimator. Recognizing that interior nodes may not represent the overall population, the paper further introduces the Augmented MII (AMII) estimator, which employs a GNN-based counterfactual predictor to correct for selection bias. Theoretical analysis implies that AMII has an intuitively smaller bias than MII estimator in specific additive model, with another interpretation based on semi-supervised adjustment provided in terms of this method. Extensive simulations on semi-synthetic social network data demonstrate that AMII generally achieves lower MSE than comparative methods, establishing it as an efficient and practical approach for causal estimation under network interference.", "questions": "1. **On the novelty of the MII estimator**  \n   Overall, I think the novelty of the proposed MII estimator could be better explained. From what I can tell, it’s basically a difference-in-means estimator applied to interior nodes, which is a pretty classical idea in causal inference. Also, the idea of leveraging interior nodes has been seen in earlier work. It would help if the authors could clarify the genuine improvement on MII estimator here in rigorous theories.\n\n2. **On theoretical justification for the advantage of MII over CAE**  \n   In lines 250–262, the paper argues that MII improves efficiency over the CAE estimator by using better cluster-relevant weights. But CAE’s weighting scheme is rooted in **cluster randomization**, which doesn’t seem to be part of the main assumptions (2.1 and 3.1). So, it feels like the two estimators might be targeting different experimental designs, making the motivation a bit insufficient. Also, there doesn’t seem to be a clear theoretical guarantee backing this claim—some clarification would be great.\n\n3. **On clarity of statements**  \n   A few parts could use clearer explanations or more precise wording:  \n   - *Line 48:* The paper says that leveraging the known graph structure is *much more appealing*, but it doesn’t really argue why.  \n   - *Line 251:* I didn’t quite follow why the model *helps build intuition* that variance analysis is more complex than bias analysis—or why it’s necessary to emphasize that intuition here.  \n   - *Lines 315–316:* The statement *they may struggle … a comparative advantage* is confusing. It’s not clear why it “struggles,” or why an $f$ -model that doesn’t estimate the interference function $h$ should be considered an advantage. Since any model $f$ can adapt, this doesn’t seem to tie closely to the covariate adjustment goal. \n   - *Line 275-276:* *We argue that ... can be violated.* It does not provide any explanation not state exactly which assumptions are violated. \n   - *Line 281-282:* *Most estimators without a regression component tend to rely heavily on interior nodes*, there have been a number of estimators on exposure effects that do not rely on interior nodes as well, thus an arbitrary assertion.\n   - The inconsistent names of $h$ as both ''transformation function'' (line 255) and ''interference function'' (line 315).\n\n4. **On covariate adjustment and coherence**  \n   Covariate adjustment can be applied to pretty much any estimator, not just MII. So it would be fairer to compare **AMII** with the augmented versions of all other estimators too. Also, Sections 3 and 4 don’t quite feel conceptually aligned—cluster-related weights and covariate adjustment seem like two separate issues, and it might help to connect them more clearly.\n\n5. **On Assumption 4.1 and model training**  \n   Assumption 4.1 says that $f$ is trained in a regression form, but in line 290 the paper says $f$ is trained using a GNN. How does that assumption hold in this case? A short explanation would make this part clearer.\n\n6. **On experiments and ablation studies**  \n   The experimental section feels a bit thin. It would really help to include a few ablation studies, for example:\n   - What happens if covariates actually affect the outcome?\n   - What if the interior nodes aren’t very representative of the full cluster?  \n   Seeing these variations would strengthen the experimental support for MII.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "The paper introduces the Mean-in-Interior (MII) and Augmented MII (AMII) estimators, fulfilling the gaps in GATE estimation based on interior nodes, and draws the attention on possible incomplete representativeness of the interior nodes to the entire cluster.", "weaknesses": "The paper features several notable weaknesses. While the authors attempt to convey dense ideas through multiple key intuitions and assertions, few are elaborated with sufficient rigor to enable non-expert readers to fully grasp them. The theoretical justification for the advantages of the MII and AMII estimators remains limited. Moreover, the rationale for viewing the MII estimator as an improvement over the CAE estimator is insufficient and not theoretically supported. Additionally, Section 3, which introduces the MII estimator, and Section 4, which discusses covariate adjustment, appear insufficiently connected in their conceptual development. Please see Questions for details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)