Data-Adaptive Exposure Thresholds under Network Interference
Authors: Vydhourie Thiyageswaran
Tyler McCormick
Jennifer Rogers Brennan
Venue: NeurIPS 2025 poster ()
Keywords: Network Interference
Experimentation
Causal Inference
Average Treatment Effect
Primary area: probabilistic_methods

Abstract: Randomized controlled trials often suffer from interference, a violation of the Stable Unit Treatment Value Assumption (SUTVA), where a unit's outcome is influenced by its neighbors' treatment assignments. This interference biases naive estimators of the average treatment effect (ATE). A popular method to achieve unbiasedness pairs the Horvitz-Thompson estimator of the ATE with a known exposure mapping, a function that identifies units in a given randomization unaffected by interference. For example, an exposure mapping may stipulate that a unit experiences no further interference if at least an $h$-fraction of its neighbors share its treatment status. However, selecting this threshold $h$ is challenging, requiring domain expertise; in its absence, fixed thresholds such as $h = 1$ are often used. In this work, we propose a data-adaptive method to select the $h$-fractional threshold that minimizes the mean-squared-error (MSE) of the Horvitz-Thompson estimator. Our approach estimates the bias and variance of the Horvitz-Thompson estimator paired with candidate thresholds by leveraging a first-order approximation, specifically, linear regression of potential outcomes on exposures. We present simulations illustrating that our method improves upon non-adaptive threshold choices, and an adapted Lepski's method. We further illustrate the performance of our estimator by running experiments with synthetic outcomes on a real village network dataset, and on a publicly-available Amazon product similarity graph. Furthermore, we demonstrate that our method remains robust to deviations from the linear potential outcomes model.

Data-Adaptive Exposure Thresholds under Network Interference
Vydhourie Thiyageswaran Department of Statistics University of Washington Seattle, WA, USA vrtt@uw.co edu
Tyler H. MeCormick Department of Statistics University of Washington Seattle, WA, USA tylermcQuv. edu
Jennifer Brennan
Google Research Kirkland, WA, USA jrbrennan@google com
Abstract
Randomized controlled trials often suffer from interference, a violation the Stable Unit Treatment Value Assumption (SUTVA), where unit's outcome is influenced by its neighbors' treatment assignments. This interference biases naive estimators of the average treatment effect (ATE). A popular method achieve unbiasedness pairs the Horvitz-Thompsoe estimator of the ATE with known exposure mapping, function that identifies units in given randomization unaffected by interference. For example, an exposure mapping may stipulate that a unit experiences no further interference if at least an h-fraction of its neighbors share its treatment status. However, selecting this threshold / is challenging, requiring domain expertise; its absence, fixed thresholds such as propose data-adaptive method to select the h-fractional threshold that minimizes the seran-squared-eee
1 are often used. In this work, we
estimator. Our approach Horvitz-Thompooe Thompson estimator paired with candidate thresholds by leveraging a first-order approximation, specifically, linear regression of potential outcomes exposures. We present simulations illustrating that our method improves upon non-adaptive threshold choices, and an adapted Lepski's method. We further illustrate the performance of our estimator by running experiments with synthetic outcomes on a real village network dataset, and on a publicly- availalee Amazon product similarity graph. Furthermore, demonstrate that our method remains robust to deviations from the linear potential outcomes model.
(MSE) of the Morvitz-Thomppe estimates the bias and variance of the Horvitz-
Introduction
Estimating the Average Treatment Effect (ATE), the difference in the average outcomes of units when all units treated versus when none are, is challenging in the presence of network interference. Under interference, the Stable Unit Treatment Value Assumption (SUTVA) [Cox] 1958 Rubin 1978
Manski 1990) is violated, as a unit's outcome is influenced by its neighbors' treatment assignments. This problem is salient in randomized controlled trials, the setting we examine. For example, estimating the adoption of new idea through random assignment of people advertisements complicated by the dissemination of information through social interactions. Consider a social media platform seeking to estimate the efficacy of a product innovation in increasing user engagement. Measuring outcomes for users in the control group is complicated by their interactions with treated
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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users. If the innovation successfully increases engagement among treated users, their friends in the control group may also exhibit increased engagement through their interactions on the platform As such discrete interactions can be modeled through network interference models, the problem of estimating average treatment effects under network interference becomes a ubiquitous one.
A fundamental challenge arises from the lack of knowledge about the exact interference pattern. Exposure mappings, as defined by Aronow and Samii [201), are functions that partition the space of treatment assignments and individual-leve features (e.g., social neighborhood structure) into distinct exposure values. These mappings encapsulate the concept of "effective treatments" introduced in [Manski, 2013]. For example, let 2 E {0, 10,1]" denote the treatment assignment vector, and W [0, I " represent the (weighted) adjacency matrix, where Way encodes the relationship between units and j. A simple exposure mapping takes the form f: (z. W) + W2, assigning each unit a weighted sum of its neighbors' treatments. (Aronow and Samii [2017] Ugander et al. (2013),
Sussman and Airoldi [2017]. Hardy et al. (2019). Auerbach and Tabord-Meehan [202]] expound upon different exposure mappings, and estimation under these settings. Following [Eckles et 2017
Toulis and Kao 2013). we characterize exposure in terms of the fraction of treated neighbors. For instance, a unit may be considered as experiencing no additional interference if at least an l-ffraction of its neighbors share its treatment assignment. This fractional neighborhood exposure mapping aligns with the fractional thresholds model [Watts] 1999]. where treated and untreated peers exert opposing influences on a unit's 's outcome. Centola and Macy [007)] illustrate this with the example of refraining from littering in a neighborhood: an individual's 's decision to avoid littering depends on the relative number of neighbors who also refrain from littering.
Under known h-ffrctional neighborhood exposure mapping, the Horvitz-Thompson estimator (Horvitz and Thompson 1952] is popular unbiased estimator of the average treatment effect. Without domain knowledge of the interference structure, however, such estimators based on non- adaptive treatment exposure conditions often suffer from high bias or variance. For instance, the Horvitz-Thompson estimator can be paired with an extreme threshold to achieve unbiasedness, requiring that unit and all its neighbors be treated (or in control). However, this can result in high variance, since the probability of such configurations is very low, especially for high-degres nodes. Conversely, setting a low threshold h introduces bias into the estimator by including units with limited treatment (resp. control) exposure in the treatment (resp. control) exposed group. We propose simple data-dependent approach to find MSE-optimal threshold for the Horvitz-Thompson estimator in the finite population setting. We compute the bias slope, an approximate rate of change of bias, which we use to estimate the bias of our estimator. We use this information together with variance estimates to select a threshold minimizing the Mean-Squared-EEre (MSE) of the estimator.
In [Basse and Airoldi, 2018 Cai et al. 2015), (known) generalized linear form of network effects on neighborhood treatments was assumed. Related to our work, Zhu et al. [024] fit functionals on the treatment assignment and exposure. In [Chin] 2019]. the author proposes regression adjustment estimators that predict potential outcomes under global treatment and control conditions. Unlike traditional regression adjustments, their approach constructs adjustment variables from functions of the treatment assignment vector, framing the learning of a more flexible exposure mapping as a feature engineering problem. While Belloni al. [2022] focus on estimating direct effects under network interference, their approach to determining the exposure radius, i.e. the m.-hop influence, shares conceptual ground with our approach of selecting optimal exposure thresholds
In this paper, we present our framework for the Horvitz-Thompeer estimator the ATE, motivated by its widespread adoption in practice. Nevertheless, our approach could be adapted for other estimators, like the Hájek, Differrnnc-inn in-aneees--eMsss...-- and Augmented Inverse Propensity Weighted (AIPW) estimators. We formulate the adaptively- threesholddd estimator for the Sifference-in-Meese is a special case of the Hájek estimator, in the Appendix [A.7.5] We note however, that the Hájek estimator is only approximately unbiased the true threshold. This makes formal characterization of its exact birs-variance tradeoff more challenging than for the Horvitz- Horvitz-Thompseee Thompson estimator.
estimator, which
Within off-policy evaluation in reinforcement learning, [Su et al. [020] address the classical problem of adaptive bandwidth selection, well studied in non-parametric statistics [Fan and Gijbels 1992
Ruppert 1997 Kallus and Zhou, 2018 Mukherjee et al. 2015). by applying Lepski's 's method
[Lepski 1992
1993 Lepski and Spokoiny| 1997 Goldenshluger and Lepski, 201T]. Our work
explores an extension of this framework to optimal bandwidth selection for the Lorvitz-Thompsee
estimator in average treatment effect estimation under network interference (see Appendix A.3) To
2

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our knowledge, this connection has not been explored previously. We compare our procedure to an adaptation of Lepski's method tailored to our problem setting, In our context, the network dependence structure can lead to violations of the monotonicity and decay rates conditions that Lepski's method relies on. Our approach offers simple and principled data-adaptive alternative designed address these challenges in our context. Philosophically, these ideas are similar. This perspective on threshold, or equivalently bandwidth, selection at the estimation stage can loosely be viewed as the "dual" of choosing cluster sizes in cluster-randomizatio approaches such as Ugander et al. [2013). Bekles et al. [2077] at the design stage.
Setup
2.1
Notation.
We use h to denote the threshold for "treatment exposure", and 1 -h for "control exposure". Therefore, for larger h, i.e. closer to one, have more restrictive setting, where only the subset of treated (resp. control) units with most of their neighbors treated (resp. control) are counted as treatment (resp. control) exposed. For smaller h, i.e. h close zero, we have less restrictive setting, larger subset of treated (resp. control) units satisfy the threshold condition. Throughout the paper we use W to denote the adjaceney matrix with W;: 0, d, = E, Way for the degree of node and when the graph is regular. We use D denote the diagonal matrix of degrees d,.
2.2 Prohlem Setup
We study the finite population setting. Additionally, we consider a fractional neighborhood exposure mapping. That is, for any treatment assignment vector E {0, 1)", , the effective influence of the graph's treatment assignment on the outcome of node is equivalent to the influence of fractional treatment assignments in the neighborhood of node i. This reduces our potential outcomes model to the following form:
(2) = y,(,, e,) = a, + (22, e4) + Ec,
where 2; {0, 1} is the treatment assignment of unit e; E [0, is the fraction treated neighbors of unit i (excluding i itself), and the are uniformly-bounde bouuded and non-random. Throughout the paper, we focus on (1 e;) = g + fe;)), but our framework generalizes. For instance, we will consider the linear potential-outcome model in some examples:
(=) = og + ((z,, ea) + = 04 + 3,4 + 768 + 6G-
(2)
We are interested in estimating the average treatment effect (ATE),
1
3e (1, (0 Σ (as + 76 + -a -
(3, + 7:),
though we could, more generally, take any difference between exposure categories.
Let Y, denote the observed outcome for unit i, E [no]. We consider the Horvitz-Thompson estimator Horvitz and Thompson 1952] for given exposure threshold
1 1
1) 1
1 Σ
14=0, 1 1- h]] P(z h]
f
Y
(4)
n
P P
Our goal then is to select the threshold h in the Horvitz- Thommpon estimator above that minimizes the MSE. More formally, given a candidate set of thresholds H, we select
h" argmin MSE(A)) = argmin Bias (i.))(fn)) + Var(f.). KEH
Stronger interference, i.e. when s0 es) more sensitive to changes in exposure e,, induces greater bias through edges that connect units with differing treatment assignments. To mitigate this bias, one

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0.0
0.2
0.4
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(a) Bias- variance tradeoff
(b) Linear regression across parameters
Figure 1: Left: bias, variance, and RMSE of AdaThresh across different thresholds for the 1000-node 2nd-power cycle graph (see Appendix (A6)) with unit-level Bernoulli randomization, and linear model outcome Y; = 10 + 10z; + 7 + é;. for fixed e; generated from (0, 1). Right: toy illustration of linear fits used to approximate exposure bias caused by including more data across different thresholds. The "jittered" datapoints signify the amount variance reduction across thresholds. Rectangular blocks depict the data regions under the MSE-optimal thresholds.
might pick higher exposure threshold h. However, increasing i reduces the exposure probabilities P(= = 1,6 2 h} and [p = 0, < - h), thereby increasing variance. To illustrate this display the biases and variances under the linear model Y, ye; + evaluated across varying thresholds in the Horvitz-Thompson estimator 44) above, in the left panel of Figure [ This trade-off is illustrated for various ratios of 7/B. The right panel of Figure [ illustrates the different fitted linear models for various B ratios, whose slopes we call the bias slope. We then use these bias slopes to compute the bias estimates for {fx } 4 as described in detail in the next section. Combining these bias estimates with the corresponding variance estimates allows us select the MSE-optimal threshold, shown by the shaded regions in the right panel of Figure (matching the left panel).
bias- trade-off,
B2,
We adopt the MSE
interested in precisely and accurately estimating the ATE, trading off bias against variance in the estimator. See Deng et al. [2024] Section 2.2] for more discussions this. We note that, while the overall trend of bias decreasing and variance increasing with h holds, neither quantity is necessarily monotonic due to the dependence between units and, consequently, their exposures. To further illustrate this, in Appendix A.6) we draw upon toy examples of circulant graphs from Ugander et al. 2013
objective
Estimating bias and variance
With the goal of obtaining an MSE-optimal threshold in mind, we find the bias slope in. which captures how the bias changes with the threshold h. More concretely, the bias slope quantifies the average change in bias as h is varied from 0 to 1, estimated by the slope of a linear fit. Generally, this linear regression coefficient %n is simply first-order approximation of how ( e4) changes with e, estimating the average change in the mean outcome per unit change in exposure. In our context, this also reflects the average change bias as we move away from the boundary exposures, i.e., from O for control or for treated, since in our setting bias arises when exposures deviate from these extremes. This interpretation motivates the use of the weights e; - 0 = e; and e; in Equation 6 corresponding to deviations from the control and treated boundaries, respectively.

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Building on this interpretation, we propose the following Horvitz- Servitz-Thomponn estimator for the bias, which integrates the bias slope over the empirical exposure distribution yield an average (absolute) bias estimate at given threshold:
- (1-6)mm
1
1, h
1 e 1 = 0, 11 h} P 0, 6, S h}
6(t%)
(6)
h}
where In. the bias slope, is the linear regression coefficient of the outcome on the exposure variable. If the potential outcomes depend strongly and positively the treatment exposures, 4, will be larger. Consequently, our estimator will capture more pronounced increase in bias caused by including more data when using smaller threshold h. We use Horvitz-Thompson-st account for the exposure distribution, which may pose challenges for Lepski's ' method (see description in Appendix 12).
bias estimator to
To estimate the variance associated with a threshold h, we use the variance estimator discussed in Aronow and Samii 20177). available in Appendix A.1]
Together, they inform the choice of threshold, and our adaptive estimator, AdaThresh. We describe our procedure in Algorithm
Algorithm 1 AdaThresh
Require:
Graph adjacency matrix W, outcome vector Y, treatment vector I: Compute exposure: e < D- W
2: Fit linear model:
32 + ye +
3: Let y be the estimated coefficient for
4: for each threshold h_ do
5: Estimate bias: Bias using 7, 2, W, and h
See Eq. (6)
6: Estimate variance: Var(h) using Y, z, W, and h
See Appendix A.
7: Compute MSE(h) Bias (h) Var(h)
8: end for
9: h arg minken MSE(h)
10:
return Fr
b> See
The bias slope from the linear fit captures the average change in bias as h is varied from 0 to 1. We do not assume an underlying linear model Y B2 + 7e, for which the sum of linear regression coefficients B + f would be the optimal estimate of the ATE. Rather, we use linear regression only to obtain the bias slope, first-order approximation of how (((:e)) changes with eç. This idea is similar in spirit to the work of Baird et al. [2018), where the authors leverage the slope of spillovers with respect to treatment saturation levels as approximation, without assuming linearity in the true seaareeenrrtingggg process.
As we first estimate the bias slope to inform our overall estimator of the ATE, sample-splitingg is one potential strategy for consistent estimation. However, since the units of interest are dependent (as modeled by the network interference), one would have to prove performance guarantees by leveraging results such as Hart and Vieu [190]]. our setting, however, as we leverage the simplicity of the model class [Van Der Vaart et al. 1996). In Appendix A.9 we discuss this using Donsker results [Van Der Vaart et al. 1996 to demonstrate that our approach does not lead to overfitting.
not necessary that we split the data
4 Theoretical Results
Denote by H the (finite) set of exposure thresholds. For example, cycle graphs (see Appendix A.6 for details), H { E (O.,,)..,d)) where d is the degree of the graph. We assume the following:
Assumption 4.1 (Unconfoundeenesss Positivity).
1. (Unconfoundedness) For all e [x]]
(u, ē.) ã (0, 1), [0, 1]} Tl (z, e4)
5

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2. (Positivity) For all i E [n], h E [0,
P( = 1, e 2 1) > 0
and P( = 0, es e h} > 0.
The first part of Assumption 4.1 states that the individual and neighborhood treatment assignments are independent of the potential outcomes. This follows from the unit-level and cluster-level Bernoulli randomization designs we consider in this paper. The second part of the assumption is also fundamental one in the causal inference literature [Rosenbaum and Rubin 1983 Petersen et al. 2012], ensuring that we have sufficient data to estimate the ATE. It also relates to statistical leverage MMhhnny et al. 2211 Martinsson and Tropp 2020 Young 2019 Pilanci and Wainwright 2015). The idea, informally. lly, is that under positivity, statistical leverage across the input data is more homogeneous. As result of this, there are no observations that have too much control over the linear regression.
Assumption 4.2 (Bounded variation reeamenn-expossue function). Let the potential outcome function be = Q; + (22,, eq) + as in (1. The function rb(2;, es) has bounded variation.
Assumption 4.3 (Bounded First-Order Interactions). For unit-level randomization, we assume that all nodes have bounded degree. Specifically, there exists finite constant dmax < oo such that, for every unit E [n]. the degree de satisfies
dj S dmaxx-
For cluster-level randomization, we assume that all clusters have bounded degrees of cross-cluster interactions. Let s; denote the number of cross-cluster connections involving the cluster to which unit i belongs. Then, there exists a finite constant Smax < oc such that, for every cluster E C, the cross-cluster degree s, satisfies
83 S 8max.
These restrictions are consistent with the observation that networks in practice tend to be sparse and clustered (Barabási 2013 Chandrasekhar 2016 Chandraaekharr al. 2020). If this assumption violated, exposure positivity may fail, and Proposition [A.5 will no longer hold. However, as discussed in Remark [4.4 below, this assumption can be weakened under certain conditions. Some examples of graph classes that fall under the categorization of Assumption /43 are expander graphs
and growth-restricted graphs [Alon] 1986 Arora et al. 2009) Gkantsidis al. [2003] Kuhn al. 2005
Krauthgamer and Lee 2003 Kowalski 2019]. This is related to requiring the graph interference structure have small k, condutatee (1.e. the k, parraittiionn generalization of the Cheeger constant) for kn growing with n. In the causal inference literature, the growth-restricted graph setting was
studied in [Ugander et al. 2013
Remark 4.4. Assumption 4.3 is also employed by Aronow and Samii [017] As we further discuss in Appendix A.8 it can be relaxed by "binning" the exposures and verifying whether the "affinity set" conditions of Chhndrassehaar et al. (2233 are satisfied. Specifically, in more general weighted settings, one can partition exposures into bins indexed by 1, 2,2..... K, each associated with an effective exposure level ēs. In this framework, node degrees need not be bounded, provided the affinity set conditions are met. These conditions subsume the approximate neighborhood interference (ANI) condition of Leung (2022) when the maximum clique size in the network remains bounded. For cases with growing clique size, we refer the reader Leung 2022) for further discussion the ANI framework.
In the following theorem, we characterize the probability of choosing the correct threshold under the best average linear fit, for which the exposure slope is >" The correct threshold At under the best average linear fit is the threshold minimizing the sum of the true variance and the true squared-bias under Y; a + 32 + We write Mr,hh, (h), M,, denote the MSE under the true average best fit line and the MSE estimated by our methods, respectively, at threshold h and unite-population size n.
Theorem 4.5. Suppose Assumptions 4.1 4.2 and 4.3 hold. Further assume unit-level Bernoulli
randomization with treatment assignment probability p. Let An == minke h IM (h) M (h" )
and let Un E R satisfy max (b,(())) < U.. which exists by Assumptions 4. 2)) and 4.2 Denote by
hū the optimal threshold under the true average best-fit fit line, and by hn the threshold chosen by our

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method. Finally, let H. be the set of exposures. Then,
p(hn hn) -
Σ
P((M,()) M,(h1)] > An/2]
NEW
(- I 3| H max exp (- mn - p) +1). exp
Anrpp11 -
-p)
+1).
8cdmax
n16Un)² Pcdmax
Cn( An 4
of
6exp
V An,2 + V (An comnas) Mn
for some constants C, C. Here, An, S 16|/011 2, (cr + (logni for some constant e₁ and i is the Fourier transform of the summands of the variance components, and Mn.2 = 41|u||e, (logn)*
If. instead, the design is cluster-level Bernoulli randomization with probability p. denote by Smax the maximum cross-cluster degree, i.e., the largest number of distinet clusters to which any given cluster is connected. Then, we can replace dmax by Smax in the bounds above.
In Theorem 4.5 we assume without loss of generality that the MSE gap lower-bounded by a constant, i.e. v A, as otherwise all candidate thresholds are optimal. The proof to the theorem is in Appendix A.S.I In this proof, we make use of the results from [Ziemann al. 2024 Theorem 3.1]. and (Shen et al. 2020 Theorem 2.1].
Our results tell that generally, for fixed n, and under a unit-level Bernoulli randomization, our approach yields a higher probability of choosing the optimal threshold when the maximum degree smaller. Under Bernoulli cluster-level randomization, our approach yields higher probability of choosing the optimal threshold when the maximum cross-cluster degree is smaller. Therefore, for well-clusterable graph, our approach would yield higher probability of choosing the optimal threshold under cluster-level randomization, as the clustering would give us a reduction in the cross- cluster degree. Not surprisingly, when n is larger, and the degree ranges are kept fixed, the probability of choosing the optimal threshold is higher. Additionally, as the variance of the variance estimator is larger, the probability of picking the right threshold is smaller. The error scales with the size of the bias, appearing in the second term in the bound. Finally, considering symmetric h for treated and control units, our approach yields a higher probability of choosing the optimal threshold when the treatment assignment probability which is maximized at p = 0.5.
0.5. This can be seen from the p(1 p) term in the bound
This gives us the following corollary. Denote the true MSE by M and the corresponding bias and optimal threshold by 6," and h respectively.
Corollary 4.6. Under the conditions of Theorem 4.5 fürther assume that (12;,, e.) =g(=,) + f(e)). and that sup., [f(e)) - -= S 8. Let U E E be such that max, 1b (f:) S U,. which exists by
Assumptions
4.1
2 and 4.2 Note that U is bounded from below. Let 5 = 1652 + 88U Then,
P(hn # h
S Σ (() MM (h)) > 2n/2]
NEH
(An / 5) 5)mp(1 p)
< 3 H3
max
exp
+ +1)
Cdmaax
(A, (166(16Um)
8)mp(1 -p) p)
+1)
exp
cdmax
On An 4
cdf - 8)
6cxp
Vnn,
(An
cdl
Mn.2
for some constants c, C. Here, An,p S 16|| 2,, {cr + (logn)* for some constant eı and ō is the Fourier transform of the summands of the variance components, and Mn.2 = A1lu||,, 2)
If, instead, the design is cluster-level Bernoulli randomization with probability p, denote by 8max the maximum cross-cluster degree, i.e., the largest number of distinct clusters to which any given cluster is connected. Then, we can replace dmax by smax in the bounds above.
7

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Figure 2: Left: graph size n 200 generated from the Stochastic Block Model (SBM). Node colors reflect ground-truth cluster membership, corresponding the underlying block membership. Right: Village network dataset, with node colors reflecting e = 1-net clustering (see [Ugander et al. 2013
for details on e-net clustering).
Aorollary4.66jtelle us that the maximum linear approximation error between the best average linear fit and the true exposure function /(-) is small enough relative the minimum MSE gap An. our estimator will be optimal with high probability for large Indeed, when the ni(,, - 5) terms large, the exp (-n(A,, - 5)) terms are small. If (xx y for x [o. 1] indeed, then with necessarily we have that with high probability for large our estimator is optimal.
5
Experiments
We now compare the performance of our approach with that of non-adaptive Torvitz-Thompson estimators HT(1) and (T).), with h = 1, and h O, respectively. Additionally, since objective is essentially an MSE-optimal bandwidth selection problem for the indicator kernel (see Appendices estimator to variant where the threshold is selected via Lepski's method Goldenshluger and Lepski 2011 Su et al.) 2020). We note that Lepski's method relies on the monotonicity of the bias and the variance work well. In setting, due to the implicit dependency structure from the graph appearing in the exposures e, the monotonicity assumption may be violated. We write out these three estimators compare against in Appendix A.2
A.3
13). we also compare
In Figure 3 we display simulation results with outcomes generated from the linear model with (2,,, e,) ((x)) f(e)), a, 10, 9(2,) 32g 102,, f(e,) -e,, with fixed e; generated from (0, 1). for a 200-node graph, generated from an SBM with 40 underlying clusters (see Figure (left panel)). We focus varying the ratio 7/3 (3 thresholding estimator, Ada AhhThrshh, generally performs better than existing estimators, interpolating between the fixed 0/1 D/1-threshold Horvitz-Thompee Horvitz-Thompson estimators when the threshold 7/8 high. We also display simulation results for 1000-node 2nd-power eycle graphs in Figure
consider fixed graph. We that adaptive
estimators, and out- performing the Lepski-based
Furthermore, cluster-randomized designs better control interference, reducing bias over wider ranges of spillover ratios. In such settings, the variance tends to dominate the bias. Since optimizing for bias leads to HT()), and optimizing for variance leads to HT(O), we observe better intermediate thresholds for longer ranges of spillover ratios under (uuster-ranoomized designs compared to unit- level designs. These patterns become even more pronounced the cycle graphs presented Figure] 7 where the node degree is held constant. From the design perspective (sce, for instance, [Viviano et al. 2023)), if spillover ratios are larger, then one might analogously perform more cluster-randomized designs to contain more of the interference and reduce bias.
5.1 Real Data
We evaluate the performance of our estimator on village (No.6) network data from (Banerjee et al.. 2013). Here, 110, with nodes representing villagers and edges indicating that the adjacent

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3x33
AdaThiicsh oracie HT HTUI
AdaThresh oradle
Ex3am
RASEVATE
2333
HT11]
HTT0)
Lepski HT
Lopski HT
Ratio
Raitio
Figure 3: RMSE (normalized by the ATE) of different Horvitz- Thompson estimators for the SBM graph in Figure 2 (left panel). Left: unit-level Ber(0.5) randomization, Right: cluster-level Ber(O.5) randomization with ground truth clusters. The error bars are the empirical 95% confidence intervals around the mean.
Ada Thresh
oracle HT
RMSEJATE
HTT00
Ada thresh oracte HT
Lopski
30
HT((0)
Figure 4: RMSE (normalized by the ATE) of different Horvitz-Thompson estimators for the village network in Figure 2 (right panel). Left: unit-level Ber(O.5) randomization. Right: cluster-level Ber(0.5) randomization with E-= 1)-tet clustering. The error bars the empirical 95% confidence intervals around the mean.
villagers have visited each other's homes. We experiments with synthetic potential outcomes, averaging over 1000 trials, under unit-level and cluster-level Bernoulli randomizations, where clus- ters were formed using an c(= 1)-net clustering (see [Ugander et al. 2013) for details on e-net clustering). We generate simulated data using the linear model with (24 ex) = g(2,) ) + f(e.). a, 10, g(+) 32, 10z, fe,) 7e4, with fixed e; generated from (0, 1). To compute the exposure probabilities, we used x 10t Monte-Carlo trials. We focus on varying the ratio 7/3 as we consider a fixed graph. Figure demonstrates that our method generally interpolates between HT(1) and HT(O), achieving intermediate performance on smaller graphs (i.e., smaller n). While it does not consistently outperform the HT(0) baseline across all settings, the results suggest that our approach remains competitive across large range of regimes and offers reasonable bias- trade-off. hypothesize that the observed imprecision and inaccuracy of our estimator stems from inaccuracies in estimating smaller exposure probabilities, particularly for higher-degree nodes. For larger n and smaller dmax, performance improves further, supporting our theoretical findings, as demonstrated in Appendix A.7 on the Amazon (DVD) products similarity network [eeskovee et al. 2007 (see Figure 6). and on various circulant graphs.
Discussion
In this paper, we focused the additive model for vb, but note that our framework is applicable more broadly. We investigated Adhhhrshh an adaptive Horvitz- -hooiiz- estimator for symmetric thresholds h and 1- for treatment and control, respectively. Our framework applies more generally with estimator thresholds h and 1 ha, respectively with h * hg. Additionally, in Appendix A.7.5 we demonstrate the performance of this approach using the Difference-ine in-Mees estimator incorporating exposure thresholds. We leave it to future work to extend our results to more general exposure estimation procedures.
[Chin] 2019), the author proposes learning the feature variables that are most predictive of the potential outcomes. One could extend our work by first learning the feature variables, as proposed by [Chin] 2019), followed by then learning the appropriate optimal threshold associated with these features, using our approach.

---

In Appendix A.7.4 we illustrate the robustness of our estimator to deviations from linearity in the potential outcomes model. We note that our framework would apply broader settings, including a direct extension to theeds-of-friendee (or further connections) or to neighborhoods that are either observed or learned through a clustering algorithm. additional interesting future direction could be to extend our work to the framework presented in Chandrasekhar et al. [023] allowing for complete graph underlying the interference structure.
Furthermore, to improve upon robustness to non-linear settings, we propose an extension using local regression estimate the rate of change of bias within the [0, 1 1 h). [h, 1] windows. As long as there is sufficient concentration of exposures in these windows, robustness is never worse. In Appendix A.7.6] we display our simulation results this setting using local linear regression estimate the rate of change of bias within the [0, 1 h), [h. 1] windows. One could also use other local regression approaches, such as kernel regression, etc., while maintaining sufficiently small model complexity to avoid needing sample-split.
This paper prioritizes minimizing the mean squared error (MSE) effectively balance the birs-variance trade-off in a Horvit=-Thompso-yyae estimator, rather than focusing on inference. However, we acknowledge the significance of characterizing inferential tools, such as confidence intervals, in specific contexts. Notably, [Aronow and Samii 2017] and (Athey et al. (0188) offer valuable insights into inferential methods for estimating average treatment effects under network interference.
Acknowledgements
The authors would like to thank Alex Kokot and Lars van der Laan for helpful discussions; Dean Eckles for pointing to reference, and Matthew Eichhorn for suggesting the Amazon product similarity network dataset.
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13

---

A
Appendix supplemental material
Notation
this supplement, write b" (mx), (h), (Ta), ((7h) to represent the bias, estimated bias, variance, estimated variance, respectively, at threshold k. We also write dea to represent the upper-bound (Assumption on the degrees of the nodes in the network.
4.3
A.1 Variance estimator
We use the variance estimator, for Horvitz-Thonpee estimator, proposed by [Aronow and Samii 2017 Eqn. 7.9].
Σ 1
1,
8(F))=
-1
n²z!
7
1=1
1/2 1,,
h)1 m²
1,, 2 hD Myj 2
11
1)
1/2
0,6, <1 n2
1
1/ 0,e, < -
1 0,e, < <1-1))1 KJXYY
00 T
1)
n n 00
1
1
(1/ = 1, h/1 = 51-
2
7/m9
10
1 1,= AY??
1(2 0,e, < K3Y?
2
2x
2
jE
Here, Tī m. are defined with respect to the threshold h. That is, n! = P(z, = 1, 2 k1. -² = P(
0,e, < 1 h}, maj
10 = P(& = 1, 2 h 2, 0, < 1 - h}, m. = P(2 = 1,6 2 h = 1,, 2 h},
11
7; P(z, 0, < 1 - h, 2, 1,e, 2 h), Tij = P(z, 0, h, 0,,, S 1-hh h).
A.2 Hlovitz-Thompsee estimators with existing approaches
We write the vanilla Horvitz-Thompso estimator threshold one) 8 the Horvitz-Thomppst estimator threshold zero 9
and the Horvitz- Thomiz- estimator with the threshold selected via Lepski's method 12 below.
A.2.1
Vanilla Horvitz- Thompson estimator (at threshold one)
1/2 = 1 a 1 0,
0 Y; 0,
(8)
P(z: -|==
P(
A.2.2
Horvitz-Thompson estimator at threshold zero
Σ 1(= 1)
" 1(z 0) 2 P( 0)
ANTO
P(z Y 1=1
Y,
(9)
A.2.3 Lepski's method
As described in
Lepski and Spokoiny
1997 we first take
I(h)= = [âm,, 2SDEV âm, + 2SDEV
(10)
Then, take
hrapou minfh H (WEnmagnd(h") # 0),
(I))
14

---

and
1 71
1/ = 1, 2 hesgou)
1 0,e; S 1 hcoposal
PLppkillT
Y
Y
(12)
P(z: e, 2 hrspsi)
P(z: ea 1 heapss)
Lepski's method requires monotonicity and decay rale conditions be satisfied. violations of these conditions.
Our setting may lead to
A.3
An equivalent formulation of the Horvitz-Thompson estimator with the exposure threshold
We can rewrite the display in in the following form:
1 - -
n -1 E.Y,
(13)
where 22, 1, that & {-1, 13. is then clear that the threshold controls how much dissimilarity between treatment status of units and their neighbors, we allow in our estimation. This allows us to reframe the problem as an optimal handwidth selection onc.
A.4 Bias and Variance estimation errors
In this subsection, write down the estimation emrors of the bias and the variance terms in the MSE estimation. For the bias terms, we have,
B(Fx) - (fx) 3. (2) 1, 2
Σ
3h X, 2 h 1:( xz))
1(x 2 h]
P(:: = 1,e h)
2,
fe = 0,e, < h)
Σ Σ
1/ <1 mi n x; 1 h 1hl (22))
P(z: = 0,e, 11-h)
1=1 2
where y the slope of the best average linear fit, and where x, ranges over the set X, of possible fractions of degree
By an abuse notation, we use ys (h+) to represent the average (across possible exposure fractions for node i)
potential outcome for unit with exposures that are at least h, while we use y; to represent (across possible exposure fractions for node i) potential outcome for unit with exposures that are most 1 h.
the average
In the variance terms,
(1+
0 (fr.
-1)
n²
y1 (h+ (h+)
n²²
I
y: (h )2
1
-1)
T
(1
Duo (11 n²
10
va(h (2s(()))) =
n²
m3
10(m):n0000
(h (15
|E[n]:1000 =
Therefore, from the above and Section A.1 we have that
15

---

sup ((..) - (fi.)
dns.
dhp
1/ = 1, 2 h
- -
4i }
122m
n²
1/2 1,
h31 (2) = 1,, KJYY,
11
-1
Ye Jy, (h+)
1) 1)
n²
1 0,0; S 1 n.Y
1
Σ
1i )z
1
-
n!
n²
1 1(= 0,0; 5
1
= 0,
1- hJyy
00
-1)
y:
Yy, 19, )l-) n²
2
A3112z = 0. 1 - (
Σ
1,a
n
sC(m]
"3 20
1)
(I (1)()))ss (Ms(h
2
1f21 = 1,0 2 h/..
112, 0,e, S h2Y?
2T
2m
i=1 se[m] 11
vb ((sh)))ssh))) bos(h)))
A.5
Proofs to Theorem
4.5 and Corollary 4.6
A.5.1
Proof to Theorem
4.5
Proof to Theorem
4.5
first consider the variance terms. Define E(o]. We have that,
P(6e- v'1 >
An/4) P((6- U+ E w²1 An/4)
(e) P(lē-
ক An/1 /1- cdhaax/1) n1)
On Ann cdfanse
exp
An,a +
Au
Mn.2
n5
where An,p S 16]1 fcr + n) 1 for some constant Gj and is the Fourier transform of the variance n
summands, and Mn, 11] 12)
The equality (a) is obtained when there exists i.j such that (h* h-) i.e. the joint exposure assignment probability of e; 2 h. and e, S 1- h. Since the contribution of these get (a). The inequality (b) is obtained from a direct application of (Shen et al. 2020 Theorem 2.1J. as it easy to see that our exposure dependencies satisfy a- e-mixing conditions and that together with Assumption the variance summands satisfy the Fourier transform conditions of (Shen et al. 2020 Theorem 2.1].
result of Assumption 4.3 Indeed, we know that x² is non-zero only
v² are bounded above by ed cduass/ /n for some constant e, we
16

---

Next, we consider the bias terms, for any constant J.
Anmp(1 - p) Jedmax +1 1
P((mn- -n1) An/J) s exp
2024 Theorem 3.1]. Below, we verify that the conditions are met in our setting. We begin with the condition that for every e IR such that v v= 1/((]e2))) 1/2 there exists h E R such that, [[(ve:)"] I A /²o" This is trivially satisfied in our setting since our exposures is bounded. Next, we consider the first part of (Ziemann el al. 2024] Condition (3.3)]. This is also trivially satisfied in our setting since our block sizes (= dmax)) are bounded by Assumption 4.3 The second part of [Ziemann et al. [204]] Condition (3.3)) is also satisfied since our de-meaned noise-class interaction variables (as defined in [Ziemann et al. 20244 (2.5)) are sub sub-gaussian by assumption of independence between the potential outcomes' residuals and the treatment design. Next, [Ziemann et al. [024] Condition (3.4)) is also trivially satisfied since take our blocks to be of size de almost, with the possible exception of the final remaining block, uniformly. Finally, is casy see that (Ziemann al. 2024 Condition (3.5) is also satisfied since the exposure variables are independent outside of the radius d'naxs since exposure variahles e, and ej for any j.j E [n] are only dependent when nodes and share a neighbor. We quickly note that the monotone partitioning (as described in [Ziemann et al. [2024] Theorem 3.1] is not necessary in our setting, the "blocking" procedure introduced by Yu 1994 holds more generally.
for some constant We use the results from Ziemann et al.
For convenience, in the following we write b", to mean (f..), B(F.). and similarly v", e to mean v² (fx), 0(f.). H is the set of exposures. Therefore, putting these together by a union bound,
pfhn h <
P( (M (b) - Mf > 4./2}
$ |H|P[ - T2 - -1 2
$ (P(62 - A./14 + P(6 -v' An/4))
$ (P(6 (()((((6))>> > A./8) + +P(( ([(0-)'²² (16(1)))))))
+P(10- -0'1> - An/1))
$ (P((%. - 7 > An/8) P((nn- - -nn) > / (16U,)²)
+ PP10- -v'1 > An/4))
Anmp(1 P)) +1
< 3/ H max
exp
8cdmax
p) Anmp((
(16Un) (16Un)eedmas Fedmax +1)
exp
Cn( An edf
6 6exp
VAn.2+ + ((
(A an Mn.2
The proof for the cluster-level Bernoulli randomization follows immediately.
A.5.2 Proof to Corollary 4.6
Proof of Corollary 4.6 Suppose sup., (e:)) - < 6. Let 5 = 168 + 880, We begin by considering
the difference between the MSE under the true best average linear fit and the true MSE:
M (h) M (1)) = (1)) 6% (1))(1))
< ((5% (Ic) 65" (b)) (h) - b (1))
< (h) -b," ((), 6%" (h)) + 26 (b))
((b) (h) - -6," (A)))) -
< (46)² + (45) = 5.
17

---

Figure 5: Circulant graphs with unit-level randomization and cluster-level randomization. Blue and red nodes represent treated and control units, respectively. Left: (Ist-power) cycle graph with Ber(0.5) unit randomization. Center: 3rd-power cycle graph with Bernoulli(.55) unit randomization. Right: 2nd-power cycle graph with Bernouli(0.55) cluster randomization, with clusters of size 5 (=2k 1).
This gives
Pfhn + hn*
Σ P(IdM.(h)) MA" (1)) An/2]
REff
Σ (1)) - (a) + M,) M, (A)) 2./2]
heff
< Σ P((AM.)) - M.(1) + (M - (K) > A.. 2)
(1) - M,( > A.. 5z
hElf
where the second inequality is obtained by applying triangle inequality.
Finally, applying Theorem 4.5 gives the result.
A.6
Toy examples: Tradeoff's in Circulant graphs
We consider unit-level and clustcr-level Bernuuli(p) randomizations in the kth-power cycle graphs, see Figure
5 We say graph a sth-power cycle graph there exists edge between each node and 2 its nearest neighbors (Ugander et al. 2013).
Proposition A.1 (Absolute bias in k-th power cycle graphs under unit-aandomizztion). When p= 1/2 and the potential outcome model is simply linear, Y; a 82z ye,, the obsolute bias the Horvitz-Thompp = 1/2k for 0, ..., 2k, kth-power cycle graph, under unit-level
estimator for a given threshold
E36 (7/k
randomization, is equal to 7
X
Proposition A.2 (Variance in k-th power eycle graphs under unit randomiammi...... Denote the degree of the nodes by d(= 2k). When p = 1/2 and the potential outcome model simply linear, i.e. Y, + B2, + yes. the variance of the Horvitz- Horvitz-Thomperr Thompson estimator for a given threshold b, in the k-th power cycle graph, under unit-level randomization is proportional to
B 7dli)2 2 + (a 3 yd(1 h))) 27/(11 k)dd)
O
(ar
7
npilh
Therefore, the optimal threshold h" here depends on 7/ B, d, the graph structure, and the number of other candidate thresholds.
From this toy example, it is not difficult to see that for unit-level Bernoulli randomization for more general graphs, the squared-bias would be O()")²), and the variance would be O(p 1²t /n). Therefore, minimizing the MSE corresponds to balancing these quantities.
kth towee cycle graphs, also consider cluster-randomizen design with cluster 2k Ugander jet al. 2013), the authors show that this clustering size minimizes variance under full neighborhood exposure. We state the approximate absolute bias and variance for this setting in the following propositions. Here, define the threshold It = & with / I,.., d.
18

---

Proposition A.3 (Absolute bias in k-th power cycle graph under celuster-randmmzaaiii When 1/2, and the potential outcome model simply linear, i.e. Y; + B2, + yye, the bias of the for a given threshold h in the k-th power cycle graph under cluster rundomization, with cluster-sizes 2k + approximately 27y - 1) for h 2 1/2.
estimator
Proposition A.4 (Variance in the k-th power cycle graph under cluster-randomizztion,, When p 1/2 and the potential outcome model simply linear, i.e. Y; = a + 3a; + yes, the variance of the Horvitz-Thompson estimator for a given threshold h, in the k-th power cycle graph under cluster-level randomization, with cluster-sizes 2k + L, is proportional to
np? (3d 2dh + 7h)² (1(1 - h))) - 322 npp2 ).
Under cluster randomization with cluster sizes 2k + 1 for the k-th power cycle graphs. the variance grows linearly the degrees of the graph. Therefore, informally, compared to unit-randomized design setting. higher node degrees lead to stronger bias than variance for fixed exposure function e.).
A.6.1 Proofs to Propositions A.1
A.2
A.4
Proof to Proposition A
When 1/2 and the potential outcome model is simply linear, i.e. Y; = + B2; + Thompson estimator for given threshold h = 1/2k for = 2, ...., 2k, in
ye, the absolute bias of the the kth ath-power cycle graph under unit randdmmanomiini equal
Σ x;Ex
1 1(x, 2 h} E X, M+ 2 h
E|1 = =x
P|a 1,e,
h
1
1f2, 11
B[1 0, xx (u
P(z: = =0,e; S 1 h1
[x; :2: X; nx $1 h]
Mex
78
1/x, 2 h}
B[1 = (.)))
24
(2h p2r
[2 2 X, n1; $ -1-h1 4]
11, S 1 11 h}
E[1 = =,
(28) p2r
[2) : us E X, nr, S kil
x
78)
Σ
2k
2k
+
2k
(r/k (r/k-117) 177) - (3 77)
Des. 2k
(2k)
rr
2k
2k
(r/k - 177 -7
DPE/
(2k)
where X = (0, 1/0,1/2,, 2/2k,, 1}.
Proof to Proposition A.2 When = 1/2 and the potential outcome model is simply linear, i.e. Y, = a + A3; + ye,, is not difficult to that the variance of the Mhorviz-TThompse estimator for given threshold h 1/2k. the k-th power cycle graph under unit-level randomization proportional
2k
[1122 2 h}
(a 1/8++1l0 2
Eiz 11)) 2 h)p2 1)
2k
+ 2(8 + (2)22) -12/2))))2 +1
Σdh (7) 1(1/2k > h)p2d
+ (a + 1/22/
1)
E (2k) 1(1/2 2 h))221
Considering the dominating terms gives us the result.
Proof to Proposition A.3 When = 1/2 and the potential outcome model is simply linear, i.e. Y, = a + Bz, +
rye, the absolute bias of the Horvitz- Thompson estimator for given threshold ht = 1/2k for I k. k 1, ..., 2k.
19

---

in the kth-pooer cycle graph under cluste--andommization is equal to is equal to
1|x4 2 h}
E[ e (x))
P(z: 1, 2 h1
It 24
X
X1EX
1
1/ 11-h}
: E X, ni; 11111-1 Bi1 = 0,e; = = =i))u((t)))
P(a= mkk=0,e 0,e; $ 1- k}
(8+
(p/(2k + 1) + -121022--1111111------0001--111- 1 - + (21 d)/d)
+ 1)
p/ + 1) 2k/ +2 2p/ + Em 2r
(8 (7)
0(22 - 22)
where X = (0, 1/2k, 2/2k, 1}. When < k, the bias scales is the same as when = k.
When = 1/2 and the potential outcome model is simply linear, ie.. Y; = a [2; + yye, it not difficult to see that the variance of the Sloovitz-Thompee estimator for given threshold h = 1/2k. in the k-th power cycle graph under
Proof to Proposition A.4
with cluster-size 2k: + 1. is proportional to
1uudd 2 h}
(3 + yu/d)
[[(3d + 1- 1-21 21)( p²
1) + (21 1) -1) 1)]
n²
d
(7( ))
1
1uudd21 1)
[(3d + - 220(
1) + (21 1)( 1)]
"
n²
p²
Σ u=0
2 (8 (8 u/d)(1 d-u
Considering the dominating terms gives us result.
A.7 Simulations
A.7.1 Experimental details and results on the Amazon product similarity graph data
We also evaluate the performance of our estimator the Amazon (DVD) products similarity network [Leskovec] al. 2007). We ran experiments with synthetic potential outcomes, averaging over 1000 trials, under level randomization and spectral cluster-level randomizations. We generate simulated data using the lincar model with (2x, ex) = g(zs) + f(es), a: = ((.) = B2; = 102,, f(e:) = yes, and 6; generated from (00, 1) under a unit-level Bernoulli(O.S) randomization design. We focus on varying the ratio 7/ as we consider a fixed graph. Figure that we consistently perform better than existing fixed estimators.
The simulations were run on a CPU. Our experiments focus on subset of (the first) 1000 nodes of the 19828- node DVD graph. In particular, we considered the 17924-node subgraph by removing all isolated nodes. To compute the exposure probabilities, we used 10 simulations. We then selected the first 1000 nodes of the 19828 to analyze. 200 replicates were run, generating random treatment assignments and the corresponding outcomes. For cach of the replicates, 1000 separate simulation runs were generated to compute the oracle MSEs. In total, this took approximately 2 hours. The graph data is available at: https://sna..stant/ http:///aap.aafoooo..cddd/aaaaa hwtp///daaaaammazoooeeeeeeeeee html
0.8
RMSE/ATE
0.6
AdaThresh
0.4-
Oracle
HT1
0.2
HTO
Ratio
Figure 6: RMSE (normalized by the ATE) across different thresholds on the Amazon product network data. The error bars are two times the standard deviation.
20

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RMSE different potertial scalings
scalings
4x.0
1x10-
2x10-1
Le-00
Sc+00
1e+01
Ie+02
10+03
1e+00
Se-00
30+00
16+02
36403
PA6.0
Robio
Figure 7: RMSE (normalized by the ATE) of different Horvitz-Thompson estimators. Left: 2nd- power cycle graph under unit-level Ber(0.5) randomization. Right: 2nd-power cycle graph under cluster-level Ber(0.5) randomization with cluster sizes 5 (=2k + 1). The error bars are two times the standard deviation.
A.7.2 Experimental details on simulated graphs
machine of Intel@ Xeon@ processors with 48 CPU cores, and S0GB of RAM. We simulated 1000 replicates, generating random treatment assignments and the corresponding outcomes, with each oracle MSE computed using 1000 separate simulation runs. The exposure probabilities under each threshold were computed as proposed in [Aronow and Samii 2017] using 20000 simulation iterations. In total, this took approximately 30 minutes on average (across the different potential outcomes, and graph settings). The code is available https://githb..com// wttpb//oo//ooooi////aaaaa. Thressh..ttt
All synthetic graph simulations were
In the following, focus key experimental results circulant graphs (see Section A.6 for discussions circulant graphs).
A.7.3 More Simulations for Horvitz-Thompest estimator with adaptive exposure thresholds
In Figure 7 we simulate outcomes from the linear model with sb ex) 9(x;) [ex), oz 10, g(2;) 82, 10z;, f(e;) = yez, and fixed e; generated from M/O0 for 1000-node 2nd-power cycle graph.
A.7.4 Simulations for non-linear potential outcomes models
We investigate the robustness of our estimator to potential outcomes models that non linelin in the exposure. In particular, consider simulations from the sigmoid, and sine exposure functions. Figure display the performance of our estimator under sigmoid (left) and sine (right) interference function. respectively, Our adaptive threshold Thompso- estimator (AdaThresh) generally improves upon other existing Horvitz Thriitz-hoompoo estimators.
A.7.5 Simulations for Difference-in-Meane estimator with adaptive exposure thresholds
We consider our approach using the Difference-in-Meenns estimator incorporating exposure thresholds:
Σe 1(2 e, 2 A)Y.
Σ 1(z = 0, S 1 n)Y
ÂDIM
(14)
Σ 1/ 1,e, 2/
Σ 1 0, 1
We use the following bias estimator:
B(A (+===<a (1 -1-
Σ
cmn1(1 0. S h) " 1( 0, h}
Σ 1 114=1,,6, 1,e 12 h}
where & the linear regression coefficient for the exposure variahle.
We the following variance estimator, decomposing it into its treatment and control parts:
8(F)
(iT + éc)
where
11 L,e, 2 111 1, [., 2
1 Σ x=l
nh
n1
21

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RMSE different potertial scalings
different potential scalings
ROSpIAT
20%
500
RAt
Figure 8: RMSE (normalized by the ATE) induced by the different Chorviz-TThompnt estimators. We take (12 e,) = g(=)) + f(e).. Left: 2nd 2nd-power eycle graph under unit-level Ber(p), p 0.5, randomization under sigmoid f(e,) = 7/(1 exp (-10(e; p))). Right: 2nd-power cycle graph under unit-level Ber(p). 0.5, randomization with f(e,) = 7(1 sin( e;)) being sine function. We focus varying the ratio 7/3 8 as we consider fixed graph, with 1000. The error bars are two times the standard deviation.
where 7o1 Ei 1(2 1, hJ. and
1
1 a E
Ya = 0,
where Σ. 1(2 0, S -1
We compare the performance of our adaptive estimator the vanilla difference-in- estimator, the difference- in-means analogue of the vamilla Corvitz-Thompson estimator, and the diffcrence-in-nen estimator with threshold plugin via Lepski's 's method. We write these out below:
vanilla difference-in-menn means estimator
Ei 112-111
> 1(2: = 0JY;
ÍDIMO
(15)
Σ 12==11
1/2 =0}
difference-in means estimator threshold h
Σ 11a=1,=111
E 1 = 0,
7DMM
(16)
Ex 1i 11==1,e=1]
1/2= = 0, 0
difference-in-means. estimator threshold hiuepas where
happak minsh H (WE) 0},
(17)
with
I(h) (fUMM 2SDEV(FUUM, fOIM, + (SDEV((FDM,))
(18)
and
1/2 1,6, A
1/2 = 0,e, Mrepass
A ApskiDM
Σ 1=1
D 2i= 1(2 = 1, Pra.pas))
Σ 1( = 0, e < 1 h }
(19)
display the performance our adaptive threshold (AdaThres) Diffcrence-in-Mern in-Maans estimators in comparison to other estimators. AdaTTresh improves upon fixed threshold Difference-in-Me estimators.
Figures and 10
A.7.6 Simulations using local linear regression
We extend our global linear regression approach to local linear regression one to estimate the rate change of the bias. We write the new bias estimators for the Horvitz- Thompson and Difference-in in-Means estimators. We illustrate the performance of the local linear regression in the settings above, well as settings that significantly
22

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10a
10"
10-1
10-1
RMSE/ATE
10
10
10-3
0.0
0.25
03
110
2.0
5.0
0.0
0.25
D3
1.0
2.0
0,25
0.5
1.0
2.0
5.0
100.0
Ratio f
AdaThresh
DIM(1)
aracle DIM
Lepski DIM
DIM(O)
Figure 9: RMSE (normalized by the ATE) under the linear model with (22,, = g(z,) + f(e)). O; = 10, g(z;) $e, 102, f(e;) ye;, and fixed generated from (O, 1), induced by the different Difference- MMans estimators. We focus on varying the ratio 2/3 as we consider fixed graph. Left: Cycle graph under unit-level Ber(O.5) randomization. Center: 2nd-power cycle graph under unit-level Ber(0.5) randomization. Right: 2nd-power cycle graph under cluster-level Ber(O.5) randomization with cluster sizes 5 (=2k + 1). The error bars are two times the standard deviation.
10a
10-1
RMSE/ATE
10-7
10
10
30
100
300
1000
0.0
0.25
50
Ratio
AdaThresh
DIM(1)
aracle DIM
Lepski
DIM((O
Figure 10: RMSE (normalized by the ATE) induced by the different Difference-in- estimators. We take (22, e,) g(2) + f(e)). Left: 2nd 2nd-power cycle graph under unit-level Ber(p), 0.5. randomization under sigmoid f(e,) = 7/ + exp (-e,)). Right: 2nd and-power cycle graph under unit-level Ber(p), focus on varying the ratio 7/8 / as we consider fixed graph. with 1000. The error bars are two times the standard deviation.
0.5, randomization with f(e,) = >(1 -sin( . ez))) being sine function. We
23

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RMMS//ATE
0.0
0.23
02
12
20
0.0
0,25
0.5
1.0
20
0.0
0.25
0.2
1.0
2.0
50
100.0
Ratio
AdaThresh
HT(0)
oracle HT
Lepski HT
HT(L)
Figure 11: RMSE (normalized by the ATE) under the linear model with (22, e4) g(2,) + fle.).
a; = 10, g(x) 3z, = 102, f(e,) = Yee, and fixed generated from _N 1), induced by the different (local) Soovitz--Thmppsee estimators. We focus on varying the ratio 7/3 we consider fixed graph, with 1000. Left: Cycle graph under unit-level Ber(0.5) randomization. Center: 2nd-power cycle graph under unit-level Ber(0.5) randomization. Right: 2nd-power cycle graph under cluster-level Ber(0.5) randomization with cluster sizes 5 (=2k + 1). The error bars are two times the standard deviation.
deviates from linearity. We note that local and global linear regression involve distinct bias-variance trade-offs. which we do not pursue here they fall outside the scope of the paper.
We write the bias estimator for the Horvitz-Thompeeet Thompson estimator
b(T%.) 1 - e1- 1(z: 1.c, 2
e: 1(2: 0, 1 P(:: 0, S h3
(1-1
1(a=1,.221
where (((h)) and 40--1) -b) the local linear regression coefficients for the exposure variable in the intervals [h, 1].
[0, - A). respectively.
Similarly, write the bias estimator for the Difference-in -n---in-Msssss estimator as:
em (h) 1 = 1,e, 2 h)
erim (1 (1-1i) 1 0,e
b(Fr)
-
12==1,, h}
Σ 1 51
where 7((h) and 7/ ((-+)) are local linear regression coefficients for the exposure variable the intervals [h, 1]. and 0, - h), respectively.
Figures
and 14 display how local linear regression bias estimates perform.
A.8 Discussion relaxing the bounded degree assumption
relax Assumption 4.3 by partitioning the exposures into exposure bins E.. b == 1, 2, .... K, each associated with an "effective exposure" čs. That is, [0, = U Eg U... Ex. with E, n E, = 0. for all i # j E [K]. In particular, the weighted graph setting, we can take the effective exposure of unit i, é, = és if and only if Es. and such that exposure positivity, for all E [n). r E {0, 1), and s [0, I(2, é, s) 0, satisfied. For instance, for uniformly sized bins with associated effective exposures é, that well approximate ex. [Es] 1/K. Let B+(i) denote the ball radius around unit such that decomposes, (i)
In more general settings,
( (i) + M (i) for all [n]. B (i) is the ball of radius around unit i with strong connections TE. and B (i) the ball radius around unit with weak connections 6." such that sup, 8" Dc. We choose our partition Bi,.. Th....yy Bx. for some K, as a function 67", and B (i), E [n]. such that the exposure bins well approximate the true exposures. Additionally, for all E [n]. we require that [B) (1). With an abuse notation, write (N) to mean the exposure unit as function of treated units in the subset / (t). Then, require also that ex ()) Ω(e (i))) Additionally, (i) must satisfy the three conditions around "affinity sets" in (Candrassckhar et al. 2023 generalizing [Aronow and Samii 2017), with an additional condition that the sum of the affinity set covariances is M(n). in
the bias terms well the variance terms in the MSE. This gives uniform control the estimated MSE

---

10a
10a
10
RMSE/ATE
10
10
0.0
0.25
0.5
10
2.0
5.0
0.
0.25
0.3
1.0
2.0
0.23
Ratio
AdaThresh
DIM(1)
oracle DIM
Lepski
DIM(O)
Figure 12: RMSE (normalized by the ATE) under the linear model with (22,,e)) = + f(e.).
a; = 10, g(4)) = 3z, = 102, f(e)) = Ye,, and fixed e; generated from NN 1), induced by the estimators. We focus varying the ratio 7/3 we consider
different (local) Difference-in-Merne a fixed graph, with n = 1000. Left: Cycle graph under unit-level Ber(0.5) randomization. Center: 2nd-power cycle graph under unit-level Ber(0.5) randomization. Right: 2nd-power cycle graph under cluster-level Ber(0.5) randomization with cluster sizes 5 (=2k + 1). The error bars are two times the standard deviation.
10a
10a
RMSE/ATE
10-1
100
500
1000
0.0
0.25
Ratio
AdaThresh
HT(()
oracle HT
Lepski HT
HT(1)
Figure 13: RMSE (normalized by the ATE) under the linear model with (222, e4) g(2)) + f(e.).
a; = 10, ((,) 3z, 102,, and fixed e; generated from (0, 1), induced by the different (local) Sarvitz-Thompsee estimators. Left: 2nd-power cycle graph under unit-level Ber(0.5) randomization under sigmoid f(e,) = 71 +1/111+0000 exp
exp (-+))). Right: 2nd-power cycle graph under unit-level Ber(O.5) randomization with f(e >(1 sin( ez))) being sine function. We focus varying the
ratio 7/3 as we consider a fixed graph, with n = 1000. The error bars are two times the standard deviation.
25

---

10a
10-2
RMSE/ATE
10
10-1
10-2
10
30 130
330
1000
0.0
0.25
Ratio
AdaThress
DIM(1) Lepski DIM
oracle DIM
(XM(()
Figure 14: RMSE (normalized by the ATE) induced by the different (local) Difference-ing -eeMas estimators. We take (z), (24) + f(e;). Left: 2nd-power cycle graph under unit-level Ber(0.5) randomization under sigmoid f(e) = 7/(1 + exp (-e:))) Right: 2nd-power cycle graph under unit-level Ber(0.5) randomization with f(ea) = 7(1 - sin ez)) being sine function. We focus on varying the ratio //8 as we consider fixed graph, with 1000. The error bars are two times the standard deviation.
of the estimator with respect the true MSE of the estimator and Eg.b=1..... ,......,K K, and all the results follow through. These conditions subsume the approximate neighborhood interference (ANT) condition in [Leung] 2022] analogously to o-mixing conditions spatial settings, such [Jenish and Prucha 2009), when maximum clique size in the network does not grow. Indeed, when the maximum clique size in the network does
not grow, we can embed the u-dependent network in [Leung] 2022] in lattice structure of random field in fixed-dimensional Euclidean space with a-mixing just in (Jenish and Prucha 2009)
A.9 Discussion "double-dipping"
Since our approach involves "double-die diiuble- into the data, we need to make sure that there is no overfitting that occurs. The authors of (Chernozhukov et al. 2018). [Belloni et al. 2015), and [Kennedy] 2022) describe this in more detail. While sample-splitting would simply take care of this in the case of independent data, it does apply
setting where the data are dependent, as modeled by the network. One could potentially leverage results under the dependent-datz setting, such as Hart and Vieu 1990), but this outside the scope of our paper. We propose to use our approach on the whole sample data and argue this via empirical process theory arguments. In particular, estimator for h has simple enough associated MSE function class. We note that it sufficient show that are not overfitting by showing that the rate of convergence of the estimated MSE under i converges to the true under the optimal threshold, under best possible linear fit, h" at least at a O(n 1/22 -
can think of our threshold /t as nuisance parameter. and we show that our
Proposition A.S. Suppose that Assumptions 2 4.2 and and44.3an are satisfied. The corresponding bias terms in the estimated and true MSE, Mh, and Mh, under "double prediction" satisfy
Va (h.)) (ho)) M1h((1))) MMC1)) 0,
E
sup 6/2</h; - km.
Suppose that Assumptions
4.2 and 4.3 are satisfied. We aim show that the corresponding biases in the MSE terms under double prediction" satisfy
E
sup 8/2c|h*
as 5 + 0, where ranges over the set X, of possible fractions of degree i.
26

---

Proof to Proposition A.5
We begin by considering the empirical process in question. We write out
h}
(h)
F| 1,
11
2
1=1
1
1,6 n 1
1)
m!m
2 1, 2 1/Hl2
= e, KYYI
1)
n² 11
1/2 = 0,0, 1 n2?
0 - 1)
n
1/2 0,e, < 1 - h [ = 0, < 1 h)YY,,
1)
m
T 5
1
(1(z = 1,e, 2 11/1(12 = 0,e, < k1)x,,s)(
10
i= JEp
7
2
1/ (a=
1/2 = 0, 1 - h3333
n
2
2n²
JE
=D
The absolute Horvizz-Toompsn bias estimate at threshold h (and at 1 h)) is
9m -1l-e))l 11)= 1, 2 h]
fn (e = 0,e, $ 1- h
b(F).) = 73
(20)
P( = 0, S 1-hh)
Then, for every h. using the identity x² - 1²² (x + 5) v). and defining Un E R such that U, 2 (1) for all h (note that there exists such a U, < 00 by Assumptions 2 4.2) we have
(f.) S 2V, (b(+)) 5' (fm) (b((x) b" ((x)
where, from Section A.4 linear fit is
the difference between the bias estimate and the true bias induced by the best average
(n - e,) = 1, 2 h)
1fx, 24
B(Fx.) - (fr.)
Σ z.EX,
I 2s X. 2 hl 7/o (1 2:))
n
P(z2 1,
1=
= 0,e 5
1/ -1 h}
Σ
Σ
I X nz: S 1 h 7r((s))
P(z: 0,c, <1
s=1 a
where "Yn is the slope of the best average lincar fit, and where x; ranges over the set X; of possible fractions of degree
We now proceed to consider each of terms in ((+r)) d" (fx) above. Each the parentheses satisfies Donsker
conditions. Indeed, under Assumptions 2 4.2 and
4.3
they are sample-mean terms with uniformly bounded
coefficients, and 4. converges to 7 a rate On 172), which we compose with 35" (h) (see Section 3.4.3.2.
in Van Der Vaart al. 1996). Thus, the terms in M (h)
(h) have bounded entropy integral. Under our
bounded-variation potential-outcome model (Assumption
4.2 we have that log Nj (c, e", L2(P) < K (t).
which gives us [[6 et,, (p)) < 82/2 Van Der Vaart et al. 1996 (see also Example 19.11 in Van der Vaart
2000
Therefore, putting these together, have the following maximal inequality for the bias terms
E sup
vn (Mhh())) Mb(hh)) - (M72(1" - Mc((1)))
(21)
6//<</h* hhnlk
J(6, 0:, L.(P)))
$ J(6, Lx())) (1+
(22)
8V7
27

---

with J". L2(P)) < 28.2
28

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Reviews:
[{"review_id": "3I1BQmqVJK", "reviewer": "Reviewer_Szy4", "summary": "This work considers estimating the average treatment effect (ATE) in the presence of network interference. A common approach to obtain unbiasedness is to define an exposure mapping that identifies units not subject to interference, and then apply a Horvitz–Thompson (HT) estimator restricted to those units. In such a case, the HT estimator would compare outcomes of treated and control units whose neighborhoods are sufficiently homogeneous (according to threshold $h$), weighting by their assignment probabilities. The core difficulty is that the threshold $h$ is typically fixed a priori and hard to choose while a mis-specified $h$ can bring bias or inflate variance. The paper’s motivation is to adaptively choose the exposure threshold rather than rely on a fixed one to best balance interference bias and statistical variance.", "questions": "- Why is the Horvitz–Thompson estimator preferred over other estimators (e.g., Hajek, AIPW, regression-adjusted)?\n- How does your method compare to direct modeling approaches that predict outcomes as a function of both treatment and exposure?", "limitations": "Only considers Horvitz–Thompson (HT) estimator for causal inference under network interference.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n- adaptively choosing the exposure threshold for HT estimator is well motivated.\n- The paper’s goal is to select $h$ that minimizes the MSE, which makes sense and leads to the bias variance trade-off.\n- Bias and variance are estimated in a data-driven way to select the optimal threshold.\n\n\nWeakness\n- The method estimates bias based on a linear regression of outcomes on exposure\n- Estimating inclusion probabilities can be challenging when the network is small or has high degrees\n- AdaThresh needs a range of exposure thresholds to be well-populated with units. Otherwise, many thresholds may include too few units to reliably estimate HT statistics or regression slopes.", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer Szy4 for the thoughtful review and questions. We would like to address the questions and concerns the reviewer has posed below.\n\nResponse to Question 1: The Horvitz-Thompson estimator is used heavily in practice! Motivated by this, we sought to address our framework through the setting of the Horvitz-Thompson estimator. Our framework, of course, could be adapted for the Hajek, Difference-in-Means, AIPW etc. estimators. We formulate the adaptively-thresholded estimator for the Difference-in-Means estimator, which is a special case of the Hajek estimator, in the Appendix (Section A.7.5.). A similar adaptation to the Hajek estimator could be used. We note however, that the Hajek estimator is only approximately unbiased at the true threshold. Therefore, characterizing the exact bias-variance tradeoff formally would be more complex compared to the Horvitz-Thompson estimator. We discuss this in lines 74-80 of the paper.\n\nResponse to Question 2:\nWe appreciate the curiosity of how the performance of our approach compares to direct modelling approaches. This, however, is not the focus of our paper. Instead, our work focuses on an adaptive optimal threshold for a specific estimator class decided upon by the practitioner a priori. In particular, our goal is to show that there is something to be gained when using an adaptive estimator, instead of a fixed estimator like HT(0) or HT(1), as in our approach. The only other viable adaptive approach that exists for that purpose is Lepski’s method which we had adapted to our settings. Therefore, we believe that the useful benchmarks for comparisons are the estimators adapted from Lepski’s method, and the fixed Horvitz-Thompson estimators. \n\nWe would also like to emphasize that, while we use linear regression to obtain a first-order approximation of how interference bias varies with exposures, which is similar in spirit to Baird et al. (2018), we go further by pairing this with a Horvitz-Thompson-style bias estimator. This approach uses the estimated regression coefficients in conjunction with exposures, enabling the bias estimator to account for the interaction between regression structure and exposure distribution present in the data.\n\nWe hope that our response has provided more clarity on the motivations and the focus of our work. We thank the reviewer once again for the thoughtful feedback, and we hope that this discussion has enhanced your perspective on our paper."}, {"review_id": "rnAlmCichc", "reviewer": "Reviewer_rfKt", "summary": "It is well known in the causal inference literature that under network interference—where the potential outcome of a unit may depend on the treatment assignments of other units—classic estimators such as the difference in means can be biased, even under the gold standard of a randomized experiment. The typical estimand in these settings is the difference in average outcomes when everyone is treated versus when no one is treated, referred to as the average treatment effect (ATE) in the current work.\n\nWhile unbiased estimation is possible using a Horvitz-Thompson estimator and a known exposure mapping, this often comes at the cost of prohibitively high variance. In this work, the authors focus on this estimator alongside a particular class of exposure mappings, where “treatment exposure” and “control exposure” are defined based on the $h$-fraction of an individual’s neighborhood assigned to treatment or control, respectively.  They propose and analyze a data-driven approach to selecting a threshold $h$ that minimizes the mean squared error of the estimator. Their method estimates bias via a simple linear regression and estimates variance using a scheme introduced by Aronow and Samii (2017). The authors analytically characterize the probability of selecting the optimal threshold under certain assumptions and empirically assess their method using simulated outcomes on both simulated and real-world networks.", "questions": "1.\t**Possible Overloaded Notation $D$**: In Section 2.1, $D$ is introduced as a degree matrix. However, in Theorem 4.5, it appears that $D$ refers instead to a bound on the number of candidate thresholds. If that is correct, it may be worth clarifying this or using different notation to avoid potential confusion.\n\n2.\t**Possible Typo in Theorem or Proof**: In Theorem 4.5 and Corollary 4.6, the result is a bound on $\\mathbb{P}(\\hat{h}_n \\neq h_n^{\\*})$ but in Appendix A.5.1 the proof starts with $\\mathbb{P}(\\hat{h}_n = h_n^{\\*})$. \n\n3.\t**Understanding the Bias Slope and Equation (6)**: I struggled to fully understand what the “bias slope” refers to and how it helps us interpret the bias of the Horvitz-Thompson estimator at a given threshold $h$. The sentence following Equation (6) on page 4 defines the term, but by that point, the bias slope has already been referenced several times without a clear definition. Introducing and defining it earlier would aid reader comprehension. Additionally, I had difficulty connecting the term “bias slope” to its definition. When I think about interpreting a regression coefficient, I typically understand it as: a unit change in exposure results in a $\\gamma$ change in the outcome. However, the last paragraph on page 4 states that the bias slope captures “the average change in bias as $h$ varies from 0 to 1,” and I wasn’t sure how to reconcile that interpretation with the regression framing. Part of my confusion may stem from not having a strong intuitive grasp of Equation (6). While it resembles a Horvitz-Thompson-style estimator, I wasn’t clear on where it comes from or why it is a reasonable estimator for the absolute bias. A bit more elaboration or context here would be helpful.\n\n4.\t**Interpreting Corollary 4.6**: I thought the authors did an excellent job explaining the implications of Theorem 4.5, despite the complexity of the expression. However, I had more difficulty following the interpretation of Corollary 4.6, particularly the explanation in lines 244–247. For example, the sentence “Corollary 4.6 tells us… for large $n$” is helpful, but I wasn’t sure where in the expression this intuition comes from. Could the authors point out which part of the bound supports that explanation?\n\n5.\t**Figure 1(a)**: I was a little confused in the bias plot as to why there is bias when $h=1.0$. When $h=1.0$, I would expect that there is no bias but very high variance because only treated units with an entirely treated (resp. untreated) neighborhood are included under the treatment (resp. control) exposure. Is the bias there because the exposure mapping itself is misspecfied compared to the way the outcomes are generated and thus the bias comes from a misspecified exposure mapping?\n\n6.\t**Figure 4 (left)**: There is a notable drop in the magnitude of the RMSE for the last ratio value. I understood this to be referring to a regime of very strong interference. I don’t understand why suddenly all the methods have lower RMSE under this regime. Do the authors have any comments on this?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "1. **Quality**: The submission appears technically sound and is well-supported by both theoretical analysis and extensive experimental results. The methods used are appropriate and straightforward, and the work is complete. The authors do a commendable job of being both careful and honest in evaluating the strengths and weaknesses of their paper.\nSpecifically, they address concerns about overfitting in the appendix, with a clearly placed overview and pointer in the main text before Section 4. They are also transparent about the proposed method outperforming baselines in some settings but not all, offering a fair discussion in Section 5.1. The Discussion section further helps clarify the scope of the proposed approach, including when it is likely to be appropriate or not.\n\n2. **Clarity**: Overall, the submission is clearly written and well-organized. The authors provide a thorough literature review and are generally careful in their explanations.\nThat said, a few areas would benefit from clearer or more detailed exposition—particularly around the bias estimator and the bias slope. The theoretical results, while solid, are somewhat difficult to parse due to the complexity of the expressions (involving multiple terms). I appreciated that the authors made an effort to offer intuition to help interpret these results; however, a few additional well-placed sentences could further aid reader understanding and better highlight the main takeaways. More detailed suggestions are included in the Questions section below.\n\n3. **Significance**: The paper presents an interesting yet simple method for selecting an optimal threshold for the Horvitz-Thompson estimator of the Average Treatment Effect in a data-driven way.\nGiven the popularity of the Horvitz-Thompson estimator—especially when paired with cluster-randomized designs in interference settings—and the widespread use of exposure mappings based on neighborhood treatment fractions, this contribution is impactful and likely to be used or extended by others in the community.\nPerhaps the biggest limitation is that such an exposure mapping is unlikely to be correctly specified in real-world applications, but this issue lies outside the scope of the current work. Regardless, since causal inference practitioners often adopt simple exposure mappings in practice, it is valuable to have a principled, data-driven approach that aims to minimize error rather than relying solely on domain knowledge to choose a threshold.\n\n4. **Originality**: This paper offers a thoughtful combination of existing ideas in a new setting, with clear articulation of both its contributions and its relationship to prior work. They highlight important properties of the widely used Horvitz-Thompson estimator in the context of causal inference under interference. While the bias-variance trade-off is a well-known issue, the authors contribute a new data-adaptive approach to selecting a key hyperparameter—specifically, the exposure threshold—and situate this contribution within the broader literature.\nThe novelty here lies not in identifying the trade-off itself, but in proposing a new method for reducing error overall and situating this method in the context of the bias-variance trade-off. It is clear how this work differs from previous contributions, and to my knowledge, this particular data-driven approach has not been proposed in this specific context before. The authors are careful with their citations and transparent about how their work builds on existing techniques.", "was_revised": true, "final_justification": "I had no major concerns in my original review, and the authors addressed my clarifying questions well. Other reviewers raised questions about the scope of the paper, the approach, and the assumptions, and I felt the authors did a good job of responding and highlighting the practicality of their approach. The paper is well-motivated, proposes a useful and reasonable approach to a widely used method in causal inference, and I am inclined to keep my original score.", "rebuttal": "We thank Reviewer rfKt very much for the encouraging feedback, and the thoughtful and detailed comments and questions. We would like to address them below:\n\nResponse to Question 1: Thank you for pointing this out. We will be clarifying and improving the notation there.\n\nResponse to Question 2: Thank you for pointing this out. We will fix the typo in the proof.\n\nResponse to Question 3: Thank you for this feedback. We agree that the notion of the “bias slope” would benefit from an earlier introduction and clearer motivation, and we will incorporate this in the revision.\nTo clarify the intuition: as you point out, the regression coefficient estimates the average change in the mean outcome per unit change in exposure. In our context, this also reflects the average change in bias as we move away from the boundary exposures, i.e., from 0 for control or 1 for treated, since bias arises (depending on the outcome function) when exposures deviate from these extremes. This interpretation motivates the use of the weights $e_i- 0 = e_i$ and $1− e_i$ in Equation (6), corresponding to deviations from the control and treated boundaries, respectively.\nEquation (6) defines a Horvitz-Thompson-style estimator for the bias, which integrates this regression slope over the empirical exposure distribution to yield an average (absolute) bias estimate at a given threshold.\n\nResponse to Question 4: This comes from the $n(\\Delta_n - \\tilde{\\delta})$ terms in the bounds in Corollary 4.6. Indeed, when these terms are larger, the $\\exp(- (n(\\Delta_n - \\tilde{\\delta}))$ terms are small. We hope this is helpful, and we are happy to add this explanation in the revision.\n\nResponse to Question 5: Thank you for pointing this out. The issue you observed was due to this particular illustrative plot having been generated using an earlier version of the code, with significantly fewer Monte Carlo trials. This was an oversight; all other plots and results in the paper had already been updated to use 1,000 Monte Carlo trials for bias computation and 10,000 for exposure probability estimation. We have now regenerated this specific plot using the updated setup. As expected, the bias at threshold 1 is numerically close to zero across all $\\gamma/\\beta$ ratios, consistent with your intuition, and bias remains near zero for $\\gamma/\\beta = 0$. We will replace the outdated figure in the revision to ensure consistency with the rest of the results.\n\nResponse to Question 6:\nThank you for pointing this out. Upon reviewing the code used to generate this plot, we discovered an inconsistency: the ATE used to normalize the RMSE was inadvertently computed using a $\\gamma/\\beta$ ratio of $10^6$, while the RMSE values themselves were computed using the correct ratio of $10^4$. This mismatch led to an artificially low normalized RMSE for the final data point.\nThe value $\\gamma/\\beta = 10^6$ was used elsewhere in the cluster-randomized setting to illustrate the effects of stronger interference, as cluster randomization helps control interference bias. Unfortunately, that parameter setting was mistakenly carried over when computing the ATE for this plot.\nWe have since corrected the figure, and the updated version behaves as expected: the normalized RMSE no longer drops sharply for the final ratio and reflects a consistent trend across all $\\gamma/\\beta$ ratios. We apologize for this oversight and will include the corrected figure in the revision. We have verified that this issue is isolated to this plot and does not affect any other results in the paper.\n\nOnce again, we very much appreciate the reviewer's thoughtful and detailed feedback."}, {"review_id": "nTmkkNsFdJ", "reviewer": "Reviewer_mQMx", "summary": "This paper considers the problem of average treatment effect (ATE) estimation under network interference.  In this setting, the treatment of neighbors can impact the outcome for an individual, and the ATE is defined as the average outcome when all units are treated (including the effect of neighbors on each other), versus the average outcome when no units are treated.  However, in a randomized trial, units often have neighbors with a mix of treatment assignments, and prior approaches to estimation use a fixed threshold to select units for use in ATE estimation:  In an extreme case, one could e.g., only consider those treated individuals for whom at least 95% of neighbors are treated, etc. This paper builds upon this line of work by proposing a data-adaptive approach for choosing these thresholds, to trade off between bias and variance.  They propose an estimator of the bias that depends on an estimate (via linear regression) of the \"bias slope\", the regression coefficient of the outcome on the fraction of neighbors exposed.  The main theoretical result (Theorem 4.5) characterizes the probability of choosing the MSE-minimizing threshold, and the approach is demonstrated on simulated and real data.", "questions": "I just have a few minor clarifying questions, to make sure that I'm understanding the paper correctly.\n* Could you provide more specific detail on what is meant by cluster-level randomization here?  How are clusters determined?  Apologies if I missed this explanation in the main text.\n* Is it fair to say that Theorem 4.5 assumes either unit- or cluster-level Bernoulli randomization?  If so, it might be worth calling that out as an assumption (e.g., \"Assumption 4.4\") a bit more explicitly.\n* Does Theorem 4.5 assume Equation (1) or the slightly more restrictive form where $\\psi(z, e) = g(z) + f(e)$?  That point was not entirely clear to me from the main text.", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Overall, I found the paper to be well-written and clearly explained, and I will enumerate strengths and weaknesses below.  I use (+) to indicate a strength, and (+/-) to indicate an area that is a bit more mixed.\n\n* (+) In terms of clarity and significance, I found the problem well-motivated, and the paper clearly positioned with respect to related work in the introduction (Section 1) and Sections 2-3.  As a caveat, my expertise is in causal inference more broadly, and I am less familiar with the literature on estimation under network interference.\n* (+) The algorithm and the potential outcome models are clear and well-motivated.  I appreciated the clarity of presentation here, e.g., Section 3 the note that the bias slope is only expected to be a first-order approximation, and that the method only assumes Equation (1), not the more restrictive Equation (2).\n* (+) I appreciated some of the intuition provided alongside Theorem 4.5, which provides some intuition on the conditions that make it easier/harder to select the optimal MSE-reducing threshold.  It might have been nice to have results that more specifically get at the resulting MSE of the method itself (accounting for possible errors), but I'm sympathetic to the fact that such an analysis might not be very tractable given the non-smooth nature of the problem.\n* (+/-) I found the technical portions of the paper to be relatively clear, modulo some points below:\n    + I see an assumption (4.1) regarding positivity, but isn't there a need for assuming unconfoundedness / randomization explicitly?  For instance, that treatment assignment of $z$ does not depend on $\\epsilon$?  Theorem 4.5 mentions unit-level Bernoulli randomization and cluster-level randomization, but the latter was not clearly defined in a self-contained way as far as I could tell, which led to some confusion on my part understanding the exact conditions where Theorem 4.5 holds.\n    + As a minor point of clarity, I found Figure 1 difficult to read given the density of information.  I might suggest (as a minor point) to use lines for Figure 1(a) to make the trends easier to see.\n* (+) I found the experiments to be a helpful addition, showcasing the fact that HT(1) and HT(0) both sometimes perform quite poorly, while the proposed approach is generally competitive with the best of the two.", "was_revised": true, "final_justification": "I am maintaining my positive score after discussion", "rebuttal": "We thank Reviewer mQMx very much for the encouraging feedback and the thoughtful comments and questions. We would like to address them below.\n\nResponse to Question 1: This corresponds to a cluster-level randomization design, where clusters are first defined, and then randomization is performed at the cluster level, i.e., if a cluster is assigned to treatment (resp. control), all units within that cluster receive treatment (resp. control).\n\nFor Figures 2, 3, and 4: The node colors in Figure 2 reflect the clustering used in our experiments for both the graph generated from the stochastic block model (SBM) and the real-world network data. For the SBM, we used the underlying block memberships (from which the graph was generated) as the clusters. For the real dataset, we applied $\\epsilon$-net clustering with $\\epsilon$ = 1. In $\\epsilon$-net clustering, nodes in the net are at least ε apart (in shortest path distance in the graph), and the union of their ε-radius neighborhoods (balls) covers the entire graph. These were described briefly in the captions of the figures. We apologize if this was unclear, and we are happy to improve on this clarity in the revision. We also refer the reviewer to Ugander et al. (2013) for a more detailed discussion of this clustering method.\nFor the k-th power cycle graphs, clusters are formed by grouping consecutive $2k + 1$ nodes. This specific clustering choice follows the design in Ugander et al. (2013), where it is shown to achieve minimum variance. We discuss this in lines 533–534 in the Appendix.\n\nRegarding Questions 2 and 3: Thank you very much for pointing this out. This is correct, and we will make this assumption more explicit in Theorem 4.5. Corollary 4.6 does assume the slightly more restrictive form of $\\psi(z,e) = g(z) + f(e)$. We will also make that clearer in the revision. Apologies for any confusion caused. Additionally, regarding clarity in Figure 1, we are happy to improve this in the revised version. \n\n Thank you very much again for all of your insightful questions and feedback."}, {"review_id": "aKPigvXAn5", "reviewer": "Reviewer_Y8AE", "summary": "This paper considers estimate the average treatment effect with special attention to the scenario the SUTVA assumption does not hold. The paper considers exposure mapping and proposes a data-adaptive method to estimate the $h$-fractional threshold  for minimizing the Horvitz-Thompson estimator’s MSE in randomized controlled trials with interference, focusing on estimating $h$ with bias and variance tradeoff of the ATE.", "questions": "(1) Can it be generalized to nonlinear potential outcome models such as generalized linear models?\n(2) The paper estimates the ATE of the form:\n\\begin{equation}\n\\tau = \\frac{1}{n} \\sum_{i = 1}^{n} y_i(1, 1) - \\frac{1}{n} \\sum_{i = 1}^{n} y_i(0, 0).\n\\end{equation}\nIs it possible to estimate the effect of the treatment itself, excluding the influence of the exposure?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths：（1）Clear objective function for threshold \\( h \\) estimation, with rigorous bounds on estimation error.  （2） Experiments show strong real-data performance.  \n\nWeakness：Limited to linear potential-outcome models only.", "was_revised": true, "final_justification": "This paper proposes a data-adaptive method to select the h-fractional threshold for exposure mappings in randomized controlled trials with interference, aiming to minimize the MSE of the Horvitz-Thompson estimator by leveraging a first-order approximation, and demonstrates its superiority through extensive simulations. I decide to raise my score to 5.", "rebuttal": "We thank Reviewer Y8AE for the thoughtful questions. We appreciate it, and would like to address the reviewer’s questions and concerns below, and hope that our response provides more clarity on the focus of our work.\n\nResponse to Concern 1 / Question 1:\n\nWe are not assuming a linear potential outcome model. Our model is much more general (please refer to eq. 1). We are simply using linear regression for a first-order approximation to the potential outcomes model for bias estimation. Only in the special case of a true linear model is this exact. We would also like to draw the reviewer’s attention to Baird et al.(2018) , where the authors leverage the slope of spillovers with respect to treatment saturation . Similar to our approach, this work uses the linear model, i.e. the slope, of the spillovers for an approximation without assuming it as the generating model. This is discussed in lines 151–157 of the paper. \n\nIn Corollary 4.6, we provide error bounds for models of the form presented on line 112. To demonstrate the method’s robustness beyond linear settings, we report simulation results for nonlinear potential outcomes models in Section A.7.4. Finally, we extend our method to incorporate local regression approaches in Section A.7.6, and apply these to nonlinear outcome models as well. This is discussed in lines 301–313 of the paper.\n \nQuestion 2: “Is it possible to estimate the effect of the treatment itself, excluding the influence of the exposure?”..\nThis is a great question. Extending our approach to other estimands, such as the direct effect, is indeed possible, though the characterization of bias and variance would be somewhat different. For example, to estimate the direct effect, one might focus on treated units whose neighbors are all assigned to control, yielding an unbiased estimate under certain conditions. In this case, rather than defining extremes as $(z_i=1,e_i \\geq h)$ and $(z_i=0, e_i \\leq 1−h)$, one might instead consider $(z_i=1,e_i \\in x_h)$ and $(z_i=0,e_i \\in x_h)$ for some interval $x_h=[x−h,x+h]$, where $x_h$ forms part of a cover of $[0,1]$. While the underlying principles remain similar, the bias–variance trade-offs do not translate in a straightforward manner between estimands, and would need to be re-analyzed in each case.\n\nWe hope our response has provided more clarity on the motivations and the focus of our work. We thank the reviewer once again for the thoughtful questions, and we hope that this discussion has enhanced your perspective on our paper."}]

Author rebuttal:

Meta-review: The paper presents a new method for estimating the bias and variance of the Horvitz-Thompson estimator, under interference, with the assumption that at least h-fraction of a unit's neighbors share its treatment. The threshold h itself is selected by their method from several candidates. The method works by "leveraging a first-order approximation, specifically, linear regression of potential outcomes on exposures".

For the most part, the reviewers praised the paper, considering the method well motivated and novel. The writing is clear and insightful, with a few questions on the figures and result interpretation being clarified during the discussion period. 


The main criticism from one of the reviewers is that the work is limited to the HT estimator. However, given how widely used the estimator is, and that many other papers in the area focus on modifications and analysis for a single estimator, I evaluate this paper to be well scoped.

All in all, the paper provides an elegant solution to a known problem, and is a valuable contribution to the field.
Decision: Accept (poster)