Title: Data-Adaptive Exposure Thresholds under Network Interference
Venue: NeurIPS 2025
Accepted: yes
Keywords: Network Interference

Experimentation

Causal Inference

Average Treatment Effect
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.


Peer reviews:
comments: 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.

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.  

Weakness：Limited to linear potential-outcome models only.

QUESTIONS:
(1) Can it be generalized to nonlinear potential outcome models such as generalized linear models?
(2) The paper estimates the ATE of the form:
\begin{equation}
\tau = \frac{1}{n} \sum_{i = 1}^{n} y_i(1, 1) - \frac{1}{n} \sum_{i = 1}^{n} y_i(0, 0).
\end{equation}
Is it possible to estimate the effect of the treatment itself, excluding the influence of the exposure?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-24

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nOverall, 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.\n\nQUESTIONS:\nI 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.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIt 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.\n\nSTRENGTHS AND WEAKNESSES:\n1. **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.\n\nQUESTIONS:\n1.\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?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\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.\n\nQUESTIONS:\n- 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?\n\nLIMITATIONS:\nOnly considers Horvitz–Thompson (HT) estimator for causal inference under network interference.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}