Active Treatment Effect Estimation via Limited Samples
Authors: Zhiheng Zhang
Haoxiang Wang
Haoxuan Li
Zhouchen Lin
Venue: ICML 2025 poster (2025)
Keywords: causality
Primary area: general_machine_learning->causality
TL;DR: Active Treatment Effect Estimation via Limited Samples

Abstract: Designing experiments for causal effect estimation remains an enduring topic in both machine learning and statistics. While much of the existing statistical literature focuses on using central limit theorems to analyze asymptotic properties of estimators, a parallel line of research has emerged around theoretical tools that provide finite-sample error bounds, offering performance on par with—or superior to—the asymptotic approaches. These finite-sample results are especially relevant in active sampling settings where the sample size is limited (for instance, under privacy or cost constraints). In this paper, we develop a finite-sample estimator with sample complexity analysis and extend its applicability to social networks. Through simulations and real-world experiments, we show that our method achieves higher estimation accuracy with fewer samples than traditional estimators endowed with asymptotic normality and other estimators backed by finite-sample guarantees.

Active Treatment Effect Estimation via Limited Samples
Zhiheng Zhang' Haoxiang Wang?
Haoxuan Li Zhouchen Lin 4in
Abstract
diverse disciplines such as medicine, economics, and the social sciences. Despite these advances, the fundamental challenge of how to design randomized experiments that estimate treatment effects both accurately and efficiently under constrained sample budgets remains underexplored.
Designing experiments for causal effect estima- tion remains an enduring topic in both machine learning and statistics. While much of the ex- isting statistical literature focuses on using cen- tral limit theorems to analyze asymptotic prop- erties of estimators, parallel line of research has emerged around theoretical tools that provide finite-sample error bounds, offering performance on par with-or superior to the asymptotic ap- proaches. These sampe results are espe- cially relevant in active sampling settings where the sample size is limited (for instance, under pri- vacy cost constraints). In this paper, we develop a finite-sample estimator with sample complex- ity analysis and extend its applicability to social networks. Through simulations and real-world experiments, we show that our method achieves higher estimation accuracy with fewer samples than traditional estimators endowed with asymp- totic normality and other estimators backed by finite-sample guarantees.
This challenge is particularly pressing for two reasons. First, from a practical standpoint, data collection is often expen- sive, time-consuming, or ethically constrained. For ex- ample, in clinical trials, it is neither economically viable nor morally justifiable to administer a new drug uniformly across all patients to measure its effect, even when detailed patient profiles are available. Second, from a statistical per- spective, analyzing the full dataset can sometimes degrade estimation quality due to overfitting and increased sensi- tivity to noise. natural strategy to mitigate these issues involves active sampling, where carefully chosen sub- set of individuals is selected to maximize information gain while minimizing sample complexity. However, existing methods often lack finite-sample guarantees, making their performance unpredictable in real-world, sssource-limitee settings.
Finite-sample guarantees are particularly crucial in active learning settings for two main reasons: (i) they provide ex- plicit, non-asymptotic assurances on the accuracy of causal estimators for any given sample size, a property that is par- ticularly valuable when data collection occurs sequentially under strict budget constraints, as emphasized by Ghadiri et al. (2024); and (ii) they offer concrete worst-case er- ror bounds, ensuring robustness in high-stakes applications such as healthcare and policy evaluation, where even mi- nor estimation errors can lead to significant real-world con- sequences. These considerations naturally lead to cen- tral question: How can we achieve the most optimal finite- sample bounds possible for causal effect estimation within an active sampling framework?
1. Introduction
Experimental design has long been regarded as the gold standard for causal effect estimation, providing a principled framework to establish causal relationships through con- trolled interventions. Building on the foundational work of Kirk (2009), researchers have developed a rich repertoire of methodologies- including randomized controlled trials (RCTs), regression-adjusted estimators, and instrumental variable techniques that have been widely adopted across
Schooo of Statistics and Data Science, Shanghai University of Finance and Economics, Shanghai 200433, P.R. China "Shool of Mathematical Sciences, Peking University 'Center for Data Sci- ence, Peking University State Key Lab of General AI, School of Intelligence Science and Technology, Peking University 'Institute for Artificial Intelligence, Peking University "Pazhou Laboratory (Huangpu), Guangzhou, China. Correspondence to: Haoxuan Li < stu, www.p.u..cuucc.. Zhouchen Lin <zlin@ pku.edu.en>.
Existing approaches provide initial insights into this prob- lem. The seminal work of Efron (1971) analyzed the trade- off between covariate balance and robustness, laying the groundwork for later methods. Harshaw et al. (2024) ex- tended this perspective by developing a Gram-Schmidt balk-based (GSW) estimator, which was the first to es- tablish sampee error bounds for average treatment ef- feet (ATE) estimation in an experimental design context.
Proceedings of the nnnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Active Treatment Effect Estimation via Limited Samples
However, their approach focuses on the full population and does not incorporate regression adjustment. More re- cently, Addanki et al. (2022); Ghadiri et al. (2024) lever- aged leverage score sampling to derive finite-sample guar- antees under active sampling. They demonstrated that ap- plying leverage score-based sampling to treatment and con- trol groups yields an e-approximation error with sample complexities of (dglo(d)) + d/e) and (d) + m')', respectively- both independent of the full sample size n.
literature overview of the finite-sample bound via active sam- pling and its variants. Section 3 introduces our framework. Section 4 presents our main result, along with two additional discussions upon lower bound (optimality) in Section 5.1 and network interference case in Section 6, respectively. Fi- nally, our method performance is demonstrated by synthetic and real-world experiments in Section 7, and we end up with a conclusion and future work in Section 8, Anonymous code is available https://gite com/ ZHzhang01/ ICML
While these results represent significant progress, existing methods do not necessarily achieve optimal dependence on the covariate dimension d. This gap is particularly critical in high nigh-dimensional settings and network dependeett scenar- ios, where each individual's outcome may be influenced by treatments administered their neighbors. Understanding the optimality of sample complexity is fundamental, as it establishes theoretical performance limits that can inform the design of more efficient estimation procedures.
2. Literature review
In the section, we start from our initial setting, Randomized experiment in causal inference, to briefly review two main technical challenges in our paper-finite-sampll guarantee and active sampling, along with the formulation of estimator: Regression- estimator. To end up, we discuss the causality estimation under network interference to prepare for our extension in Section 6.
To address these challenges, we introduce a novel active sampling strategy for treatment effect estimation that inte- grates partitioning and subsampling under explicit sample constraints. Our method, termed RWAS, achieves sample complexity of only ( ensuring both efficiency and theo- retical robustness. Unlike previous approaches, our method comes with finite-sample guarantees, making it well-suited for scenarios where large-scale experimentation is infeasi- ble due to cost, ethical concerns, or logistical constraints. Furthermore, our framework naturally extends to network interference settings. addressing the additional complexities introduced by treatment spillover effects in networked envi- ronments. Empirical evaluations synthetic and real-world datasets demonstrate that RWAS consistently outperforms existing baseline methods, achieving higher estimation ac- curacy with fewer samples. These findings establish our method as a theoretically sound and practically effective solution for sample-efficient causal inference. Our key con- tributions follows:
Randomized experiment in causal inference. Different from observational studies (Rosenbaum & Rubin, 1983; Wang et al., 2023; Zhang et al., 2023b; Zhang, 2024; Zhang & Su, 2024; Wang et al., 2025), randomization in exper- iments and optimization has been considered an impor- tant way provide robustness and balance covariates in causal inference (Efron, 1971; Kallus, 2018; Zhang, 2022; Li et al., 2023a;b; Zhou et al., 2025), especially in finite sam- ples (Bertsimas et al., 2015; Basse et al., 2023; Bai, 2023), For the first time, Harshaw et al. (2024) explicitly reveals the balanee-robustness trade-off under the Gram-Schmidt design. Apart from that, rerandomization is also a proper way balance covariates (Morgan & Rubin, 2015; Li & Ding, 2020), but time efficiency is an important issue to be considered (Johansson & Schultzberg, 2022).
Finite-sample guarantee in experimental design. The effect of finite sample size to the experimental design has been discussed for a long time (Kacewicz & Milanese, 1992; Way et al., 2010). However, to our knowledge, Harshaw et al. (2024) is the exploring paper to thoroughly induce the finite sample bound in experimental design. Stepping forward, Addanki et al. (2022); Ghadiri et al. (2024) extend the bound, taking account of the sampling procedure.
We propose an efficient active sampling method, RWAS, with a sample complexity constrained by (d/s), ac- companied by sinite-sample error guarantees.
We demonstrate the (near) optimality of our sample size and extend the framework to handle network inter- ference scenarios.
We empirically validate our method on synthetic and real-world datasets, showing that it consistently outper- forms existing basclines in terms of both accuracy and efficiency.
Active sampling in experimental design. As a form of importance sampling, active sampling proposed in Chen & Price (2019) selects samples through weights assigned to each sample point. Later work extends active sampling and linear regression to various settings, including online experiments (Fontaine et al., 2021) or other norm spaces Musco et al. (2022). In parallel, leverage score is another effective importance sampling method (Mahoney et al.,
Our paper is organized as follows. Section 2 provides the
'mm is a constant more minor than the full sample size.
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Active Treatment Effect Estimation via Limited Samples
2011; Drineas al., 2012), employed in causal inference in Addanki et al. (2022); Ghadiri et al. (2024). See Tokdar & Kass (2010); Tabandeh et al. (2022) for reviews on impor- tance sampling. Another remarkable type of method is to treat the problem as an active learning task (Jesson et al., 2021; Toth et al., 2022; Zhang et al., 2023a).
between the potential outcome when the treatment is chosen as 0 or 1. Namely, we denote the ITE of i-th individual as 4 :- (1(1) - (O). Moreover, T = (to, f indicates the vector of individual treatment effect (ITE) which serves as the ground truth.
Definition 3.3. (Average Treatment Effect) (ATE) We de- note the average treatment effect as := 1. Ei=4
Regression adjustment in causal inference. Regression adjustment is common way to improve efficiency in causal inference (Li & Ding, 2020; Bulbulia et al., 2021; Zhang et al., 2020). However, few works derive finite sample bound for causal estimators with regression adjustment, especially in finite population (Mou et al., 2022), except for Ghadiri et al. (2024) used as our baseline in the following text.
Y((L) Y((O)).
Our question is: Given the covariate information of n sam- ples, which is denotes as X E R how can we accurately estimate the individual treatment effect ((t, ((t))==) and aver- age treatment effect (7), using as few sample points (s < n) as possible, with non-asymptotic guarantee?
Causal inference in networked data. Conducting causal inference in networked data, also known as causal inference with interference, has been long aware of both statistically and experimentally (Hudgens & Halloran, 2008; Vander- Weele et al., 2012; Ogburn & VanderWeele, 2014; Su & Zhang, 2023; Zhang & Wang, 2024; Li et al., 2024; Yang et al., 2024). Exposure mapping is a classical way to model the neighborhood impact on individual outcome (Aronow & Samii, 2017; Leung, 2022). Although there has been some works on randomized tests such as A/B testing with interference (Basse& & Airoldi, 2018; Basse et al., 2019), to our knowledge, there hasn't 't been work so far dealing with interference with limited samples, nor deriving finite sample bound for causal inference with interference.
4. Method
We achieve a new active sampling algorithm with finite- sample guarantees by adopting a superior beyond-leverage- score sampling approach. The implementation process con- sists of two critical steps: (i) Refinement of Design- Design-Based Results: We first adapt the comprehensive theoretical results from Chen & Price (2019) within the design-based setting and formulate them into novel (Iterative Re- Re-weighting in design-based Set- ting) IRD algorithm. This algorithm can effectively control fitting errors with high probability while requiring a significantly smaller sample complexity. (ii) In- tegration into Experimental Design: We then incorporate the IRD algorithm into the experimental design, leveraging the regression coefficients obtained from IRD to adjust classical finite-sample estimators, thereby enhancing the accuracy of finite-sample bounds. This structured approach allows us to develop more efficient and theoretically grounded active sampling framework with strong finite-sample guarantees.
3. Framework and notation
In this section, we define the basic notation of the article. For each individual i, we use X, E Red to denote the co- variates of i-th individual. We represent the total sample information by X, where X = =[x,, Xx,,..,,]]" is an n * d matrix. Xi, denotes the (ith) elements of matrix X, namely, the j-th coordinate of X,. By performing column-wise Gram-Schmice orthogonalization on X, we assume without loss of generality that the columns of X is a set of orthogonal basis in R". Here n represents the number of samples and d represents the sample dimension. Moreover, Y1(1), (0) denotes the potential outcome on each individual when the treatment chosen to be 0 or The observed data Y; = 1(z; = (1); + 1(z; = (0). We denote (0)
4.1. Theoretical results on active regression
We first consider the linear case and define the linear func- tion family as F. namely: H:=(h: ://::<(x;)) h(X)= x,3), where 33 is an arbitrary d x 1 vector. Our goal is to derive a reweighting strategy that distinguishes individuals who have a sigmificant impact on the fitting process. This approach serves as refinement of Chen & Price (2019), in which we transfer the strategy (Definition 5 in Chen & Price (2019)) to the design-based setting, namely, in the finite population, all randomness originates solely from the assignment process rather than from inherent data variability.
(e)))))e())) and (1) = (1) For each integer k. we use [k) to replace (1,2, More- over, we denote the treatment assignment as 2 E 1) and Z : {21, 22, Z.. for each individual.
Definition 4.1. (Good E-reweighting sampling strategy, finite-sample version) Consider a sampling procedure with m(s n.) iterations. Let e == (21, Ea). In each iteration E [m], we extract one sample whose index is r(i) = j with probability P.y. where Psg is a probability derived from Algorithm 2. We introduce the weighting coefficient
Assumption 3.1. (SUTVA assumption) Each individual's treatment could not affect other individuals' outcomes (Im- bens& & Rubin, 2015).
Definition 3.2. (Individual Treatment Effect) (ITE) The individual treatment effect (TTE) is denoted as the different
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Active Treatment Effect Estimation via Limited Samples
as w, P/PPr where 0g O. We say it is "good e-reweighting sampling strategy" if (i) Defining a matrix A with elements A(i,j) = V ( E [mm]] e [d], where X denotes the j-th coordinate of X, Let X(-) denote the eigenvalue of the matrix. Then the eigenvalues of A A can be bounded: with at least 1 - probability,
for the proof details. Lemma 4.2 indicates the approxi- mation error could be controlled via such active sampling process in Algorithm 2. Furthermore, we emphasize that IRD (Algorithm 2) is superior to previous active sampling techniques such as leverage score sampling (Ghadiri et al., 2024; Addanki et al., 2022) in terms of sample complexity. We formulate it in Lemma 4.3.
X(A E [1 - E1, + E.]. (ii)
05/ // + 52) S and
Lemma 4.3. With probability at least 8, where 8 is a small constant, Algorithm 2 is good e-reweighting sam- pling strategy and is terminated within ((d/s3) iterations.
moreover, Vi E [n], we have ae max h
X S
83.
is a novel active sampling procedure, blessed by four properties: (i) Preserving Geometric Structure. By con- straining the eigenvalues of A A near 1, the algorithm ensures that key geometric properties (e.g., the covariance structure) of the original data are retained in the reduced sample. This allows the small subset of points to be a reliable proxy for the entire dataset. (ii) Selecting Influ- ential Points. The sampling probabilities Pi typically reflect leverage scores or importance weights, emphasiz- ing the most critical data points for the regression. As result, even limited number of samples can capture the most "informative" observations in the feature space. (iii) Correcting for Sampling Bias. The reweighting term w/ Pir(e) compensates for uneven sampling probabili- ties. It assigns smaller weights to frequently sampled points and larger weights to rarely sampled points, thereby miti- gating bias and controlling variance under finite-sample regime. (iv) Enforcing Bounded Influence. The additional constraints in (ii) particularly limiting Σ aj and bound- ing max; | / f --reeentt any single point (or small group of points) from dominating the estimation. Combined with the geometric and statistical safeguards above, this bounded influence preserves estimation accu- racy despite a significantly reduced sample size. Blessed with properties (i)-(iv), we provide a variant of Theorem of Chen & Price (2019) as follows.
The proof is shown in Appendix B. Such sample complexity is also demonstrated to be nearly optimal (up to a constant coefficient) in Section 5.
After this preparation, we propose the total active sampling algorithm in experimental design, which is called ATE es- timation via Re- Re-Weighting Active Sampling (RWAS esti- mator). Our inspiration comes from Harshaw et al. (2024); Ghadiri et al. (2024); Addanki et al. (2022). Unlike the objective of Harshaw et al. (2024), we seek to apply their GSW, a well-known estimator with finite-sample guarantees mentioned in our introduction, in sinite-sample setting. We then leverage the coefficients and related covariates learned from our aforementioned IRD algorithm to perform regression adjustment, thereby improving the estimator's 's ac- curacy. However, since the IRD algorithm assigns different importance weights to each individual during its iterative process, using a subset of covariates to adjust the individ- ual's 's own potential outcome often biased Ghadiri et al. (2024). Therefore, we follow Ghadiri et al. (2024)'s idea by splitting the entire population into two parts, {S, and the total procedure is detailed in Algorithm 1. Here S is used for Bernoulli active sampling to learn the regression co- efficients (line 1-4), while S is used to design GSW (line 5). Based on this partition, we can construct an unbiased esti- mator (line 6-7). Note that our main difference from Ghadiri et al. (2024); Addanki al. (2022) lies in adopting more efficient strategy (in terms of sample complexity) in S com- pared to Leveage score sampling, enabling us to achieve the lowest sample complexity so far for regression-adjusse estimator that induces finite-sample guarantees'.
Lemma 4.2 (approximation error of active sam- pling). Given dataset
(i)een and de- fine fopt = arg min Σ (Y; - 1(x.))² More- NEW
=
over, for a good e-reweighing sampling strategy,
fuct = arg min hET
Σ wx(((()) - h(X h(x,())) Then
4.2. Finite-sample bound of ITE & ATE estimation
EiE (I,..., m). with probability 1 1 8.
In this part, we aim to provide the upper bound of ATE under Algorithm 1. For preparation, it is natural first to present a coarse upper bound of ITE estimation.
Paj(ract (((aac(())-) (Xp) popt X (x,))²
< n
1
popt
1=1
Lemma 4.4 (TTE estimation upper bound). Conditioning that Vi, Wi 2 w > 0. Under Algorithm 1, the ITE estimation
Here 1 = (1 + 82)es X (1 (1-61
E1) is a small 2onstant with related to Eι, 61,62,65 E3. We refer readers to Appendix A
'In card Σie. X., where E
Algorithm 1, we define X. ==
[m], which is the average of
HHee || || denotes the Frohenius norm of the matrix.
the corresponding row values.
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Active Treatment Effect Estimation via Limited Samples
Algorithm 1 RWAS estimator
Require: X Ruod Y Y E R" e (0.1).
1: Y = ( = 1) - [ o) where IP = 1) 0.5.
2: Let X = 1 X - where X is the column-mean matrix of X.
3: Select |S| samples via IRD and induces the weights {wi, W2,,
4: Compute the coefficient in IRD sampling via B arg min ggRa+1 IS - -S where S is diagonal matrix
with the j-th element 8j = vuz if there exists j such that r(i) = j. and 8j = if j $ S. Let Boet be the first d-dimension of B removing the intercept.
5: Let f3 = 2 ( = II = 0))/[5| be the HT estimator.
6: Uniformly sample m' samples 5m E S = [n]/S. S.1..11110 |Smil= = m'. Obtain ng by HT estimator and deploy GSW design on S. with parameter Φ
7: Compute Fact,1 = ig
[Sml Eiesm. (X; Xsmm sm "gut and let fact,2 îs.
Ensure: Pact = (1S) + (Sct,
Sample complexity
Algorithm 2 IRD, modified from Chen & Price (2019)
Addanki et al. (2022)
(dlog(d) + d/e) Onn)
Require: Data X = (XI...., Xx)
Harshaw et al. (2024)
Initialize: ? min (VEE///4I,,,, 13, 0, Bo
Ghadiri et al. (2024)
(dllgg de² m')
0.4y -2d/y, uy 2d/7.
Ours
Old/s m')
2: while u, 1 8d/7 do
3: Identify the coefficients D = tr(ul - Bi) +
Table 1. Comparison of active sampling method in ATE estimation with finite-sample guarantee. Here E denotes the relative approxi- mation error, and m' is independent constant lower than n.
tr(B, - -1)-1 -1))-41--
(7/ (72/2d(1 -2))
4: Ps x [w, B₂) (B. - -11 X2 /
5: Sample rii) = j with probability Pij. and let
X,(1) = Xx;
bound [ - tll2 could be upper bounded by
Update the matrix B Bi + (/(
Xr(s) X T88
(1+ min
|aain|Ixx {x - 1
d dul
Update the upper and lower bounds u = u, +
p 1 2,'1/(1- 7), 1, 7/(1 7). i+1:
w
8: end while
with (1 - 5) probability. Here wi is constant.
Ensure: Terminates at i = and produces {wi 0:/ (urri (i(m)) for each iteration.
Here 17 is identified in Lemma 4.2. We defer the proof in Appendix C, where we also justify that the condition in Lemma 4.4 is natural to satisfy. Lemma 4.4 establishes the relationship between the ITE estimation error, fitting precision and the dataset scale. According to it, we further construct the finite-sample ATE bound as follows:
Theorem 4.5 reveals that the ATE finite-sample estimation error inherently consists of three parts, deriving from two partitions: for S, it induces the upper bound according to the naive HT estimator; on the other hand, for 8, it causes another two kinds of upper bound, corresponding to the GSW design guarantee and Lemma 4.4, respectively.
Theorem 4.5 (ATE estimation upper bound). Under Algo- rithm 1, the variance of ATE estimator could be bounded by
Recall that a fundamental result is that when we select n' samples from the whole population (n samples), the up- per bound of ATE via the traditional Morvitz-Thomppe (HT) estimator without covariates is E(f 7)2 < n'n + n'n (Ghadiri et al., 2024; Harshaw al., 2024), where T is the expectation of T. For comparison, the first term is optimized via regression-based design technique and for- malized as errors,Gsw and errors STE using m' samples, and the second term is inherited by errors, with an diminishing scale of individual set from n to (d/Es). We defer more thorough comparison with bounds in previous literature in
[TTaact - - S errorg + errors, + erTOrS,ITE, where
errors = 14
En²
l
X - 11
1
2 |$1
|
errors,asw min 3
m'nØ
(m')²
errors,Ie
++1 1) m'n
min B
x (t
t)
d 1
m nw
with (1 - 8) probability under Algorithm 1.
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Active Treatment Effect Estimation via Limited Samples
IPW- Isoed Ealinator
Select (1) sumple to comtrrat GSW based Exiintter
required for accurate estimation is optimal and cannot be reduced further without sacrificing precision.
6. Additional discussion: SUTVA assupmtion
Active regression ((d) samples (V(1)- ((() X
In the preceding chapters, we introduced an active sampling algorithm with a sample complexity of ((d) (up to con- stant factor) and established its near-optimality by deriving a corresponding lower bound for the problem. However, these results rely fundamentally on Assumption 3.1, which asserts that the treatment assigned to one node does not influence the outcomes of other nodes. It is well recognized, however, that this assumption may be violated in many social net- work settings. For instance, as discussed in the introduction with the COVID-19 vaccine case, an individual's s health out- come can be directly affected by the treatment status (e.g.. vaccination) of those in their immediate surroundings.
Use Coeffisicne ingrorenave GSW cstimatoe seffiiency
Figure I. Illustration of our main algorithm.
Appendix I.
5. Additional discussion: lower bound
In the above section, we propose the RWAS estimator and then analyze its precision. Hence, natural question arises: what is the lower bound of the query complexity based the dimension d? This section establishes a lower-bound conditioning that constantly bounds the estimation error, under a super-pone popelppopulttion perspective.
This section extends our previous findings to the network setting by relaxing the SUTVA assumption. Specifically, we develop an active sampling algorithm that achieves a sample error bound when treatments and outcomes ex- hibit network- based dependencies. This advancement lays the groundwork for analyzing and designing experimental strategies in social network contexts, where interference among interconnected units must be considered.
Theorem 5.1 (Lower bound). V any fixed dimension d, for any E E (0. 0.1) sufficiently small, there exists a feasible set of (YL, Yo} such that for any algorithm whose output + satisfying I -| S 0.1 with probability 0.75, need at least 2.86d/e sample queries.
The concept in the above paragraph is inherited. More- over, the additional notation is the arbitrary neighborhood interference exposure mapping:
The detailed explanation is deferred into Appendix E. It demonstrates that for active sampling algorithms in exper- imental design, achieving satisfactory precision level re- quires a sample complexity at the linear complexity level, which is not optimizable. This lower bound and the upper bound differ only by a constant factor, making it nearly optimal.
Definition 6.1. (Arbitrary neighborhood interfer- ence (Kandiros et al., 2024)) We define the ar- bitrary neighborhood exposure mapping d; as (T) =- (j N(i) U [i) l 1}. representing the index set of treated neighborhoods. Here, (i) denotes the set of nodes that share an edge with node i. Additionally, we denote the adjacency matrix as H..
Remark 5.2. Despite the seemingly counterintuitive result that estimation errors can be controlled within 0. with some probability while still requiring a sample size of O(d/e), this outcome is mathematically grounded. Specifically, even under mild estimation еитоr conditions, there exist instances where the estimation process becomes particularly challeng- ing, requiring significantly more samples. We illustrate this with a construction where the individual treatment effect (ITE) r(X,) is defined as a function (Xx) plus Gaussian noise u.. This setup, while meeting the error bound, demands a sample size of at least Od/e). The intuition behind this lies in the existence of a large number of function mappings with nontrivial distance between them, which complicates the estimation process. As detailed in proof, this condi- tion, alongside the mutual information bound derived using Fano's 's inequality and the Shannon- -hannon-Hartley theorem, results in the necessity of a sample size proportional to the dimen- sionality d and inversely proportional to the error tolerance Thus, our analysis confirms that the sample complexity
Assumption 6.2. (Additive Potential Outcome (Kandiros et al., 2024)) For every unit i [n] and interventions T T", if de (T') = d, ()",, then y (T") (T"). Equivalently, for each unit é E [m]. there exist coefficients ag,s E R for be expressed as
S C (i)) such that the potential outcome function y) can
Y(TT) = Σ oss
II
II ty). JES
SCA(()
jES
In this sense, our causal estimands are defined as
7, = --)(() Y,(Z,,o)), -= 1 Σ n.
iE|n]
Analogous to Kandiros et al. (2024), we could artificially set the definition of exposure mapping d,( (Z) to degenerate to
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Active Treatment Effect Estimation via Limited Samples
the well-known concepts such as global ATE, direct effect and spill-over effect. Following the conflict graph construc- tion strategy in Kandiros et al. (2024), the conflict graph design is in Algorithm 4, which is deferred in Appendix H. Under such design, the current estimator is represented as
periments: (i) HT estimator, which employs the traditional
Horvitz-Thompson estimator (Horvitz & Thompson, 1952) taking advantage of the inverse propensity score. (ii) Hajek estimator, another commonly-used estimator in ATE estima- tion attributed to Hajek (1971). (iii) Classic Regression Adjustment (CRA) estimator, which adopts the method in Li& Ding (2020). In (i)-(ii,, the estimators are operated under finite samples m among the total population via the random Bernoulli trial with probability 0.5. (iv) Gram- Schimit Walk design (GSW) The HT estimator following the GSW design in Harshaw et al. (2024)*. (v) Regression Adjusted Thropso- (RAHT) estimator proposed in Ghadiri et al. (2024), a method combining GSW design and leverage score sampling. (vi) Sample-constrained (SC) estimator in Addanki al. (2022). Also, Vectors by Mou et al. (2022), which also adopts similar cross-adjustment strategy. To ensure faimesss we keep the same amount of selected samples in all these settings except GSW-Adj. For GSW-Adj, the sample size is selected as the maximal num- ber of samples allocated for GSW design among all other methods. To evaluate the performances of methods in finite samples, we repeat the method holding the sample fixed, which is generated in the initial of the experiment.
n E (Y, x333 12
1 [aaD]
1 [[
P (E()))
P (
(1)
Here ((LA) = {U;= en and U, * for all E Ng (i)) ac-
cording to Kandiros et al. (2024).
Then, we provide the thorough active sampling algorithm un- der such network interference network, which is called ATE estimation via Conflict-Graphe Active Sampling (CGAS) in Algorithm 3. Here X ssl (X,, HH) is the aggregated function, which could be realized by graph neural network such as Leung (2022); Ma & Tresp (2021); ; Ma et al. (2022).
We provide the result as follows:
Proposition 6.3. Under Algorithm 6, the variance of estimator could be bounded by El(fact - -)
26 C 25A(HL) (3 ||lll + (1 + min | ]
mdd 11 P. ) )3, where C is a constant.
1-51
7.1. Synthetic Experiment
We defer the proof into Appendix F. Proposition 6.3 indi- cates that our active sampling technique is intuitive and generalizable method that seamlessly integrates with net- work settings. Specifically, it allows for direct regression adjustment on the latest fimite-sample results to enhance esti- mation accuracy. In contrast to the conclusions in Section 4, we do not claim that our results are guaranteed unbiased. This, in turn, reinforces the key insight from Kandiros et al. (2024): in the absence of additional structural assumptions about the network, estimators inherently face a birs-variance trade-off-ore optimizing for minimal variance alone may intro- duce significant bias. Striking an optimal balance between these two competing factors remains an important direction for our future work.
To estimate ATE, the procedure and data- generatinn process of the experiment are illustrated as follows:
ATE Dataset Denote the sample size as n and set the amount of covariates d 50. The matrix of covariates X E Raxd is generated in three steps. First, generate matrix X E RRaxd with each entry sampled from uniform distri- bution in [0, 0.01] independently. Then, a Gram-Schmidt orthogonalization process is performed on the column space of X to generate an orthogonal matrix Q E x satisfy- ing Q Q = Ia. Finally, set X = n/10 Q to recover the column norm. The potential outcome vector for the control y0) is generated uniformly at random from [0,5]. and the individual treatment effect vector t satisfies t = Xb + with each element of b E Rd be uniform random number in [0. 1]. and r E IR" follows a mean zero Gaussian distribu- tion with a standard deviation sd = 0.2. Eventually, y(1) is generated by = y(1) - -y()) The ground truth of ATE is set as 1 Σ tg, with t (fi....,
7. Experiment
In this section, we conduct synthetic and real-world experi- ments to demonstrate the unbiasedness and efficiency of our estimator. We aim to present our RWAS estimator could out- perform the competitive baseline in terms of unbiasedness and precision, Whether these methods achieve a theoretical finite-sample error bound (e.g., Ghadiri et al. (2024); Ad- danki et al. (2022)) or only possess asymptotic normality (e.g., HT estimator).
1 vE[n]
Experiment Procedure Generate an ATE dataset with size n at the start to serve as pre-determined finite samples. Then, run each relevant method r rounds and get the esti-
Most of the baselines are inherited from Ghadiri et al. (2024). The difference is that we focus on ATE estimation based on limited, finite samples instead of the whole population. Specifically, we compare the following baselines in our ex-
*We also conduct naive regression adjustment upon it if its performance could he better, for better competiveness of such baseline.
7

---

Active Treatment Effect Estimation via Limited Samples
Algorithm 3 CGAS
Require: e E R H. Y y( E
(0.1).
1: Y ((,) = 1) - (( = =))(!") where P(Z = 1) = 0.5.
2: Select [S] samples via IRD (?,,, } and induces the weights {wi, U2 ... w
3: Compute the coefficient in active reweighing sampling via 3act = argmin JSY - SX33],, where S is a diagonal matrix with the j-th element 5j = vw; if there exists j such that r(i) j. and 5j = Of S.
4: Set an estimator řs to estimate Ligs (1) Y((0) (could be manually adjusted).
5: Uniformly sample m' samples Sm'' S [n]/S. |Smrl m' .oor the complement set, conduct CGD
in Algorithm 4 and employ the assignments in to the estimator in (1). Namely, iş
[smm Eies (Y;
X ( [Been)) 1[
((E P(E 6: Replicate Line in Algorithm 1, Ensure: Fact = (1S)fact,1 + (8]fact,2)/1
Table 2. Upper: Synthetic experiment: Error(sd) of ATE estimations. Prop. = Proportion of samples used in estimation. sd= Standard
deviation of estimation. The bolded number denotes the smallest absolute bias. The reported errors are multiplied by 10. Lower:
Real-world experiments: Error(sd) of ATE estimations, For the final line, we report the result with scale e-3
Sample size
Prop.
HT
Hajek
CRA
GSW
RAHT
SC
4-Vectors
RWAS (Ours)
0.2
1.99 (1.24)
1.89 (1.09)
2.62 (0.66)
2.39 (1.17)
1.81 (1.19)
2.54 (1.54)
1.99 (1.43)
1.84 (1.16)
1000
0.5
1.14 (0.74)
1.03 (0.50)
1.02 (0.39)
0.80 (0.64)
0.80 (0.56)
0.80 (0.68)
0.92 (0.61)
0.71 (0.47)
0.8
0.86 (0.54)
0.69 (0.24)
0.64 (0.26)
0.40 (0.44)
0.43 (0.28)
0.49 (0.52)
0.51 (0.33)
0.40 (0.26)
0.2
2.01 (1.41)
1.76 (0.99)
1.58 (0.74)
1.54 (1.24)
1.91 (1.30)
1.76 (1.43)
1.43 (0.97)
1.29 (0.88)
2000
0.5
0.95 (0.66)
0.91 (0.40)
0,70 (0.35)
0.58 (0.62)
0,70 (0:49)
0.71 (0.42)
0.75 (0.49)
0.59 (0.41)
0.8
0,66 (0.45)
0,52 (0.21)
0,43 (0.22)
0.30 (0.37)
0.32 (0.23)
0.38 (0.35)
0.39 (0.42)
0.30 (0.22)
Dataset
Prop.
HT
Hajek
CRA
GSW
RAHT
SC
4- 4-Vectors (1.94)
RWAS (Ours)
0.2
2.35 (1.92)
2,40 (2.10)
0.80 (0.72)
1.15 (1.00)
1.05 (0.62)
1.08 (0.96)
0.99 (0.80)
Boston
0.5
1.98 (1.63)
2.02 (1.81)
0.68 (0.56)
0.95 (0.84)
0.89 (0.50)
0.91 (0.80)
1.95 (1.65)
0.65 (1,55)
0.8
1.79 (1.45)
1.80 (1.65)
0.54 (0.43)
0.78 (0.69)
0.75 (0.41)
0.75 (0.65)
1.76 (1.54)
0.74 (0.43)
0.2
0.42 (0.38)
0.40 (0.35)
0.06 (0.04)
0.08 (0.06)
0.08 (0.06)
0.95 (0.23)
0.44 (0.33)
0.07 (0.06)
IHDP
0.5
0.35 (0.31)
0.32 (0.29)
0.04 (0.03)
0.07 (0.05)
0.06 (0.05)
0.75 (0.18)
0.36 (0.27)
0.06 (0.05)
0.8
0.29 (0.26)
0.26 (0.25)
0.02 (0.01)
0.05 (0.03)
0.05 (0.04)
0.62 (0.12)
0.30 (0.21)
0.05 (0.04)
0.2
2.10 (1.68)
1.95 (1.50)
1.45 (1.09)
1.45 (1.35)
1.61 (1.23)
1.92 (1.58)
2.07 (1.74)
1.65 (1.54)
Lalonde
0,5
1.82 (1.48)
1.73 (1.34)
1.15 (0.87)
1.36 (1.09)
1.30 (1.00)
1.57 (1.32)
1.80 (1.50)
1.13 (0.88)
0.8
1.60 (1.22)
1.48 (1.16)
0.94 (0.70)
1.08 (0.87)
1.05 (0.79)
1.32 (1.05)
1.57 (1.25)
1.04 (0.70)
0.2
1,90 (1.42)
2,00 (1.37)
1.55 (1.23)
1.80 (1.43)
1.70 (1.38)
1.88 (1.50)
1.85 (1.53)
1.55 (1.43)
Twins
0.5
1.65 (1.24)
1.73 (1.18)
1.35 (1.08)
1.55 (1.27)
1.48 (1.22)
1.61 (1.35)
1.60 (1.38)
1.21 (1.32)
0.8
1.42 (1.04)
1,51 (0.97)
1.21 (0.99)
1.34 (1.03)
1.24 (0.99)
1.39 (1.10)
1.40 (1.12)
1.21 (1.04)
mate fy for ATE 7 in round j. The performance is estimated
Table 3. Characteristics of different real-world datasets. To enable evaluation, we true homogeneous ATE for each datasel based on relevant results. No. Feature: Dimension of features (covari- ates). Semi-synthetic means be computed based on manually pre-defined fixed shift.
through Error = Σ f - |T
Result Analysis Table 7 illustrates the performance of methods with different sample sizes and proportions of selected samples. Compared with classical estimators under the Bernoulli trial, our estimator generally has lower estimation error, with an increasing gap as the proportion of selected samples enlarges. Our method has an inherited trade-off between covariate balance and robustness. In finite samples, robustness is reached once sufficient samples are selected in the experiment. Therefore, our method achieves more balanced covariates among treatment and control groups compared with a completely randomized assignment, which benefits the performance. Among the estimators with balancing assignments (GSW-Adj.RAN) RAHT, RWHS, our
Boston
IHDP
Twins
LaLonde
Sample Size
506
747
32,120
445
No. Feature
25
50
10
True ATE
0
-40116
6,4 x e
Semi-synthetic
estimator generally has lower error compared with RAHT, which indicates our method is pretty competitive among relevant methods.
8

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Active Treatment Effeet Estimation via Limited Samples
7.2. Real-world Experiment
Acknowledgements
We evaluate the performances of methods on the following real-world datasets: Boston Dataset Harrison Jr & Rubinfeld (1978), IHDP Dataset Multisite (1990); Dorie (2016), Twins Dataset Almond et al. (2005) and LaLonde Dataset LaLonne (1986), whose basic information are shown in Table 7.1.
This work is supported by National Key R&D Program of China (2022Z01600300) and the NSF China (No. 62276004, 623B2002).
Impact Statement
illustrated that our method outperforms the previous base- lines in most cases: (i) Exponential Decay in Estimation Error and Variance with Increasing Active Samples. As the number of actively sampled instances increases, both the estimation error and variance exhibit exponential de- cay trend. In particular, when the sample size is relatively small (e.g., only 20% proportion), the approximation error in causal queries is susceptible to the number samples. This further reinforces our motivation to seek the optimal sample complexity active sampling algorithms- -inne sample size significantly impacts model performance. (ii) Compa- rable Performance of CRA and RWAS, but with Theoret- ical Limitations. In many scenarios, CRA performs com- parably to our RWAS, suggesting it can be highly effective in practice. However, such strong empirical performance remains largely anecdotal, as CRA lacks a well-established saite- theoretical analysis. This highlights key lim- itation of CRA when contrasted with theoretically grounded active sampling techniques. (iii) GSW's Performance and Its Trade-offs. Additionally, the performance of GSW is worth noting. As acknowledged by Ghadiri et al. (2024), the GSW approach often surpasses RAHT and, at times (such as in the Lalonde and Twins dataset), even outper- forms our RWAS. However, RAHT and RWAS remain of clear research significance because GSW typically requires substantially higher computational complexity. Moreover, since GSW relies on offline balancing of covariates across the entire population, it is inherently less adaptable to online settings. In contrast, both RAHT and RWAS are more read- ily extendable to online applications, making them more versatile in dynamic experimental designs.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
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Active Treatment Effeet Estimation via Limited Samples
Supplementary Material for "Active Treatment Effect Estimation via Limited Samples"
For the sake of simplicity and without introducing any misleading implications, we denote the additional sampled quantity m' in GSW as m the main text.
A. The proof of lemma. 4.2
Proof. We follow the notation and analogy in Chen & Price (2019). Notice that
juct = arg min WEF 1=1
wi (Y, 1(X (x)))))))
(2)
We consider the coefficient of function fact, If use d vector ar.)) denote the coefficient of function on each orthonormal basis, then have
a(faat) arg min ||A (hh) mll12 [I(ATA)= A)-'AT (w m2) a(h)
(3)
a(fopt) = [I(A A)- (w fm]l22.
Here A is identified in our main text, (sopl ray): poot
(v W2 ... sopt X x) T) and o denotes the Hadamard product. Then we have
V Wm
Yn
(Yaa),
fm
pact (X:) porl (x:)) 1 [|(a(Jacty [[ < 1 [[a(Satt a(Sopt)
(4)
According to Eq. (3)-(4)
Eylla(fary) pact) - a |12 = E A) A w (Ym - -mm)l1
< <R,I((A A) 1 A w (Ym fm)ll
$ F ( A)
-11 * E, Σ
w/ X (((()s popt (X))))
- -Y(r))
(5)
j=1
m
E. (1 A) E,
w x2 ( (popl (X (Xx())-Yr())"
where E, denotes the expectation taken upon sampling. Since (1, 1...., 1)n is included in the column space of X, the last equality comes from the fact that
aiPik Pi
E,w; X ((( (TOP
(X (()) - YrO))
[R.([oo((X, (Xa) - Yv) 0.
(6)
k=1
Hence
Er Ealla((actt ae (
m
<AT
() A)
E,
max
Wi
x2
0₁ Par(i)
(Xpmp(XX - Yr(o))
(e(m]
1=1
(AT (LATA)-- A) E,
Σ X2
Σ a, popt (Xx) YYk22
max ie(ml,kE|n
W;
k=
=E, (JICA A) 11
C4
E,
max kE|m)
Σ x2,
copt (xp) - -Ya)²
1=1
Ø=1
12

---

Active Treatment Effeet Estimation via Limited Samples
Observe that [B, A) -B= - on event X(ATA) A) (1 E1, + 81), and
max Ws té|m],kE[u]
Σ x2
max
Σ
ai
x S c3.
(8)
ke|m)],E[n]]
Pir(t)
j=1
j=1
Therefore, recalling Eq. (4), we finally get B. The proof of lemma 4.3
62+ X|| n(1 - E1)
E, fact ford
(for 1x))
(9)
1m
on an event with probability at least 1 - 6, which proves Lemma 4.2.
We prove Algorithm. 2 good e-reweighting sampling strategy with high probability based on Chen Price (2019).
Proof. It is equivalent to prove that Algorithm. 2 contains two main properties with high probability in main text. For the first property, notice the construction A(i,)) uez & i E [m], E [d] of matrix A, we have
m
72
A
X (1) X Bm 2d(1 72)
(10)
Pi)())
Here X(() = (X.()),,..,, Xr11).) and Bm is defined in Algorithm 2.
According to lemma. G.3, we have
72 mm
72 2d(1 - 72) Um)
A(A A) E
(11)
2d(1
-22)
Moreover, due to Algorithm 2, we have
Um + mm
m
(12)
4;
1+7
On the observations (11), (12) above, in order to substitute Uma Im in the equations, aim to bridge the relationship m
between Σ 2 and the coefficient 24(1 as follows.
Since the while loop in Algorithm 2 stops the m-th iteration, have
1
+ 4d
8d
7 2d(1 - -22)
Um
1m
Hence
(13)
T 1 7
1 1+
7
7
+
22
2=1
In the (m 1)-th iteration, from the condition of the while loop, have
m
4d
2d(1 - 72)
(14)
72
Moreover, since the trace of a matrix equals to the sum of its eigenvalues, we have
d
AB +
2
(2d)² >
4d = dy
1=
Um
\x(Bmm) mm
(um 1m)
(15)
13

---

Active Treatment Effeet Estimation via Limited Samples
Therefore
2d(1 - )2)
7
2d 72)
1
(16)
72
4;
4;
4
72
d
Combined with Eqn. (12), (14) and (16), we have
2d(1 - 72)
m
7
m
7
Um
Um
m
(17)
72
4;
2;
d'
+
147
1-7
Combined with Eqn. (11), we have
X(A A) E
Im
Mm
1-7
147
(18)
umttlp T-y 147
um +a
Im
In
1-
Moreover, due to
m
7
Egn. (16)
m
7
1
Egn. (13)
2d(1 -22)
2
2d
Um
+;
440
+;
+ u10
1-7
72
d
+
(19)
According to lemma. G.2, with probability (1 1-0), 6), we have
Mm Ty
E
[1, 87.. and hence
X(A E
14
T-
144
(20)
1
87
2d 1yy
Note that 7 S 3- We have
I-77
Lower bound of (20)
> 1 - 47
1
(21)
Upper bound of (20) <
1-92
1-7
5
2(1 + +7)
S +
72
2
72
2(1 + 7) - 2(1 + -2
2 2n
1+y
d
x2 Therefore, let e min 24273 yields the first property of lemma 4.3.
x2, Par(s)
For the second property, recalling the main text, we aim to prove Σ a; S 1+ and vi E [m], a; max
Σ
Eg.
From Eq. (15), om < < 1. Therefore,
2
7 E 11,1
72 2d
(22)
2d(1
72)
72
T=1
i=1
Moreover, note that the column of X, ||eyl| 1, and K, max Σ
x²
Σ
x d.
1=1
kEin],jE|d
14

---

Active Treatment Effeet Estimation via Limited Samples
d
X R.j Pir(i)
0j
max
&
1=1
7
72
+
Σ x
4
2d(1
max k
XT r(s)
[1uI - B,)-1 + (B; - 1,1)-11 xx(3)
(() j=1
7.3
K;
(23)
2d(1 - -22)
Amin flult B,) + min (BI--,1)-1)
73
2d(1 - -2)
1
"; Ni-,; 14-1;
73
1
9d
d7
< 2d(1 - --²)
* d
= E32
1- 72
where the last inequality comes from Claim 18 in Chen & Price (2019).
Hence Algorithm 2 both satisfies the two properties as in Definition 4.1. Combined with lemma G.2, the proof is done.
C. The proof of Lemma 4.4
Proof. Consider each given fixed Y, we have that with high probability (1 - 8),
=
:act:= : 2(
W(t 2 m0).
(24)
Hence
[xBact - 11
x W
W (t+ ~0 - |t
2
2
$
[ (X
W ) T W- - -)ee 2 + [X X XTZZoo
Xx3 - 112 + | X) x W(Z
$(1+ + +0++mmi min
| $12
4 W X)
(25)
<(1+)) min
min; Wi
| -0- al?
d d|| min, W
<(1 + c) min
x 8 2
mad || 11e (max,,) X1,))
<(1 + c) min
E1
Here, the penultimate inequality is deferred to Appendix G.5.
D. The proof of Theorem 4.5
Proof of Theorem 4.5. The unbiasedness Notice that Y is sampled from Bernoulli sampling for each individual. We conduct the expansion via the actively sampled S. We emphasize that the whole randomness of our proposed estimator Tact is derived from three sources: (i) the random treatment assignment, (ii) the stochastic reweighing vector (w)()) Jiema] whose index (r(1))se)m) = S is random, and (iii) the uniform sampling process within the random complementary set S. Then
Blfact 7) := EgEeefac T|S)
(26)
Here S is obtained from Algorithm 2. Recalling that r3 and řg the unbiased estimation of r, hence
15

---

Active Treatment Effeet Estimation via Limited Samples
Fact
S
Σ
(X;
X
+ Σ
(X; - Xs)
xs)"Jam Jone.
(27)
Sm..
ics
leads to (fix S. B
Es El Fant
-|5]=-
--
1
EsE
|S|(
Σ
(X; X s. gacc) |s)
iE Sm
(28)
1
Es
E
|
Xx X Xsm) ) gact.
mn
S
Since S arbitrarily uniformly selected from [n]/S, it naturally leads
E
|S
Σ (x: - Ss.. ) ) =0
(29)
S
Hence Efll(fact - r|S) = and thus B(fact T) = 0.
The variance Conditioning each random Y, it leads to (also conditioning the event where Algorithm 2 could make an s-error guarantee). Notice that
S
S Ts, where Σ with (8,S), n/S.
Ts +
(30)
implies
B[(fact - -))
EE [(1)) - Tg) +S| - -Ts))² Error + Error + Error3,
(31)
n²
where
Error
1.12
E
[[(tss
fs
1
Σ
(X, - Xmm) Brat) -TB] 12 [s(]]
n
m
VES...
|.512 JE'
(32)
Error
is
Ts 15).
n12
S|
Error3 :=
(fact,1 - Ts)1S 18
( 0.
n22
Notice that ig is derived from the Gram-Schmidt design in Harshaw et al. (2024). We take advantage of the Lemma G.6 as follows:
Error,
According to Lemma G.6, given the best coefficient 3
arg min BERd
It implies that when |X usll2 م ||1|3]
Oact is fixed, we have
Error1
|S| 2
Xg33
uB
2
38
X
Xg
Bust
-(65 -t 11
(
<
+
2 +
m|S1@
m2
Φ)
m|S1
(33)
S
(1111))
| - us
2
+
5
|135
112
+
1 X - Xs) Bust - (1 (t=)s))
n
m
mn
Here the definition of tt is analogous to that of X.g. 3. We additionally set
X
m X
2
3 arg min BERd
111
|1
3
arg min [[[(X X)33 (t --))112 1|| B
n
16

---

Active Treatment Effeet Estimation via Limited Samples
then
|
1
c²
1 1xx² [x - Xss Buct - -(e -ts)
2
(33) S
|
usll2
18
+
ver xb
m²
(34)
|S|
|x - 1ell2 +
8²
1
3
||
[x
X5) Bact
(+º -ts t.g 11:
n
mn Φ
m
mn
For the third term in the upper bound in (34), according to Algorithm 2, it leads to that with (1 8) probability,
( x -xs) Buct -01
(o¹ (t=
1
(x - x) (t-t) [)
(35)
Finally, we aim establish the upper bound of the RHS of (35).
Errory We could derive the upper bound of the variance of the HT estimator according to Harshaw et al. (2024); Ghadiri et al. (2024):
|||
|1 (Y:(0) + Y(II)
|1iess Y
Error I
(36)
n²
n²
Therefore,
E[Errorz] Egg[EErorz((]] < (BISI)) +100)2 s Icllpullse/nnt
(37)
Combining with (32), (35), (37), it achieves that with (1 6) probability,
[f((eett T)]
- []]
(38)
Cd 1 11
+ min
[
|X3
-m12 +
1312
1+e
min
X 3
112
d dl
En2
9
mm
2 1
3
12
wmn
E. The Proof of lower bound
The sketch of proof. We consider the following constructions: 7(X,) = L(X,) + μ.. Here L(-) is the pre-fixed function, selected from the linear family L satisfying [| = 15. is the i.i.d. Gaussian noise satisfying considered as a uniform distribution under the d dimensional Euclidean space. We consider that X is sampled from D, and then we construct the potential outcome (X))) On this basis, we extract a specific subset L C L:
NO, 2) D is
Lemma E.1. (Sparsity of L) Chen & Price (2019) 3 a subset {L1, L.) L S L with 2 20.7d such that || P 1, 1, Lilloo < 1. Moreover, [[ Lj] p 2 0.2.
The sketch of proof of Lemma E. We provide the construction as follows: In this procedure, we start with n 0 and let L be all functions mapping [d] to {+1) While L nonempty, we pick any function h in L. remove from U all functions within distance 0.2 of h, increment n by one, and name that function f.- After functions remain,
return the set L (L1,... L.). In the initialization step, |U'| 24, then in the above process, according to Chen & Price (2019), we claim this process remove C9.01d < 20.081d each round, and hence it follows that 2 2d 22000/00. = 2.919d
We consider the mutual information between the selected linear function family and our algorithm's output, i.e., estimator, f. the one hand, we provide the lower bound such mutual information by Fano's 's inequality. Specifically, leads to
1(Ly;t)= H (L, ()) 2 log(s) 1 - log((s (s- 1)/4 > 1.43.
(39)
*We define (l2( == By ("
17

---

Active Treatment Effeet Estimation via Limited Samples
On the other hand, aim provide the upper bound according the Shannon-Hartley theorem:
1ff; SI((TI, T2, (2....)):LL)))
Σ
[(r (X,))
71....i-11)
iE|s
maxLEx
(X:)²
(40)
2 log 1+
1/8
=2 log(1 (1 +E)
88
2
Combined with (39) and (40), it implies that
2.86d
2
(41)
The result follows.
F. The Proof of Proposition 6.3 Proof. Notice that
[E(.0))
ig |5
E(r, (Y,
Pr (F(t.))
Pr (F(,,)))
iESm"
|S| (Fact,1 () + |5| (Fact,2 T), then it follows that
On this basis, it leads to fact
2|51 n² (fact.1 - +) +
2|512 n² (fact,2 -)²
E El((fatt 7)
The first part could be bounded by leads to
2d zaaa
C, where C is constant. For the second part, according Kandiros et al. (2024), it
2
25H(H)
Σ Σ (X, - 3.0gaar
n²
|S|2
10.1
According to similar analogy in Theorem 4.4, it follows that |x( = 5
md
| X ||
(1 + E) min
X
_
81
G. Auxiliary lemmas
Lemma G.1 (Random vector regression (Ghadiri et al., 2024)). Let (y))))()) E T and y E R be random vector such that for each i e [n]. ys is independently and with equal probability is either equal to y (0) b" arg ming [Xb y11.. Let μ = (1)) + y()) Then
or y (1)
Moreover, let
E
12bb
-H1l2] s sd ( y(0)
min Xb
-1ll2
Lemma G.2. (Lee & Sun (2018); Chen & Price (2019)) There exists constant C, such that with least probability (), Algorithm. 2 can be terminated by C 0 iterations and Nmm. Lemma G.3. (Lee Sun (2018); Batson et al. (2009)) The eigenvalues of B, in Algorithm. 2 is bounded, namely X(B)) u.)
Tm E [1,1+ 87).
18

---

Active Treatment Efffet Estimation via Limited Samples
2
10.9
0..95(11
4y) - 3.41y" for Different
20
0.0000
-0.0005
-0,0010
-0,0015
d=1
-0,0020
10
20
30
d 50
0.00
0.02
0.04
0.06
0.08
0.10
Figure 2. Illustration for Lemma G.4.
Lemma G.4.
7/2
27²
Vy E (0,0.1),111 (1+ 2d 72)
S 33.417²
(42)
(1-
(1
5)(1 47)
Lemma G.5.
1 11
Wiinn 2
d a²
Proof. The matrix E Rx now has elements: A(i,j) = VwiXigh where is an m matrix with bounded entries:
Xr E [a, b], for some constants a, b 0.
We aim find the minimum wt that ensures the eigenvalues of A A satisfy: A(A A) E -1, + t1), with least
1 - 8 probability.
Step 1: Matrix A and its structure The matrix A now has entries: A(i,j) = Vw,
The matrix product A' A given by: A A E a where æ; is the i-th row of X treated as a vector in Rd Thus, AT T is weighted sum of the outer products of the rows of X.
Step 2: Eigenvalue properties of AT A The eigenvalues of AT A depend on the row vectors æ; and their weights w. Specifically: A(A A) X (E w.
Using the Rayleigh quotient for eigenvalues: Xmax (AT A) = sup 1 || A
This eigenvalue depends on the magnitude and alignment of the weighted row vectors / a, If [|>;11 is bounded, this affects the scaling.
Step 3: Row norm of æ; The norm of each row æ; is bounded because Xi, E [a, 6). The maximum and minimum row norms satisfy: || E [d a², d b3), where: a2: The minimum row norm when all X = a. B:: The maximum row norm when all X
Step Bounds on AT A A The matrix AT T A can be written as: T A D w.a...
The cigenvalues of A A are bounded as: Amin (ATA) A) 2 min, (ws x (13), Amax (A AA) Σn w,||,,113. (A
Substituting the bounds on ||a;;1ß: Amin (ATA) 2 min; (w: d a²) Amax (ATA) Σ m
i= W; d b.
Step 5: Ensuring X(ATA A) E 1 81, + e1] To ensure the eigenvalues lie within []
E1), we require: - S
Amin (A A) and Amax (A A) < 11
Using the bounds: 1. For Amin (AT A): E < min, (w; d a²) d-a?) This gives: Wiin 2
1-21 d-
19

---

Active Treatment Effeet Estimation via Limited Samples
Algorithm 4 Conffict-Graph--Deeee (CGD, following Kandiros et al. (2024))
Require:
Importance ordering x, maximum eigenvalue X(H), and conflict graph H. 1: Set sampling parameter r =
2:
Sample desired exposure variables
U Un
independently and identically
U
C1
with probability r-2
eo with probability r-2A(R)
with probability r-X(H)
3: Initialize the intervention vector f- 0.
4: while i 1......n do
5: if UU; E en) and U for all E NVg (i) then
6: Update intervention vector: set Zj (x) (5) for all j (i) where U; = ck-
7: end if
8: end while
Ensure: Random intervention Z E Z {0, 1)
Method
Variance
Sample complexity
HT estimator
GSW design (Harshaw et al., 2024)
minbers m
|Xb
m
T
RAHT (Ghadiri al., 2024)
XXb
1112 mm
6² 1T
|b' 100h log d/0)
Oidlog(d)=s})
+(1
11 + (X
X-b
Leverage (Addanki 2022)
(mainly consider absolute bias, and relics Gaussian noise assumption)
O(dlogi d/2) O(d/c + m²)
errorg errorg. csw + errorg.m
Table 4. Comparison of different active sampling for estimating ATE inherited from Ghadiri al. (2024). 0 < & < 1, < << are
parameters in GSW design. Moreover, is maximum norm over rows X. S, denotes the standard variance
Lemma G.6. (Harshaw et al. (2024); Ghadiri el al. (2024)) Let Fs be an estimate obtained by the Horvitz- Thoompsn estimator on the Gram-Schmine Schmidt walk design. Let be fixed/preassigned vector and < 6 < 1. Then, the partial observation regression-adjuts Horvitz- Thommpoo estimator with m samples an unbiased estimator with variance of:
2
(t - (t-
E [(f - -}
Xbb h 11/2
+
X
(43)
mn Φ
m (1
m
where
C2 (1 の)
b" = arg min
|Xb -112 +
bb
(44)
beRd
Lemma G.7. C0.01d < 20.08dd
Proof. To establish smaller constant for the inequality (6.61a) < 2nd we use upper bound involving the binary entropy function. A well-known result states that for E N and 0 < 1, the binomial coefficient (m) is bounded above by 2nH()), where (p) logz (p) - (1 p) log, p) is the binary entropy. In our case, with = 0.01, we compute H (0.01) x 0.0807. Consequently, for sufficiently large d, (o.01d) < 2d11(.001) z 20.0807d Hence, by choosing any 0.0807, we guarantee (0.6id) < 2nd
H. Auxiliary algorithms
Seen Algorithm and Algorithm 5.
Comparison with previous bounds Seen in Table 4.
20

---

Active Treatment Effect Estimation via Limited Samples
Algorithm GSW design (following Harshaw et al. (2024))
Require: Matrix X E Baxd
1, vector p E R"
Initialize an index j 1. Select initial pivot unit uniformly random from [n]. Set +- max X llll- Let
VΦ e;
B (a+() be matrix such that B =
where e, the i, the basis vector of dimension
x Xx:
while 200 & {11,+11]" do
Create the set + { i [n] 2. < <1
if k S then
Select new pivot from S uniformly at random.
end if
Compute step direction as (f) +- arg min, { [Bul2 u, for all i S, 1}. Let = {8 E R 2 + (i) E 1]m}
Set < max A and 8 + min A. Pick a random step size 6, that is equal to 5 with probability 85 + +8 and equal 5 with probability 8+ / (5+ + +
Update fractional assignment: z(++1 ( + )6u5) Increment the index j +.
end while
Ensure: Return the assignment vector z(0) f1],
21


Reviews:
[{"review_id": "78AuYtdKiy", "reviewer": "Reviewer_kg3L", "summary": "This paper developed a finite-sample estimator with sample complexity analysis for causal effect estimation. The paper demonstrated the near-optimality of the sample size, and further extended the framework to social networks. Numerical experiments with simulated and real-world data supported the effectiveness of the proposed estimator.\n\n\n### update after rebuttal:\nI have read through the authors' response and will be keeping my original scores.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **Acknowledgement and General Response to `RW kg3L`, `RW PucS` , `RW RneH`, `RW fVjN`** ###\nThrilled to receive such a positive reception from dear reviewers! Also, sincerely thank all reviewers for their insightful suggestions! In this general response, we carefully synthesized the reviewers’ suggestions to facilitate the AC’s assessment and, more importantly, use them as a blueprint to guarantee the quality of the camera-ready version. We have addressed these points:\n\n1. **Detailed and intuitive explanation of motivation, methods and algorithms** (`RW PucS`, `RW RneH`). In our rebuttal and our revised version, we provide _**(i) insightful motivation and intuition, (ii) additional comparison with previous literature and (iii) detailed interpretation of the main algorithm**_, especially for audiences who are unfamiliar with causality or active sampling. \n\n2. **Experiments** (`RW kg3L`, `RW fVjN`).  We provide additional performance comparisons upon the dimension $d$, the size $n$, the network interference, etc., to check in which circumstance the method is more advantageous. These empirical results back up our theories and intuitions.\n\n3. **Mention some misleading statements and notations** (`RW kg3L`, `RW PucS`, `RW fVjN`). Thanks for the meticulousness. Our improved version has made a few isolated definitions more transparent and intuitive.\n\nIf have any further questions, please don’t hesitate to contact us directly. We will respond as promptly as possible and do our utmost to address every concern thoroughly. Let’s enjoy this fruitful discussion together.\n\n----\n### **For `RWkg3L`: Thanks and please kindly find below our concise and clear rebuttal addressing your concerns.** ###\nTruly appreciate your strong endorsement! Here is our response:\n> **Q1: the synthetic experimental design appears to be sound. In the experiments, there are several settings where CRA / GSW actually outperformed the proposed estimator. Can you provide more intuition or justifications as to why the proposed estimators tend to perform better or worse?**\n\nThanks for your advice! More intuitions are as follows: \n1. **The advantage of our RWAS algorithm over traditional CRA/GSW methods is especially larger when the sample dimension $d$ is large**. This is because we have proven, through bounding arguments, that the sample complexity of RWAS is relatively optimal during the active sampling process that achieves a (1+\\epsilon) relative fitting error, which is O(d/\\epsilon). As a result, when the sample dimension dd is relatively large, our error is minorer than that of other methods.\n\n2. **The data structure (i.e., whether there exists a subset with efficient global representational capability) plays an important role in comparing the performance of these algorithms**. Compared to the GSW algorithm, our key differences are that \n    * We adopt the IRD active sampling strategy (yielding a subset whose $X^\\top X$ distribution closely mirrors the overall distribution), thus obtaining regression coefficients from $X$ to the individual treatment effect (ITE = Y(1)-Y(0)); and \n    * We apply regression adjustment to GSW itself (leveraging the regression coefficients from the former step).\n Intuitively, suppose the covariate distribution of this subset closely approximates the overall distribution. In that case, (i) our regression-adjusted GSW will be more efficient, and (ii) our IRD will produce a smaller fitting error for estimating the overall ATE.\n\n3. **Motivated by your inspiration, we provide the following supplementary experiments** on the performance differences as d varies, which aligns with the above intuition.\n\n| d     | HT         | Hajek      | CRA         | GSW         | RAHT        | SC          | 4-Vectors   | RWAS (Ours)   |\n|-------|------------|------------|-------------|-------------|-------------|-------------|-------------|---------------|\n| 10    | 1.98 (0.78) | 1.27 (0.53) | 0.98 (0.40) | **0.93 (0.38)** | 1.08 (0.47) | 1.13 (0.65) | 1.15 (0.53) | 1.08 (0.46)    |\n| 20    | 3.67 (0.88) | 2.88 (0.57) | **2.10 (0.49)** | 2.28 (0.70) | 2.41 (0.59) | 2.50 (0.73) | 2.46 (0.62) | 2.19 (0.55)    |\n| 50    | 1.14 (0.74) | 1.03 (0.50) | 1.02 (0.39) | 0.80 (0.64) | 0.80 (0.56) | 0.80 (0.68) | 0.92 (0.61) | **0.71 (0.47)** |\n| 100   | 1.82 (0.95) | 1.62 (0.86) | 2.22 (0.53) | 1.78 (0.87) | 1.53 (0.83) | 1.73 (0.90) | 1.80 (0.85) | **1.51 (0.82)** |\n\nThe results indicate that our RWAS methods dominate the previous methods, especially when $d$ is large, which is relevant in practice. The code is available at https://anonymous.4open.science/r/Causal-Active-Sample-B0C5/README.md.\n\n> **Q2: Minor typo.**\n\nSincerely, thanks for your careful review! We revise this topo from \"Set-ting\" to \"Setting\".", "claims_and_evidence": "Overall the proposed framework is well-described and the main claims in the paper are well-supported. The main theoretical results include the ATE estimation error upper bound and a matching lower bound under a superpopulation perspective. The proof sketch provided for the upper bound is clearly presented.", "methods_and_evaluation_criteria": "The upper bound was established via two main steps: the IRD algorithm adapted from Chen & Price (2019)'s design-based setting results, and using the obtained coefficients from IRD to adjust the finite-sample estimator. The lower bound shows that this upper bound is near-optimal. Based on the experiments with synthetic and real-world data, the proposed estimator did not substantially outperform all baselines, but achieved comparable or better results in most of the scenarios.", "theoretical_claims": "To my best knowledge the upper bound proof is correct.", "experimental_designs_or_analyses": "Based on the presented content in the main paper, the experimental design appears to be sound.", "relation_to_broader_scientific_literature": "The broader literature was well-addressed.", "essential_references_not_discussed": "I did not notice missing essential references.", "other_strengths_and_weaknesses": "The paper was well-organized, claims and proof sketches were clearly stated.", "other_comments_or_suggestions": "Minor typo: section 4 1st paragraph, Iterative Reweighting in Design-based \"Set- ting\"", "questions_for_authors": "In the experiments, there are several settings where CRA / GSW actually outperformed the proposed estimator. Can you provide more intuition or justifications when the proposed estimators tend to perform better or worse?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "h3PvrcTQIr", "reviewer": "Reviewer_PucS", "summary": "The authors considered the problem of estimating the causal effect in an active sampling framework. In particular, they proposed a method called RWAS which attains the sample query of $O(d/\\epsilon)$ to achieve $\\epsilon$-approximation error where $d$ is the number of covariates. Moreover, they also provided a lower bound, showing the optimality of the proposed method up to some constant. They performed experiments on synthetic and real datasets showing that the proposed method has a better performance compared to baseline methods.\n\n### Update after rebuttal\nI checked the proof of the lower bound given in the rebuttal and it looks correct to me. I suggested adding these explanations to the paper. I adjusted my score. However, I still believe that the paper is not well-written. Therefore, I could not recommend a clear acceptance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review and valuable questions! We address your questions point-to-point in the following.\n\n> **Q1: However, the authors should provide more explanation or context about the proposed method.**\n\n1. **Insightful motivation**. \n\n**Motivating example**. Suppose $X$ is $n \\times 2$, and we want a subset to approximate $X^\\top X$ for regression. Random sampling may cluster points along a single direction, leading to rank deficiency or poor conditioning. In contrast, Algorithm 2 adapts sampling probabilities to promote diversity: it increases the chance of selecting points in underrepresented directions by updating $B_i$. This ensures that new samples help span missing dimensions. After a few rounds, the selected subset captures both primary axes and preserves the geometry.\n\nTo avoid over-representing outliers, constraints on $\\alpha_i$ limit how much influence any point can have. As a result, the spectrum of the reweighted matrix $A^\\top A$ remains close to that of the full dataset (identity matrix after normalisation). We avoid redundancy while preserving rare but critical directions—achieving near-optimal regression performance with far fewer, well-chosen samples.\n\n\n\n2. **Detailed interpretation of main algorithm**\n\n| **Step**     | **Description** |\n|----------------------|-----------------|\n| Line 1       | Subsequently assign the Bernoulli trial to the cases selected by IRD. |\n| Lines 2–4    | The IRD Algorithm (Alg. 2) is deployed to select a subset $S$ representative to the covariate distribution of the entire samples and derive the estimation on adjustment parameter $\\hat{\\beta}^{act}$. |\n| Line 5       | Deduce the Horvitz-Thompson estimator on $S$ following the Bernoulli trial assigned in line 1. |\n| Lines 6–7    | Randomly sample a subset from the complementary set of $S$ and deploy the GSW design to derive a balanced assignment. Adjust the HT estimator to improve efficiency. |\n| Line 8       | Weight the causal effect estimators on representative set $S$ and set $\\bar{S}_{m'}$ to yield the combined estimator using well-balanced assignments and well-representative samples. |\n\nAnother line of motivation, refer to `RW RneH` for the **Q4**.\n\n\n\n\n3. **Additional comparision**\n* **The technical novelty (i)**: First, we refine the previous result in Chen & Price (2019) to adapt to a finite-sample setting. It resorts to the refinement of the parameter in the $\\epsilon$-reweighting strategy, compared to that of Chen & Price (2019), Definition 2.1. Intuitively, consider the OLS estimator regressing $Y$ on $X$ for any arbitrary weight $w_i$. We have\n$E[w_i (\\hat{Y}_i^{ols} - Y_i)] = 0$, so the expectation of the noise term vanishes. However, in finite samples, such sum may usually deviate from zero, necessitating further refinement of the weighting parameters.\n\n* **The technical novelty (ii)**: It brings significant challenges since, compared with the setting in Chen & Price (2019), each individual's outcome is not fixed as $Y_i$, instead, the outcome is randomly chosen as $Y_i(0), Y_i(1)$, jumping between the real world and the counterfactual world. \n\n> **Q2: technical novelties compares to Chen & Price (2019)?**\n\nKindly see **Q1-Additional comparison**\n\n> **Q3: Explain Algorithm 1**\n\nKindly see **Q1-Detailed interpretation of main algorithm**.\n\n>**Q4: The lines 149-157's writing, and the meaning of Condition (ii).**\n\n\nThe second condition (ii) is intended to impose an upper bound on the total weighting factors $\\alpha_i$. Specifically, this condition ensures that the cumulative weights are controlled, preventing any single sample or a small group of samples from excessively dominating the overall estimator. \n\nTo clarify the mathematical intuition, we provide the following rewritten version of the paragraph. Briefly, it includes (i) Spectral approximation condition, (ii) Bounded cumulative weight condition and (iii) Single-point influence condition. Due to space limitations, we kindly refer you to the anonymous link: https://anonymous.4open.science/r/Causal-Active-Sample-B0C5/README.md.\n\n> **Q5: The parameter analysis of the lower bound**\n\nOverall, whether setting it as $|\\hat{\\tau}-\\tau|\\leq \\epsilon$ or $0.1$ are both correct (as long as $\\epsilon < 0.1$); We sincerely appreciate your suggestion—explicitly including the $\\epsilon$ parameter in the estimation error bound improves clarity and helps readers better understand the overall process and how the Shannon-Hartley upper bound applies to the estimation error.\n\n> **Q6: Notation meaning**\n\n“∗” is a “null option”; also refer to the dear reviewers to Algorithm 4 in the appendix. the notation $\\mathcal{N}_b^{\\pi}(i)$ is deferred in the above anonymous link.\n***\nWe are eager to hear your feedback. We’d deeply appreciate it if you could let us know whether your concerns have been addressed.", "claims_and_evidence": "The paper is a theoretical work and proofs are provided for theorems or lemmas in the paper. However, the paper is not well-written and it is hard to follow some parts of it For instance, some terms or definitions are not defined. Moreover, it is often assumed that the reader is completely knowledgable about all aspects of the problem. However, the authors should give more explanation or context about the proposed method.", "methods_and_evaluation_criteria": "The main metric for comparing the methods is a sample query to achieve an $\\epsilon$-approximation error which is a reasonable evaluation criterion.", "theoretical_claims": "Unfortunately, I only had time to check the proof of Lemma 4.2.", "experimental_designs_or_analyses": "I did not check the codes, but I read the experiment section. Based on the text, it seems that the experimental results are sound and the authors also provided some explanations for the plots.", "relation_to_broader_scientific_literature": "Identifying causal effects is one of the main goals in many areas of empirical science. One of the main approaches in causal effect identification is to perform experiments. This paper considers active sampling in experiment design in order to provide order-optimal methods in estimating the causal effect.", "essential_references_not_discussed": "I am not an expert in the specific area of active sampling for causal inference. Therefore, I am not sure whether any important reference is missed.", "other_strengths_and_weaknesses": "Strengths:\n- The proposed method provides an order optimal method for active sampling to estimate the causal effect. For this purpose, they also gave a lower bound on the number of queries of any algorithm to achieve an $\\epsilon$-approximation error. \n\n- The experimental results showed that the proposed method has better performance compared to previous work in most real datasets.\n\nWeakness:\n- Some parts of the paper is not well written and some notations or terms are used without defining them which make hard to read the paper or validate the proofs.", "other_comments_or_suggestions": "I suggest revising the whole paper and making sure that all the terms are defined. Moreover, please provide a general description of the proposed method first and then describe the details of each part. In the current version, for instance, it is hard to understand how Algorithm 1 works such as what is the purpose of GSW design. As another example, the term HT is used before saying that it refers to Horvitz-Thompson estimator.", "questions_for_authors": "- What are technical novelties compared to Chen & Price (2019)? It seems that the results are adoptation of results in Chen & Price (2019) to the current setting. The explanation of the proposed method in lines 134-142 is vague. For instance, it is not clear \"This approach serves as a refinement of Chen & Price (2019), in which we transfer the strategy (Definition 5 in Chen & Price (2019)) to the design-based setting.\"\n- Please explain Algorithm 1 line by line and explain the intuition behind each part there. \n-  The lines 149-157 are not written well. For instance, \"We say it is a “good $\\epsilon$-reweighting sampling strategy” if (i) Defining a matrix\". Moreover, what is the second condition (ii) here?\n- Regarding the statement of Theorem 5.1, first please fix \"$\\forall$ any\" to \"For any\". Moreover, the statement is a little bit weird as it says that \"for any algorithm whose output $\\hat{\\tau}$ satisfying $|\\hat{\\tau}-\\tau|\\leq 0.1$ with probability 0.75\", then sample queries is at least $2.86d/\\epsilon$ but $|\\hat{\\tau}-\\tau|\\leq 0.1$ does not depend on $\\epsilon$ and it is just needed to be less than 0.1.\n- In line 287, what is the notation \"*\"? In line, what is the notation $\\tilde{\\mathcal{N}}(i)$?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "I0ZxAtnJkr", "reviewer": "Reviewer_RneH", "summary": "Experimental design for estimating treatment effects does not generally have strong finite-sample guarantees, especially as the dimensionality of the covariates grows. Recent works implement experimental design based on leverage scores. This work proposes an alternative approach called IRD, which helps achieve a sample complexity for the estimation error that is linear in the covariate dimensionality. The method is validated with a variety of standard semi-synthetic experiments.\n\n**Update after rebuttal**: after considering the additional results provided, I have decided to increase my score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your review and comments! Here are the responses to all of your questions.\n> **Q1: It would be helpful to see results as a function of covariate dimensionality.**\n\nThanks for your suggestion! We provide the following supplementary experiments on the performance differences as d varies with $n=1000$ samples. The advantage of our RWAS algorithm over traditional CRA/GSW methods is especially apparent when the sample dimension d is large, as a result of the optimal sample complexity in active sampling process.\n| d     | HT         | Hajek      | CRA         | GSW         | RAHT        | SC          | 4-Vectors   | RWAS (Ours)   |\n|-------|------------|------------|-------------|-------------|-------------|-------------|-------------|---------------|\n| 10    | 1.98 (0.78) | 1.27 (0.53) | 0.98 (0.40) | **0.93 (0.38)** | 1.08 (0.47) | 1.13 (0.65) | 1.15 (0.53) | 1.08 (0.46)    |\n| 20    | 3.67 (0.88) | 2.88 (0.57) | **2.10 (0.49)** | 2.28 (0.70) | 2.41 (0.59) | 2.50 (0.73) | 2.46 (0.62) | 2.19 (0.55)    |\n| 50    | 1.14 (0.74) | 1.03 (0.50) | 1.02 (0.39) | 0.80 (0.64) | 0.80 (0.56) | 0.80 (0.68) | 0.92 (0.61) | **0.71 (0.47)** |\n| 100   | 1.82 (0.95) | 1.62 (0.86) | 2.22 (0.53) | 1.78 (0.87) | 1.53 (0.83) | 1.73 (0.90) | 1.80 (0.85) | **1.51 (0.82)** |\n\n> **Q2: Further description on the core novelties of this method like the adaptation of IRD.**\n\nThanks for your concern. Please see the response to `RW PucS`.\n\n> **Q3: Revision on the word like outstanding**\n\nThank you for this helpful note. In response, we have carefully revised the wording in the beginning of Section 4 to remove exaggerated adjectives such as \"outstanding\" and \"superior\". We now use more neutral, objective language and anchor our claims directly to specific empirical results and theoretical guarantees. This revision aims to improve clarity and expression throughout the manuscript. We sincerely appreciate your feedback on this matter.\n\n>**Q4: How to emphasize the role of partitioning & subsampling**\n\nThanks for your concern! The overall target of partitioning and subsampling is to find efficient causal effect estimators with limited cost on sample collection and computation, but focusing on different components.\n\n* Subsampling finds the most representative samples in terms of covariate distribution by selecting and up-weighting ``influential'' directions in the data.\n\n* Partitioning seeks for well-balanced assignments and provide unbiased effect estimators with controllable expected error.\n\nIn Algorithm 1, partitioning and subsampling are deployed in two disjoint subset of the entire samples, one of which applies GSW with regression adjustment, and the other applies IRD. Through weighting and averaging, we obtain an unbiased estimate with the total sample size being O(d).\n\nBesides, the key idea is to improve the MSE through well-balanced assignments and representative samples. As stated in line 65, two types of techniques are applied to achieve these goals: partitioning and subsampling. The IRD algorithm selects the representative samples to ensure a controllable estimation error. At the same time, the GSW design is applied to achieve balanced assignments by partitioning the samples into treatment and control groups. This motivates the following thought, serving as another line of motivation: \n\n* We can use O(1) samples for the GSW regression-adjusted estimator.\n* However, learning the regression coefficient requires O(d) samples.\n* Therefore, we divide the data into two disjoint sets, one of which applies GSW with regression adjustment, and the other applies IRD, using O(d) samples to learn the regression coefficient from the first set.\n* These two sets each construct an estimator, and after weighting and averaging, we obtain an unbiased estimate, with the total sample size being O(d). Note that using either set alone as an estimator may be biased because we actively select individuals corresponding to X with good representational properties for the estimation. This group is not independent and is randomly selected from the entire sample.\n\n> **Q5: Does this method easily extend to multiple treatments?**\n\nThanks for your question. It is not complicated to propose a solution for multiple-treatment extension. For example, the estimator on the treatment effects between two treatment levels $Z=z_1$ and $Z=z_2$ can be constructed through our method restricting treatment allocation on $\\{z_1,z_2\\}$ as long as the positivity assumption $P(Z_i=z)>0$ holds for targeted levels. On the other hand, it is possible to find methods to compare different treatment pairs through one sampling process, or combine multiple processes to further increase efficiency, which requires further research. It might be highly nono-trivial due to the proof related to the martingale in the GSW design.", "claims_and_evidence": "The contribution is clear and the method is sufficiently demonstrated on a variety of standard benchmarks. It would be helpful to see results that specifically highlight the finite-sample guarantee as a function covariate dimensionality. Without this, there is little intuition on the supposed performance benefits.", "methods_and_evaluation_criteria": "The core novelties of this method like the adaptation of IRD are never fully described. Section 4.1 gives some intuition and a broad overview of the goals for the proposed method, but the steps in Algorithms 1 and 2 are not justified in detail. In my view, the authors need to put more effort into clearly explaining the motivation behind the new algorithm, beyond referencing recent works that they are building upon.", "theoretical_claims": "I did not check the lengthy appendices that contain all of the proofs. However, I took a brief look and there are no obvious errors.", "experimental_designs_or_analyses": "The paper presents numerous standard benchmarks in treatment-effect estimation like IHDP and the Boston dataset, for different subsample sizes.", "relation_to_broader_scientific_literature": "The contributions are clearly stated in relation to recent works like those of Ghadiri et al. and Harshaw et al.", "essential_references_not_discussed": "References are sufficient.", "other_strengths_and_weaknesses": "The problem of active sampling for treatment-effect estimation with high-dimensional covariates is clearly significant. The solution appears to have clear benefits over other recent works. It would be very helpful to better describe the method so that readers can understand the key contributions.", "other_comments_or_suggestions": "Some of the language is inappropriate for a venue like ICML. For instance, in the beginning of Section 4, the authors use exaggerated adjectives like \"outstanding\" and \"superior\" without reference to specific claims or objectives.", "questions_for_authors": "1. Specifically what role do partitioning and subsampling play in the proposed method?\n\n2. Does this method easily extend to multiple treatments?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gMr8w8cSoO", "reviewer": "Reviewer_fVjN", "summary": "The paper proposes a new active learning strategy for experimental design and an accompanying ATE estimator, derived from literature on estimators with finite sample guarantees. This is especially relevant to cases where experimental sampling must be constrained due to cost or other concerns. Additionally, the paper briefly discusses an extension to scenarios where SUTVA is violated.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your advice! Here is our response point by point:\n\n> **Q1: The proof of Lemma 4.4 borrows notation from Ghadiri et al. without defining it or referencing that paper, making it hard to follow without previously reading the other paper. In particular, $2 y=t+Z \\bigodot \\mu$. Also, it does not appear that $\\mu$ is formally defined in the main body of the text, but is presumably consistent with the definition from Ghadiri et al.**\n\nThanks for the careful review and wonderful comment! Here $\\mu$ indicates that the vector composes of $Y_i(0)+Y_i(1)$. and $\\bigodot$ denots the Hadamard product. We agree that, even though we had previously stated that these local notations were adopted from Ghadiri et al., it is important to make this explicit again in the main text or appendix. We have revised the manuscript accordingly with our utmost care and sincerity, which will be presented with full rigor and clarity in the final camera-ready version.\n\n> **Q2: Add network experiment**\n\nThanks for your advice. First, we have conducted large-scale experiments across a range of values for both $d$ and $n$, as shown in  `RW kg3L`. \nSecond, we have fully implemented your suggestion and carried out additional experiments under settings with network interference. Specifically, we followed both the simulation setup and the real-world dataset configuration used in the work by Lu et al. [1]. We replicate their basic settings wholly. We refer reader for the experimental results in the link https://anonymous.4open.science/r/Causal-Active-Sample-B0C5/README.md.\n\n> **Q3: Expeiremnt focuses on d.**\n\nRefer to `RW kg3L`. \n\n> **Q4: Notation issue**\nThanks for your comment! We will fix this notational issue by choosing another symbol to denote the row of the vector or matrix to avoid potential incomprehension.\n\n\n>**Q5: The proposed ATE estimator in Addanki et al. instead recursively partitions the data using the GSW algorithm from Harshaw et al., so presumably the sample complexity would be different.**\n\n\nThanks for your comment. The sample complexity for Addanki et al is $O(dlog(d)+d/\\epsilon)$ for the ITE estimation; for ATE estimation, in their excellent paper, the complexity could be controlled by \"ATE estimation. For ATE estimation we give a randomized algorithm that selects at most $s$ individuals for treatment/control assignment and obtains an error of $\\widetilde{O}\\left(\\sigma / \\sqrt{s}+\\left(\\left\\|\\boldsymbol{\\beta}^1\\right\\|+\\left\\|\\boldsymbol{\\beta}^0\\right\\|\\right) / s\\right)$,\" We will revise it in the final version. Thanks for your wonderful comment!\n\n----\n[1] Adjusting auxiliary variables under approximate neighborhood interference, X Lu, Y Wang, Z Zhang, arXiv preprint arXiv:2411.19789.", "claims_and_evidence": "The key contributions stated in the introduction are all supported by clear evidence.", "methods_and_evaluation_criteria": "The proposed methods make sense for the problem.", "theoretical_claims": "I only briefly reviewed the proofs, and do not have any immediate concerns about correctness.\n\nThe proof of Lemma 4.4 borrows notation from Ghadiri et al without defining it or referencing that paper, making it hard to follow without previously reading the other paper. In particular, $2y = t + Z \\bigodot \\mu$. Also, it does not appear that $\\mu$ is formally defined in the main body of the text, but is presumably consistent with the definition from Ghadiri et al.", "experimental_designs_or_analyses": "For the primary proposed algorithm, the experimental section was sound and the baselines were consistent with other papers in this area. \n\nHowever, there is no experimental section for the SUTVA relaxed CGAS method. This does not impact the primary argument of the paper, but perhaps hints that the Section 6 material is better saved for an expanded sequel with more analysis.\n\nOne suggestion: The difference in sample complexity between the proposed RWAS method and the previously proposed RAHT method is primarily a function of the feature space size $d$, so I suspect further experiments investigating empirical performance differences as $d$ varies may be interesting. Further reinforcing this, in Table 2, the Twins data set shows the largest different between RWAS and RAHT and also happens to have the largest feature space.", "relation_to_broader_scientific_literature": "The paper adapts the proposed method from Ghadiri et al to include a more efficient sampling strategy, derived from the more theoretical work of Chen and Price. The proposed estimator shows improved sample efficiency both theoretically and empirically. \n\nWithin the much broader literature, experiments with limited sample sizes are quite prevalent due to a number of reasons, including high costs or ethical concerns. The proposed methodology seems like a potentially valuable solution to this common scenario.", "essential_references_not_discussed": "Nothing notable is omitted from my knowledge.", "other_strengths_and_weaknesses": "The paper does a good job of presenting the problem and discussing the solution. It was relatively easy to follow their arguments and evaluation.\n\nThere are some minor notational issues, and while section 6 appears to be quite valuable, it does not appear to have the space it deserves, including empirical validation.", "other_comments_or_suggestions": "I think there may be notational issues with the index $j$ in definition 4.1. Initially we extract the sample with index $r(i) = j$, so presumably $X_{r(i)} = X_j$, which is used to create $P_{ij}$. In a following sentence where $A$ is defined, $j$ now denotes the $j$th coordinate of $X_i$, and we confusingly have $X_{r(i),j}$ despite previously defining $r(i) = j$ a few sentences prior.\n\nin the section 6 header, 'assumption' is misspelled as assupmtion", "questions_for_authors": "In Table 1, the sample complexity for Addanki et al. is stated as $O(d log(d) + d/\\epsilon)$, which is the sample complexity for leverage sampling. However, only the proposed ITE estimator in that paper relies on leverage sampling. Their proposed ATE estimator instead recursively partitions the data using the GSW algorithm from Harshaw et al., so presumably the sample complexity would be different. Am I mistaken?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Almost all reviewers are favorable to the paper. 

There is one issue with the lower bound with the way it was stated. Authors show that a specific data generating process could explain the counterintuitive-looking claim in the lower bound (I read through it and seems satisfactory). I encourage the authors to update the statement of the theorem for lower bound as they have promised.

Authors show that given n units with covariates to be assigned for treatment and control group, in the active learning setting authors propose a way to sub select only $O(d/epsilon)$ units for assignment and also propose a way to partition them also to achieve an ATE estimation error of at most $epsilon$ when the potential outcomes follows a linear model. Authors exploit a sub selection procedure of Chen and Price from 2019 that actively sub selects covariates for labeling for linear regression that is representative of the full set of covariates and further combining it with partitioning using Gram-Schmidt Walk type experimental design of Harshaw et al 2024 that preserves covariate balancing. 

Given the results of Chen and Price 2019 and its advantages over leverage score sampling (employed in previous ATE active estimation works), the results here are not totally surprising. However, the final result establishes near optimal sample complexity tightening existing results. 

Therefore I recommend accept.
Decision: Accept (poster)