Enhancing Treatment Effect Estimation via Active Learning: A Counterfactual Covering Perspective
Authors: Hechuan Wen
Tong Chen
Mingming Gong
Li Kheng Chai
Shazia Sadiq
Hongzhi Yin
Venue: ICML 2025 poster (2025)
Keywords: Causal Inference; Treatment Effect Estimation; Active Learning
Primary area: general_machine_learning->causality
TL;DR: Data valuation for selection under limited budget for treatment effect estimation

Abstract: Although numerous complex algorithms for treatment effect estimation have been developed in recent years, their effectiveness remains limited when handling insufficiently labeled training sets due to the high cost of labeling the post-treatment effect, e.g., the expensive tumor imaging or biopsy procedures needed to evaluate treatment effects. Therefore, it becomes essential to actively incorporate more high-quality labeled data, all while adhering to a constrained labeling budget. To enable data-efficient treatment effect estimation, we formalize the problem through rigorous theoretical analysis within the active learning context, where the derived key measures -- factual and counterfactual covering radii determine the risk upper bound. To reduce the bound, we propose a greedy radius reduction algorithm, which excels under an idealized, balanced data distribution. To generalize to more realistic data distributions, we further propose FCCM, which transforms the optimization objective into the Factual and Counterfactual Coverage Maximization to ensure effective radius reduction during data acquisition. Furthermore, benchmarking FCCM against other baselines demonstrates its superiority across both fully synthetic and semi-synthetic datasets. Code: https://github.com/uqhwen2/FCCM.

Enhancing Treatment Effect Estimation via Active Learning: A Counterfactual Covering Perspective
Hechuan Wen
Tong Chen Mingming Gong? 23 Li Kheng Chai
Shazia Sadiq Hongzhi Yin
Abstract
across diverse domains, including randomized control trials (RCTs) in medication (Pilat et al., 2015), A/B testing for business mecision-making (Kohavi & Longbotham, 2015) and government policy evaluation (MacKay, 2020), etc. However, real-world scenarios often involve the trade-off between cost and return, e.g., the high cost of tumor imaging or biopsy at scale frequently limits the amount of treatment effects (labels) collected from the individuals given partic- ular drug, which in turn impacts the accuracy of estimation. Such challenge highlights the need for designing data- efficient treatment effect estimation methods, which can be described as the following optimization problem:
Although numerous complex algorithms for treat- ment effect estimation have been developed in recent years, their effectiveness remains limited when handling insufficiently labeled training sets due the high cost of labeling the post-treatment effect, e.g., the expensive tumor imaging biopsy procedures needed to evaluate treatment effects. Therefore, it becomes essential to actively incor- porate more high-quality labeled data, all while adhering to a constrained labeling budget. To en- able data-efficient treatment effect estimation, we formalize the problem through rigorous theoret- ical analysis within the active learning context, where the derived key measures factual and counterfactual covering radii determine the risk upper bound. To reduce the bound, we propose a greedy radius reduction algorithm, which ex- cels under an idealized, balanced data distribution. To generalize to more realistic data distributions, further propose FCCM, which transforms the optimization objective into the Factual and Coun- terfactual Coverage Maximization to ensure ef- fective radius reduction during data acquisition. Furthermore, benchmarking FCCM against other baselines demonstrates its superiority across both fully synthetic and semi-synthetic datasets. Code: https:/ //tg//giieee //gittuttbb/// com/ wttpwn//wweeeeec /FCCM.
min efss SCD
s.t. |
S B,
(I)
where is risk metric e.g. precision in estimation of het- erogeneous effect (PEHE) (Shalit et al., 2017), D is the pool set containing all candidate samples, I's is the regression model trained on subset S, and B is the labeling budget.
Depending on whether or not the treatment indicator is readily available alongside the covariates in D, there are two branches of research related to the formulation in Eq. (1). If is not given, one build the dataset S via active experi- mental design (Addanki et al., 2022; Connolly et al., 2023; Ghadiri et al., 2024), i.e., subset of DD consisting only of covariates is selected, which is then partitioned to receive treatments, and subsequently annotated with treatment ef- fects. Otherwise, given the pool set D consisting of both the covariates and the treatment indicator (Sundin al., 2019; Qin et al., 2021; Jesson et al., 2021b; Wen et al., 2025), the algorithm only deals with selection from the pool set, then let the oracle label the selected samples. Although both aim to facilitate reliable, data-efficient predictions by selectively expanding the labeled dataset for training, studies on the lat- ter are searcer compared with those on the former. Despite the exper-expooration of the latter setting, has been widely encountered in real-world applications, such as collecting customer preferences from those who have already received different services, tracking patients' side effects from those who have already been administered different drugs, etc. Therefore, in this paper, we study the latter scenario where both the covariates and the treatment indicator are known in the pool set, while in the presence of high treatment effect
1. Introduction
Understanding the causal effects of interventions essential for making informed decisions, positioning treatment effect estimation as fundamental tool with broad applications
'School of EECS, The University of Queensland, Australia "School of Mathematics and Statistics. The University of Mel- bourne, Australia 'eppartment of Machine Learning, Mohamed bin Zayed University of Artificial Intelligence, United Arab Emi- rates 'Health and Wellbeing Queensland, Australia. Correspon- dence to: Hongzhi Yin <i. @uq.edu.au>>
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
annotation cost and a limited labeling budget, only subset of them are selected for the oracle to label.
et al., 2021b), the proposed theorem outlines the model- independent reducible quantities the optimization alter- native, i.e., factual and counterfactual covering radius. Our theorem further generalizes QHTE by additionally account- ing for the distribution alignment with the counterfactual covering radius. 2). In contrast to MACAL (Wen et al., 2025), we propose two model-independent algorithms, both of which obtain optimality via single data point acquisi- tion instead of pairwise queries, to minimize the covering radius- based objective: a greedy radius reduction method that works well in idealized distributions, and a greedy fac- tual and counterfactual coverage maximization method- - FCCM that allows for greater flexibility on the data distri- bution of the pool set; 3). We demonstrate the superiority of FCCM on real-world covariates with extensive performance comparisons and qualitative visualizations.
Unlike traditional optimization problems where abundant training data is assumed for treatment effect estimation (Shalit et al., 2017; Shi et al., 2019; Jesson et al., 2020; Wang et al., 2024), Eq. (1) is in-essence an active learning (AL) problem (Settles, 2009) and considers the practical issue of the scarcity of labeled training data, in conjunction with limited labeling budget expand the dataset for training more generalizable model. Despite the simple formulation, solving the problem is NP-hard due to the combinatorial nature of selecting the optimal subset of data points to la- bel (Tsang et al., 2005; Settles, 2009). From a general AL perspective, by simplifying the pre-assigned treatment as a feature variable, AL-based regression methods (Gal al., 2017; Sener & Savarese, 2018; Ash et al., 2019) might be directly applicable to Eq. (1). However, straightforward adoption of AL for Eq. (1) is suboptimal as it omits the distribution alignment between different treatment groups during data acquisition.
2. Preliminaries
Treatment effect estimation. In this paper, we estimate the treatment effect under the potential outcome framework (Imbens Rubin, 2015). Let the covariate, treatment, and treatment outcome space be denoted as X, T, and V, respec- tively. We train the treatment effect estimator SD : X - Y based on the dataset D = {(x, t,, 3:) }:)) where x,, fi, Vi are respectively the feature vectors, treatment assigmment, and treatment outcome that correspond to the i-th individual. Without loss of generality, we consider binary treatments E {0, 1}. and use y and y to denote the potential outcomes of corresponding treatments. The ground truth individual treatment effect (ITH) for individual is defined as (Shalit et al., 2017):
Thus far, some designated AL approaches have been pro- posed to address the constrained regression problem in Eq. (1) with observational data. Qin et al. (2021) formalize theoretical framework QHTE that directly extends the treat- ment effect estimation risk upper bound (Shalit et al., 2017) by the core-set (Tsang et al., 2005) approach; however, the derived optimizable objectives do not account for the im- portance of promoting distribution alignment during label acquisition. Following that, Jesson et al. (2021b) propose 1pBBALD from an information theory perspective, which reduces the distributional discrepancy between treatment groups by scaling the acquisition criterion with the inverse of counterfactual uncertainty. However, such a method re- lies heavily on the accuracy of the quantified uncertainty and the training of complex estimators, e.g., deep kernel learning (Wilson al., 2016). To get the best of both worlds, Wen et al. (2025) devise a simple yet performant algorithm MACAL, which considers the reduction of distributional dis- crepancy while remaining model-indeepnnent during data acquisition. However, MACAL has to query the data in pairs (i.e.. one each from treated/control group), which hin- ders its generalizability when optimality can be achieved by querying from one treatment group. Additionally, it hard to obtain the overall risk upper bound convergence despite the convergence analysis on sub-objectives.
7(x) = - x
(2)
To evaluate the performance of the trained model /p, we adopt the expected precision in estimation of heterogeneous effect (PEHE) (Hill, 2011), which is the go-to choice for various treatment effect estimation tasks (Shalit et al., 2017; Louizos et al., 2017; Shi et al., 2019; Jesson et al., 2021b; b; Wang et al., 2024):
Definition 2.1. The expected PEHE of the estimator f with squared loss metric {(-) is defined as:
<PEHE (f) = / {(x: fp(x))xx
(3)
where E(x; f) = (f(x) - -xx))) T(x) is the ground truth effect defined in (2), and f(x) = f(xxt = 1) - f(x t = 0) is its estimation. The lower the PEHE value, the better the model performance.
Contribution. Considering the aforementioned theoreti- cal and practical limitations of existing methods for data- efficient treatment effect estimation, this paper presents a three-fold contribution to this area of research. 1). We establish a theoretical framework rooted in the active learn- ing paradigm, specifically tailored address Eq. (1) which is not directly optimizable. Unlike upBALD (Jesson
For the identifiability of the treatment effect 7(x), the fol- lowing assumptions from the causal inference literature are the sufficient conditions to let it hold (Shalit et al., 2017: Pearl, 2009):
2

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Assumption 2.2 (Consistency). Only one potential outcome is seen by each unit given the treatment i,, i.e., y y if t = 0 or y y if f = 1.
where {(; is in Definition 2.1 (incalculable without the counterfactual outcomes), and [ss a loss function for the labeled training set 8 with observed potential outcomes.
Assumption 2.3 (Strong Ignorability). The independence relation { y IL t x and the conditional proba- bility 0 < p(( = 1 x) < 1 hold for all x.
The expected model risk EPEHE (fs) at the population level with trained estimator fs on subset S is controlled by three terms as shown in Eq. (4). The generalization error is bounded w.r.t. the size n as in the general machine learning research (Vapnik, 1999). Commonly, the counterfactual effect for the same unit is rarely observable, e.g.. a patient undergoes only one type of surgery, or customer experi- ences only single version of the software in an A/B test. Consequently, rendering the term [(x;; fs) non-computable. Nonetheless, the key is that we can bound the gap between the critical yet incalculable term with the training loss on S in the remaining two terms. By assuming zero training loss in the same fashion as (Sener & Savarese, 2018), we can explicitly formalize Eq. (1) into:
Integrating active learning. In situations where the labels y/ in 'D are unavailable, we thus introduce AL to build a labeled training set out of D. To enhance treatment effect estimators via AL, we 1). feed the oracle-labeled dataset S= [(x;, ti y:) ii for training estimator fs: 2). evaluate the performance of the model; determine if training can be terminated based on performance or labeling budget; 4). if no to the above, select the unlabeled subset S* from pool set D = (x,, t=) for the oracle to label; 5). expand the fully labeled subset with newly labeled data points S and return to step 1). This recursive procedure terminates when the desired performance is reached or the labeling budget is exhausted, and to achieve the objective in Eq. (1), the key is to identify the best strategy to construct S.
1 [s]
min SCD
E(x fs)
(xi, yi, Mi, (s)
[8]
(5)
Related work. In addition to the analysis in Section 1, we include a more detailed review of existing work on treatment effect estimation, AL, and data-efficient treatment effect estimation in Appendix B.
s.t.
18|
S B,
where the subset generalization gap A in (fEHE(fs) is main- tained as the final objective.
To optimize the objective in Eq. (5) that is incalculable be- cause of {xx;; fs), we instead focus on identifying reducible quantities by capping the objective with probabilistic risk upper bound outlined in Theorem 3.4. The key reducible quantities, i.e., factual covering radius &(c,t), and the coun- terfactual covering radius (1.1-1) are defined below.
3. Bounds: A Counterfactual Covering Perspective
We upper-bound Eq. (1) in general form under the AL paradigm, using a formulation similar to that defined in (Sener& & Savarese, 2018) for the classification problem. Thus, the population risk- Crens (fs) ) is constrained as:
The covering radius 6(2,2)) is defined as the radius of the smallest ball centered at the labeled samples from treatment group t, such that the union of these balls covers all samples within group t', Vt' {t, -)), with fac- tual covering induced by 1' = ( and counterfactual covering induced by t' 1- t.
Definition 3.1.
EPENE (fs)
n fs)
1 n
=EPEHE
(fs)
> n
E(x
Σ
{(x;: fs)
n
1 |s|
[s|
[x Uo tgi fs)
+
1
(x Mo tyi fs)
To derive Theorem 3.4, we further assume the Lipschitz continuity and the existence of the constant k, with the discussions on their practicability given in Appendix C.5.
[S|
[8]
EPEHE (fs) - 1 Σ f(x
+
Assumption 3.2 (Lipschitz Continuity). Assume that the conditional probability density function p' p'()/x) is X Lipschitz, the squared loss function is X--Lipschitz and I. is further bounded by L..
n
Generalization Error
S
f(x S
1x 1y 4Jss)
+
Assumption 3.3 (Constant K). Let 7 = (h/h : X + R} be a family of functions and ʃ X' x T + y be the hypothesis. Assume that there exists a constant k > 0, such that hy (x, t) == (x, t) E H.
Subset Generalization Gap
(x,), Mys fi)sss))
Theorem 3.4. Let x be sampled i.i.d. n times from do- main X. Under Assumption 3.2 and Assumption 3.3, with probability at least - %, where 7 E (0, the subset
S
j=1
Empirical Training Loss
3

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Gt=1
t
Anquired t=
Acquired t
Partual Coveritn on 1
Pactunl Covecing on =
-Couteetartual Covering on t
Countertattul Covering on t=1
6
4
2
2
0
A
-2
-2
-2
-4
-
-6
-6
-5.0
-2.5
0.0
2.5
5.0
-5.0
-2.5
0.0
2.5 5.0
-5.0
-2.5
0.0
2.5
5.0
-5.0
-2.5
0.0
2.5
5.0
(a) FC on ==
(b) CFC on f =
(c) FC on =
0
(d) CFC on f =
Figure 1. Visualization of the factual covering (FC) and the counterfactual covering (CFC) on the datasct by the acquired samples from each group. Note that each covering is constrained by the full coverage on the desired dataset with the minimum radius.
generalization gap A is upper-bounder as:
away from the center of the group O, the radius ((1,)) larger. Thus, it is evident that the objective is to minimize the covering radius much as possible within the labeling budget, thereby reducing the risk upper bound. Intuitively, the radius reduction given limited centers is analogous to the k- E-Center problem (Wolf, 201 1). However, under the treat- ment effect estimation setting, the derived bound introduces the counterfactual covering radius which fundamentally dif- fers from the classical k-Center problem, where the centers only cover data points from the same class (i.e., treatment group in our case), as per Figure ((a) and 1(c).
1
E(x: {s) - |
(xx Vb. 45 fs)
n 1=1
L3 log
Σ A0(1.1)
Kt (E(c.)) + )(111-)) 2 2kH +
2n
(6)
(Ac + 3 (xLL)), and KH with (P- being the in- tegral probability metric induced by H. and (x) denotes the density distribution of treatment group
where the constants Ki
s (p=
4. Methodologies
In this section, we propose two greedy algorithms to mini- mize the four covering radii in Eq, (7). The first algorithm draws direct inspiration from the Corollary 3.5 and extends the core-set solution (Tsang et al., 2005; Sener & Savarese, 2018) into the counterfactual covering perspective, while the second algorithm provides more flexibility on the data dis- tribution and relaxes the full coverage constraint to achieve stronger radius reduction under the same labeling budget.
Proof of the theorem is provided in Appendix A.1. To enhance interpretation, we visualize the factual and counter- factual covering radius in Figure with four labeled samples from random two-dimensional toy dataset. For example, Figure ((a) shows the factual covering with radius 6(1,)) such that the union of the circles centered at the two labeled t t= samples covers all samples from group t = 1. Noting that the covering radius ((c,e") decreases monotonically as the size of labeled S grows under the AL paradigm, we present further corollary to reveal the convergence of the subset generalization gap A given the fixed-size pool set D. Corollary 3.5 (Informal). Let be fixed, given di, de- creases monotonically as S grows under the AL paradigm, then, with probability at least 1 7, we have:
4.1. Factual and Counterfactual Radii Reduction
Denote x Dt, and S, as the individual covariate vector, pool set, and labeled training set for the treatment group f, respectively. Note that d(.-) is distance metric, and S, is proxy collection which is explicitly explained in Appendix C. Motivated by the Corollary 3.5, the objective is to find the optimal subset S that minimizes the sum of the factual and the counterfactual covering radii as follows:
A =(6)..))) + (66 + (6) + (b, (7)
Qualitatively, the factual covering radius is inversely related to the sample diversity, e.g., if the two marked samples in Figure I(a) are overlapped (low diversity), the radius ((1.1) is larger. The counterfaetual covering radius is also inversely related to the distributional discrepancy, e.g., if the two marked = samples in Figure 1(b) are further
min S S=SpUS,,|$IEn 8(1,1) + 5(1,0) + 6(0.0) + 5
(8)
and for E (0, ((.))) max \. minjes, d(x!, x/). 6(1,1-)) = max minjes, d(x!-,,x)) xx)).
However, to say the least, this minimization as difficult

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Algorithm 1 Greedy Radius Reduction (Sketch)
1: Input: Di, Do; randomly initialized St, So: budget B;
Bound
pseudo operator I := arg max min .(
Normalized Values
0.8
61.0
0.9
2: Sinit <- S1 So S <- S S1 < Ø,, So 28
6..0
0.8
0.6
6
#Labels in S is not used in querying.
0.7
3: while |S| < | + B do
0.6
4: Calculate 8 t
(). (00,0) 1
5: 6 == (1,1) or = oc000 then
6: Find point a to reduce the factual radius via r.
0.0
0.4
1000
2000
3000
4000
1000
2000
3000
4000
5000
7: else
Training Size
Training Size
8:
Find proxy point a' reduce the counterfactual radius via I, and then compute corresponded
(a) Identical Distribution
(b) CMNIST
9: end if
Figure 2. Visuals of the radius reduction and the descent of the Bound under ideal and realistic data distributions by Algorithm 1.
10: S - S U {a} #a is not labeled.
11: end while
12: Output: S
identical distribution scenario, in Figure 2(a), the factual and counterfactual covering radii decline synchronously (five plots fully overlap with each other) under a quick risk con- vergence to zero, because Eq. (8) is reduced to the simple k- A-Center problem (where Assumption 2.3 surely satisfies) which guarantees 2 - OPT approximation and the greedy nature of Algorithm forces the synchronous reduction. However, real-world data distribution commonly witnesses large group-wise discrepancies, e-g., in CMNIST bench- mark (Jesson al., 2021a), significantly slower bound convergence is observed in Figure 2(b) due to the difficulty of consistently reducing the counterfactual covering radii 8(100) and d(0,1)- Note that the synchronous reduction of ((.1) and (0,0) is also forced by the greedy nature of Algo- rithm 1.
as solving the classic k-Center problem that is NP-hard (Cook et al., 1998). The minimization of the factual cov- ering radii, i... 6(1,)) and 6(0.0). is essentially the classical k-Center problem, while the minimization of the counter- factual covering radii, i.e., 8(,0) and 0(o,1), involves the unconventional covering of one group by the centers from the other group, e.g., covering all samples within group 1 = by the centers from group / = 0 or vise versa, as visually depicted in Figure 1(b) and Icd).
To solve Eq. (8), we provide greedy radius reduction method in Algorithm (sketch, with details in Appendix C.I), where the largest radius among the four radii is priori- tized to be reduced. Recall Assumption 2.3 (Strong Ignor- ability), the counterfactual covering radii returned by our proposed algorithm can be effectively reduced under this assumption for the data distribution between the treatment groups, i.e., 0 < p(1 = 1[x) < 1. Furthermore, denoting the minimal covering radius under the optimal solution as OPT& we give the theoretical guarantee for Algorithm 1 as follows:
Discussions: Given the probability threshold y and fixed- size pool set D, Theorem 3.4 provides adjustable upper- bound which depends on the declining covering radii as S grows. However, the derivation of the upper bound is constrained by the full coverage on the pool set, which chal- lenges the counterfactual covering radius reduction due to the uncontrollable real-world data distribution. In light of this, we further explore the possibility of relaxing the full coverage constraint to a more flexible condition, i.e., ap- proximating the full coverage given the relatively smaller fixed covering radius. This underlying mentality resonates with the duality discussion of the core-set and the max cov- erage problem in (Yehuda et al., 2022). In the following section, we propose the factual and counterfactual coverage maximization (FCCM) solution to solve the radius reduc- tion problem under compromised data distributions while maintaining high satisfaction with the constraint.
Theorem 4.1. Under Assumption 2.3, the sum of the cov- ering radii returned by Algorithm 1 is upper-bounded by 2 ELE(0,)(OPPT)
+ OPT b
Proof is provided in Appendix A.5. It is noted that data distribution can strongly affect the effectiveness of radius reduction by Algorithm 1. For example, if two treatment groups share identical distributions, Eq. (8) simply reduces to the conventional k-Center problem, for which a quick convergence of the covering radii is foresecable. However, partially overlapped data distributions between groups can bring exponential challenges in effectively minimizing the sum of radii. In Figure 2, we illustrate the reduction of radii and their sum (denoted as Bound) in the normalized form against the size of acquired data, e.g.. 1.0 represents the max value, 0.5 represents half of the max value. Under the
4.2. Counterfactual-nnterrre Coverage Maximization
Let the factual and the counterfactual covering balls, for treatment group t be defined follows:
Definition 4.2. Given the fixed covering radius 6(,r) > 0.
5

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Algorithm 2 FCCM
50
till 100% (%)
1: Input: Covariate matrix X E D, random Simit S f- Sinit; radius 8(L,9) Vt, E {0, 1}; weight ar budget B
Bound Reduction (%6
95
Coverage (%)
2: E(e)) = ((xxx') : x' E (x')),,, (() = (xx, (xx)x') : x' E (cr 8 Eqe,)) U (Q.1-))
P(A5**)
30
PLAy))
1, xf E (x')xx)
85
PlA3-9
W(x'x
PLA(5
a, x' E A'cp(xt)
10
Weighted G G=
(V= = _ = gr U . W= W)
75
95%
Cocerage Gap Curve
Bound Reduction Curve
3: for E Sinic do
100
2500
500D
100
2500
5000
Training Size
Training Size
4: Remove the edges to the covered vertices:
(a) Coverage CMNIST
(b) Gain vs. Loss
((x'xx) E Eq..: (x', x²) E E(,11)) (v'xx) E
E(L,), Vt E {0,1))}
Figure 3. Visualization of the high coverage by Algorithm 2 on CMNIST, and reduction gain over mean coverage loss by Algo- rithm 2 when compared to Algorithm 1.
5: end for
6: while [S] S |Stnit| + B do
7: S 5 Su [r], where e is the vertice with the highest
scaled out-degree in graph 9 (see Appendix C.2). 8: Remove the edges to the covered vertices:
To solve (11), we propose a greedy solution - factual and counterfactual coverage maximization (FCCM) in Algo- rithm 2. Specifically, FCCM constructs weighted graph g with the node V by the entire covariate matrix X e D. and each node of E Dr builds the directed edge e(v', u²) (with unit weight) pointing to the node u' within the ball (v")) thus constructing a directed graph G. Then, graph G. is further expanded by adding the weighted edges e(v', u u'=)) (with weight a)) for of E DD pointing to the node w/-( within the counterfactual ball ((=.) Once 9 and G are both obtained, they are naturally connected to build the final weighted graph G. Note that FCCM differs from ProbCover (Ychuda et al., 2022) which only works on single class/group, whereas FCCM not only handles binary treatment groups but also integrates the distinctive counterfactual covering to solve Eq. (11).
(xx'xx) E Eu..: (x(,') E &(,11)) (v', x²) E
(, Vt {0,
9: end while
10: Output: S
the covering ball A(t,e) centered at x E St is defined as: (x) = {x' E Dr ||x - x'Il < )( Vt' E {t, 1t,1-1 - t). with the factual covering ball induced by = and the counterfactual covering ball induced by = t.
Therefore, let the union of the factual covering balls be Ab UXES, (xx and the factual coverage be LA'l P(4/) = [S+] E (0, 1]: let the union of the counterfactual covering balls be ACFF = UXES, (x) and the coun- Lecel [0, 1]. Then,
Unlike Algorithm 1, which can be regarded as a top-down approach that optimizes the risk upper bound throughout the AL process with the equality constraint satisfied at each step. Algorithm 2 adopts bottom-up approach, working on satisfying the equality constraint under a fixed bound. To further give Algorithm 2 a theoretical guarantee in Theorem 4.4 (with proof provided in Appendix A.6), we assume that the full factual and counterfactual coverage are possible by the returned labeled set S as follows:
terfactual coverage be P(A)r) Ti-e
we further define the mean coverage P(A) and sum of the radii Ssm:
P(A) 4
+ ") + P(AEE))
8amm = +(.11) + 5(1,)) + 60.00) + 60011)
(9)
With the underlying full coverage constraint, Eq. (8) can be explicitly expressed as:
Assumption 4.3. Given the fixed covering radius ō(t,e) and ((.1-1)) there exists the optimal solution S S S* for treatment group t, Vt E {0, 13. such that Ar U D.
min SED
6mm
S.L. P(A) - P(A)-1= 0.
(10)
Theorem 4.4. Under Assumption 4.3, Algorithm (1 - -- approximation for the full coverage constraint on the equally weighted graph and unsculed out-degree.
a
Noting that in Section 4. we discuss the dilemma of reduc- ing -() due to the large discrepancy that exists in the realistic dataset. To further suppress the interested bound- deam. transform Eq. (10) into the mean coverage max- imization in Eq. (11) to maximally satisfy the equality constraint given smaller radius for the bound:
Exploratory analysis: Following our discussions on Algo- rithm 1, we conduct the exploratory analysis on the CM- NIST benchmark (Jesson et al., 2021a) in Figure 3(a) to demonstrate the high coverage efficiency by Algorithm (11) on the real-world covariates. Further experiment on the per-
max P(A).
SED
6

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
100
100
Table I. Summary of the Acquisition Setup and Testing
Coverage
85
80
75-
Dataset
Start
Length
Steps
Pool
Val
Test
70
65
50-
TOY
ALL*
50
7200
2880
1600
55
50
IBM
ALL*
50
50
2891
3180
6250
35
CMNIST
ALL*
50
50
16706
10500
18000
0.05
0.10
0.05
0.30
0.15
0.4
6
(a) TOY
(b) IBM
(c) CMNIST
if the distribution discrepancy between treatment groups is
large, it is then viable to identify a different value for each covering radius to maintain the high coverage under the lin- ear complexity due the independence and monotonicity.
Figure 4. Estimating a smaller range for the covering radius around the 95% coverage threshold by Algorithm 2.
formance compared to Algorithm is accessible in Figure 3(b), where both the high reduction of the sum of the radii and the low compromise of the coverage are observed, e.g., maximally 25% reduction (red line) with 1% mean coverage loss (blue line) around the training size of 2500.
5. Experiments
Datasets: Toy - a simulated 2-dimensional toy dataset based on 16,000 randomly generated samples. IBM (Shi- moni et al., 2018) -a high-dimensional tabular dataset based on the publicly available Linked Births and Infant Deaths Database. Bach simulation contains 25,000 samples with 177 real-world covariates randomly selected from cohort of 100,000 individuals; CMNIST (Jesson et al., 2021a)- This dataset contains 60,000 image samples (10 classes) of size 28x28, which are adapted from MINIST (LeCun, 1998) benchmark. Further simulation details are deferred to the Appendix C.3.
4.3. Approximating the Covering Radius
The hyperparameters, i.e., ((22)) and -() are crucial for the success of obtaining a relatively lower upper bound with- out heavily intruding into the full coverage constraint. For simplicity, we set the same size for all four radii and denote them as ó uniformly, in Figure 4, we experiment different values of (normalized w.r.t. the max distance between points), and calculate the converged coverage on the grow- ing training set given the pre-set covering radius. Thus, we estimate narrower range for the relatively small radius with an achievable high mean coverage around the 95% thresh- old, to reduce the search space for further hyperparameter tuning.
Metric: The PEHE defined in Definition 2.1 with the squared root empirical expression: VEPEHE v " ") - T/)//N, is used for measuring the risk of the estimator the individual level.
Baselines: We compare FCCM to two groups of models, namely, the general AI. model from the broader research field: BADGE (Ash et al., 2019), BAIT (Ash et al., 2021). and LCMD (Holzmüller et al., 2023). And the designated model for treatment effect estimation with AL: QHTE (Qin et al., 2021), Causal-Bald (Jesson et al., 2021b) (with vari- ants ,BALD, pBALD, and ppBALD), and MACAL (Wen et al., 2025) are the designated algorithms proposed to deal with the treatment effect estimation with AL.
Uniform 8: The rationale to use uniform is to avoid mak- ing the search space prohibitively large- - on the order of Omm)) if each radius has m candidate settings. The key insight to reduce the search complexity is that, by the def- inition of the mean coverage P(A) in Eq. (9), Definition 3.1, and Definition 4.2, each radius is independent and each sub-term of P(A) increases monotonically with its corre- sponding radius, which leads to the mean coverage P(A) increases monotonically. Thus, by the independence and monotonicity, the search for four radii stays in the same direction, making the initial search space ((m) to identify the smallest radius for satisfying the 95% mean coverage threshold.
Estimators: The same setup as described in (Jesson et al., 2021b; Wen et al., 2025) is adopted here by utilizing the following two open- estimators: DUE-DNN (Van Amersfoort et al., 2021) for tabular data, and DUE- CNN (Van Amersfoort al., 2021) for image data. Model training details can be found in Appendix C.6.
Different 5: Though the shared value among four radii allows for effective and efficient initial hyperparameter tun- ing, it could potentially fail if the distribution discrepancy between the two treatment groups very large. This is because the counterfactual covering radius can be far larger than the factual covering radius, i.e., 6(1,1-)) > δ(..)). to let the counterfactual coverage P(ACp) to get close to full, lowering the utility of the uniform radius value. As such,
Evaluation scheme: The details of the data acquisition setup is summarized in Table 1, where we initialize the training set S with the entire labeled samples (denoted as ALL*) from group L = and start acquisition only on the sample from t = 1. which simulates scenarios with a significant number of missing counterfactual samples. Then, fixed step length is enforced at each acquisition step with fifty data acquisition steps. Note that for Toy dataset we
7

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
FCCM
5
0.55
5
J-BALD
p-BALD
0.50
Mo-BALL
0.45
QHTE
3
0.40
MACAL
LCMD
0.35
2
BADGE
0.30
BAIT
20%
40%
60%
80%
100%
20%
40%
60%
80%
100%
20%
40%
60%
80%
100%
Consumption the Budget
Consumption the Budget
Consumption Budget
(a) TOY
(b) IBM
(c) CMNIST
Figure 5. All plots are the mean values averaged from 10 simulations assnciated with the standard deviation the eroo bar. Note that all models at 0% exhibit the same performance given the fixed estimators and are thus neglected. The performance under 2% granularity presented in Appendix C.7.
Overiaz Regist
154
12
19
20
1
9
118
20
10
I'
4
12
1
1
I'4
17
14
6
19
12
10
9 14
20
305
(a) Data Distribution
(b) FCCM
(c) ppBALD
(d) BAIT
Figure 6. Visualizations of the toy dataset distribution, and the actual acquisition of the data by FCCM, μpBALD, and BAIT. The size of the data point and color from (b) to (c) is adjusted to its associated density, with deeper color representing higher density and vice versa.
do single data acquisition to show fine- aie-granularrty results, and batch-mode acquisition for the other two datasets. Each evaluation is done by the estimator trained from the last best checkpoint without completely re-train from seratch.
by prioritizing the data acquisition toward the overlapping region with high density, while others cannot do both essen- tially.
Given the above qualitative analysis, it is explainable that the other method designed for data-efficiens treatment ef- feet estimation (DTEE), e.g. μpBALD, underperforms our method. For example, although μpBALD bias the data ac- quisition toward the overlapping region, it does not further embed the property to query from the high-density region, and thus accounts for less risk. The same mentality ap- plies the other DTEE methods, e.g., MACAL, QHTE, etc. Also, it is observed that the baselines from the general AL field, e.g., LCMD and BAIT, underperform our method by not directly considering the data acquisition toward the overlapping region, which is pivotal for treatment effect estimation. However, it interestingly observed that BAIT can outperform many DTEE basclines CMNIST, but underperforms TOY and IBM. Furthermore, the current DTEE baselines cannot consistently outperform the general AL methods across various datasets.
5.1. Risk Evaluations
In Figure 5, it is observed that our proposed method is gen- erally served the risk lower bound in all three datasets. Its outstanding performance empirically proves the supe- riority of our method which considers the joint coverage on the factual and counterfactual data throughout the pro- cess of the querying. This acquisition scheme leads to a lower estimation risk that can be further explained quali- tatively with the two underlying properties: i). querying from high-density region; ii). considering the satisfactions of the positivity assumption alongside. The first property leverages the generalizability of the trained estimator to the acquired samples' neighborhoods, where a high-density neighborhood accounts for higher loss. The second property considers the pivotal overlapping assumption for treatment effect estimation, for which robust estimation toward an individual can be derived by pairing such an individual's factual counterfactual part from the overlapping region. Thus, our proposed method outperforms the other baselines
5.2. Acquisition Visualization
The original distribution of the two-dimensional toy dataset is given in Figure 6(a) for reference, where the overlapping
8

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
region is plotted in dashed purple line by kernel density es- timation (Scotr, 2015). From Figure 6(c) to 6(d), the density of each data point is calculated in by Line 2 in Algorithm 2, the higher the out-degree of each node, the denser it is as visually observed in the overlapped region. Under the fine granularity of querying one sample at a time, it's ob- served that in Figure 6(b) FCCM consistently delivers the high priority to query from the overlapping region with high density, while HoBALD is witnessed to have more queries fall outside the overlapping region and less priority on query from the high-density region (query 2o.2 is fall at the edge of the distribution, where sparse and no overlapping is ob- served) as shown in Figure 6(c). Also, BAIT's s acquisition spreads out the entire data on group / without consid- cring the presence of the counterfactual samples and thus queries the least from the desired region as shown in Figure 6(d). Their associated performance by 20 acquired samples (40% consumption of budget) also seen with a significant gap as illustrated in Figure 5(a).
of the underlying rationale. Additionally, the sensitivity analysis of a and the covering radii ((1,1) and (1,0) is pre- sented in Appendix C.8. There, we generally observe that a non-zero weight a is influential for dataset with less over- lap, and that a larger covering radius can lead to greater risk reduction in the early stage, albeit at the cost of reduced performance in the later stage.
6. Conclusion
We formalize the data-efficient treatment effect estimation problem under a solid theoretical framework, where the convergence of the risk upper bound is governed by the reduction of the derived factual and counterfactual covering radii. To reduce the bound, we propose greedy radius re- duction algorithm, which is 2 OPT-like under an idealized data distribution assumption. To generalize to more realistic data distributions for higher radius reduction, we further pro- pose FCCM, which transforms the optimization objective into the factual and counterfactual coverage maximization with a (1- apprroxiiatoiiiionnnn to the full coverage constraint. Also, benchmarking with other baselines further proves the superiority of FCCM on solving the data-efficient treatment effect estimation problem.
5.3. Ablation and Sensitivity Study
Table 2. Ablation Study of the Counterfactual Covering Radii
Consumption of the Total Budget
Dataset
Method
Limitation: FCCM is designed to better handle the par- tially overlapped data for a quicker bound reduction while maintaining high coverage (Figure 3(b)). As such, in scenar- ios where the two treatment groups have non-overlapping regions in the raw feature space (e.g., biased treatment as- signment), the data acquisition of FCCM will be challenged as there no overlapping, counterfactual pairs be iden- tified. One possible remedy is to operate FCCM in the latent space where the inter- group distributions are aligned by methods like (Shalit et al., 2017: Zhang et al., 2020; Wang et al., 2024), but it also offsets the model-independent advantage of FCCM.
1/5
2/5
3/5
4/5
5/5
FCCM-
4.7680
2.2496
1.3372
1.0545
0,9024
TOY
FCCM
4.7664
1.8655
1.0978
0.8637
0.7565
Gain
+0%
+17%
+18%
+18%
+16%
FCCM-
0.4745
0.4088
0.3797
0.3512
0.3291
IBM
FCCM
0.4813
0.4132
0.3845
0.3507
0.3286
Gain
-1%
-1%
-1%
+0%
+0%
FCCM-
2.9250
2.6627
2,4652
2.3105
2,2073
CMNIST
FCCM
2.2207
1.8681
1.7735
1.7344
1.6790
Gain
+24%
+30%
+28%
+25%
+24%
The ablation study is conducted to study the effect of max- imizing the counterfactual coverage, e.g., P( Es- sentially, two models are compared, our proposed method FCCM, and the FCCM (by setting counterfactual covering radii .(1.0) = 0 and o(..1) = O, and this action tailors the ProbCover (Ychuda et al., 2022) to align with the context for binary-class AL). In Table 2, we capture five stages of the total query steps, and calculate the performance gain by CGPEHEjcGM---/eaaeee
Acknowledgements
The Australian Research Council partially supports this work under the streams of Future Fellowship (Grant No. FT210100624). Discovery Early Career Researcher Award (Grant No. DE230101033 and DF210100124), Discov- ery Project (Grant No. DP24010110S 108, DP240101814 and DP240102088), Industrial Transformation Training Cen- tre (Grant No. IC200100022). and Linkage Project (Grant No. LP230200892 and LP240200546). Additional support from Health and Wellbeing Queensland is also gratefully acknowledged. We also thank the reviewers and area chair for their insightful suggestions during the rebuttal and dis- cussion phase.
x 100%. It is noted that the per- formance gain on Toy and CMNIST datasets are signifi- cant, however, with neck-to-neck performance observed on the IBM dataset. This phenomenon is explainable since the treated and control distributions on IBM are heavily overlapped such that the high-density region on the treated sample (t- = 1) is in fact the high-density region on the coun- terfactual side (t = 0). We provide the visualization of the IBM and CMNIST in Appendix C.4 for further discussions
VFPEHEECCM
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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Impact Statement
Fujii, K. and Kashima, H. Budgeted stream-based active learning via adaptive submodular maximization. Ad- vances in Neural Information Processing Systems, 29, 2016.
This paper presents work whose goal is to advance the field of Causal Inference. Our work entails many potential so- cictal benefits, including enhanced accuracy treatment estimation for new drugs and improved robustness in causal inference models under data sparsity.
Gal, Y., Islam, R., and Ghahramani, Z. Deep bayesian active learning with image data. In International Conference on Machine Learning, pp. 1 1183 1192. PMLR, 2017.
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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
A. Theory
A.1. Proof of Theorem 3.4
Assumption 3.2 (Lipschitz Continuity). Assume that the conditional probability function pf (y)x) is X--Liscchizz the squared loss function A Ai-iisscitz and 1 is further bounded by Lt.
RR be a family of functions and X x T - y be the hypothesis. Assume that there exists constant K > O, such that h (x,t) == ,/,(x,t) H.
Assumption 3.3 (Constant 6). Let H {hihh X
Theorem 3.4. Let x be sampled i..d. n times from domain X. Under Assumption 3.2 and Assumption 3.3, with probability at least where r E (0, 1), the subset generalization gap is upper-bounded as:
[s|
5(x fs)
1
Σ
1l (x Da. 44 fs)
[8]
(12)
Li log +
Σ (E(1.))
Kt (ō1e,0) 0(c,--1) 2xx + V
2n
where the constants K4 = 2 + },LLE), and = K IPMu (pm= 0xx)(x)) with 1PM probability metric induced by H and p² (x) denotes the density distribution of trealment group I.
denotes the integral
Proof of Theorem 3.4. The proof is done in three main steps: Firstly, we bound the expected value of the interested term - {xx;; fis) with factual and counterfactual loss, namely, cy and CCF the domain of the pool set. Secondly, constrain the EF with the factual covering radius ō(G.0) and the 6CF with the counterfactual covering radius ((c,1-)). Lastly, we conclude the probabilistic bound by Hoeffding's inequality.
E
E(x, (s)
2Er +ccr)
(13a)
Es, + t + 1 fhelt))
Es. + 6 + 1 {r(l))
Σ 8E(0.1}
2
Σ 1E(0.1)
2 2
+ 2 kM
(13b)
Σ (0.1)
2(x + (.LL)) (B(ee) + 6()) + 2 Kz
(13c)
The inequality (13a) is by Lemma A.2, the inequality (13b) is by Lemma A.3 and A.5, the equality in (13c) is by the zero training loss assumption and the fact that < 1t E {0, 1}. With Hoeffding's inequality, we have:
n
20) =exp exp (-
2ne
TP
Ex;; fs)
E
E(x;; fs)
(13d)
L
we solve for the bounding gap e V least 1 - 7, the following inequality holds:
By setting exp
2nc
1.7 log 2m
Thus, we can derive that with probability at
Lf log 7 2m
Exx;: fs) < E
E(x;: fs)
+
(13e)
13

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
To conclude, have:
18]
1
{(x: fs)
2 (x,, 44 fs)
(13f)
[S]
1=1
==1
a
E(x; fs)
(13g)
72
I log
E
5 1s) +
(13h)
2n
L log 1
1 {ALLi)) (ō(e.)) + )(a1-1)) + 2 kx + ( +
(13i)
2n
(0,1)
The equality (13g) is by the zero training loss assumption, the inequality (13h) is by incorporating (13c), the final incquality (13i) concluded by incorporating (13c).
Discussion on the bound: Given the i.i.d. sampled pool set from the domain X, noted that the final probabilistic bound, if the sampled two distributions were identical and selected the entire pool set for training, simply by definition, we have both the factual and counterfactual covering radius be completely zero, and also the distributional discrepancy term counted by IPM be completed zero, leaving the tightness of the risk upper bound solely depends on the size of the pool set n and for the given probability threshold 7/
A.2. Proof of Lemma A.2
Definition A.1. Given the loss metric the expected factual loss ey and counterfactual loss cCF are defined a manner consistent with (Shalit et al., 2017) as follows:
I 1(x, ((xx,)((x,) l)xxdl,
xxXT
(14)
<cr = - 1(x, 1 t)dxdd..
Lemma A.2. Let x to be sampled i..dd. n times from the domain X. With the two-headed trained model fs = ( on the selected subset S, where f X + y are the estimators for the treatment effect y respectively. The expected value of 1 Eis E(x;; fs) is upper-bounded as follows:
E
E(x (s)
s 2((eF +cF))
(15)
Proof of Lemma A.2. Note that the overall estimator is denoted as fs. which is a two-headed model with f* representing the estimation for the treatment effect y' under treatment assignment t. With a slight abuse of notation, we write the expected loss over x where y is implicitly assumed be function of x, i.e., each x has an associated label y, as is typical in
14

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
supervised learning setting:
1 7
E (s)
B [ [s)
(16a)
71
Exxx [5(x: fs)l
(16b)
|x (r(x) +f(x))
(16c)
Jxx ((V'' - 0) 1 (f(=((x) f (x))) pxx)xx
(16d)
(('=1 (x)) - (y/=0) (x) / p(x)dx
(16e)
Swap -0 and f(-1(xx
2
(1) f (x)) 2 - - - ((=0 )x()))
(161)
2 x
(y |=1 - fa= (x)) p(x, = 1)dx (y'=1
=)dx
(16g)
x
Apply p(x) J p(a, t)de
x(x f(xx = 0)
/
2
(y yy-0 (x) = 0) +
(g - 10==0- (x) = 1)dx
(16h)
x
x
(1/=1= - (x)) 1)dx + (x)
0)dx +
(16i)
x
Re-arrange and this term equals p by definition
(y) (y'11 (x) 0)dx+
(x)))
(y (y' (x)) p(x, 1)dx
(16j)
x
Re-arrange and term equals 2 by definition
2((r + (CF).
(16k)
Lemma A.2 indicates an interesting decomposition of the expected loss into factual and counterfactual errors, which is a prelude to our final probabilistic bound under AL paradigm. is worth mentioning that a similar intermediate conclusion appears to align with observations in (Shalit et al., 2017).
Discussion:
A.3. Proof of Lemma A.3
Lemma A.3. Denote the expected loss on subset S₁ as es., the factual covering radius as 8(,,y) let the constant p(l) be the marginal probability, Assume that the conditional probability density function p (ylx) is X Lipschit, the squared loss function l Xt- Xi-Lipschitz and I is further bounded by La, the expected factual loss is bounded as follows:
1 Σ M: (Es. ( + 1nggghh (,,1t)) (0
CF S
(17)
Proof of Lemma A.3. We start with the Tower Law for the key of the proof, let the estimation be ÿ = f(x) given the fixed treatment we have the expected loss be decomposed in the general form:
Ex (y -xx())) = Ex Ey (u f(x))² x]]
(18a)
Exx 1/s (w f(x))) ply xidr]
(18b)
15

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Denote p' (v|xx p(y]x,() and (x) p(x((), we apply the Tower Law conclusion in Eq. (18b) by Tow Law and further bound the expected factual error cyp for the treatment group t:
- Ex
(y (xx))
(19a)
Bys (y - (x)) xx
= -xxtt
(19b)
- -Ex*
E5p0 (uix)dy)
(19c)
J (1 -p' (vlxx) + p' (1yx2'))d
= =xxx
(19d)
Add up Zoro
S 12p0 (ul (ulx')dy + 1/ (1'oo'))))))))))))( 1² (vl ))dy
= xx
(19e)
We decompose the first term within the expectation in (19e) into the followings, by the selected x' E Se that covers the x from group within the factual radius ((,,)), we bound the term as:
J Eop' (u1x ( Jdy / (1 - ty + (ullx')))))
(20a)
| ( - -by (' I by p' ('llx')
(20b)
S bc,,)de + / (ulx))d
(20c)
= - + (x)
(20d)
where the inequality in (20c) is because:
J (1y - 12 (yll)) S x1x-x' /
19 1/
D' d'y
(21a)
x x'
< x x'1 / Anp'vuli>
(21b)
s o(0,0) I, (ubx
(21c)
=du,od / (ulx')dy
(21d)
= =
(21c)
for which the equality in (21e) is because the integral of the density across the domain is
/ ('xxx))dy
(22)
The second term within the expectation in (19e) is bounded by:
J 12 (, - p'(x)x) & 1x -xx / 1t
('((())) - (x(()))
Jdy
(23a)
x- x'
s dc ( iyyy
(23b)
1 -flnesrell
(23c)
16

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Combining all the inequalities together, we have:
6 = 1x 1 Gyp' (ob')) I (,00'(0/))) p'((' (yl
(24a)
1
S Ex (x)(x²) + (1) +
(24b)
1 3 {x111))) es + (1. +
(24c)
where the last equality holds due Eles, (x') = es, and the invariance of constants under expectation, i,e., E(c) for any constant
Given that Mt=1 = p(t = 1) and Mi=0 = p(t = 0), where M4=1 + #4=0 = 1, we conclude the proof by expanding the expected factual loss F by definition:
1(x, t)p(xx) (opp(x,tddee t)xxdt
(25a)
XXT
/
(== (x)p(xxt 1)dx /
(== (x)(xx)xttt))) 0)dx
(25b)
x
/ 1= (x)p (x)p(t = 1)dx +
p==0 (x)p (x))== = 0)dx
(25c)
1
p(t = 1) Ex (w (x))
+ +plt =0) Ex (12 - (x)))
(25d)
M4=1 Exi-1 Eyc-1 (w (x))
xx]
+ 14=0 Exc-o Ey [(V (x=0 |x]
(25e)
Σ H4 cs, + ( tEf0,1)
+ 3 3(r
(25f)
Discussion: Lemma A.3 establishes a general upper bound on the factual loss under the core-set paradigm, which has been explored in prior studies (Sener & Savarese, 2018; Qin et al., 2021). Building on this foundation, we provide more rigorous and comprehensive proof strengthen the theoretical underpinnings. Also, visualize the factual covering radius in Figure I(a) and (c) to enhance interpretation, where the full coverage the same class is required.
A.4. Proof of Lemma A.5
Definition A.4. Let H (h|h: metric (IPM) between two data distributions pt=1(x) and p==0
h X - [R} be a family of functions. The distribution distance measure contre--nnteee integral probability
(x) over the domain X is defined as:
sup 1 ((x)(x'= (x) (x)xx NEW
IPM (x)"" (x))
(26)
Lemma A.5. Denote the expected loss on subset S, as cs., the constant 14 = p(t). counterfactual covering radius as -ty. Assume that the conditional probability densiry function p' (ylxx X Xe-Lipschitz, the squared loss function I is X--Lipschitz and is further bounded by La. Also, let H [hih X ] be family of functions and f X x T T be the hypothesis. Assume that there exists constant > O, such that (x(xxt) = (x,(xx)) H. The counterfactual expected loss is bounded as follows:
Σ p1-t cs. + & - {((L)t) + KH. LE(0,1)
ECF S
(27)
where constant KH = K. IPM (x), (x)) describes the distributional discrepancy between the treatment groups' distritbuions (p' (x) and p² 0==0xx)) (x) over the domain X, e.g., its i.i.d realization set S induced by AL.
Proof of Lemma A.5. For notation simplicity, we denote 1, = (y (x))) and denote -x-) = p(x | - f), for
17

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
px-((x) we have the counterfactual loss on the group L as:
cC -Ex--
(v - ((x)))
(28a)
Eyr [1V - f((x)) x
(28b)
= -x
y byp' (D((u)))
(28c)
-Ex1
y by (vlx)) - (xlxx) + p((lxx)) dy
(28d)
SS (ulx)) Jdy + IS (o'(vlx) -pp'-pp'ooa (ulxx) dy
-Ex--
(28e)
Note, that in (28d) we introduce the selected x' E St from the distribution (x) (x' is not directly sampled from p'(x) but from S,), which be consider counterfactual sample w.r.t. (x). That is, the sample from the group ( is covered by the counterfactual sample x' E St from group t within the counterfactual radius ((t,1-1)-
Similar to the deduction in Appendix A.3, the first term within the expectation in (28e) is bounded by:
J 1o0' (ulx = / ( - 12 + D'''''s
D.'''ylle (u)xx)ds
(29a)
S (1, - (y)) + I fyr D' (ulx')
(29b)
S 01 / by ('llxx
(29c)
01 + -Es, (x).
(29d)
The incquality in (29c) is because:
110%/ (1, - (1) s x xx1 S 1 1y
pt ((1xx))
(29c)
y
x'
$ xx x' / (1
(29f)
61,1-1)) f (1p'()l
(29g)
6(,1-1)21 / p¹ (y)x')dy
(29h)
-1)
(29i)
The equality in (29i) is because the integral of the density across the domain is
[ ff (y)x((dy
(29j)
The second term within the expectation in (28e) is bounded by:
J 1,, (p(x)) - p'(ulx()) dy = 1x - x'1 / zt lp' (u(x)) - 1p'(ppp (2(x1'')) dy
(30a)
1x - x'1
S 61 ( fydy
(30b)
1 1) 1.p
(30c)
18

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Combining all the inequalities together, we have:
Cor =Exi-e U/8 (yl'() + / 12 (p'(ul))-pt ('((xx))
dy 3 0
(31a)
EEx- (x)) + 6 -1 d +
(31b)
-e (x') + ((.1-1) Ao + *1++zhol?? zhali)
(31c)
where the first term is the counterfactual loss on the subset St induced by AL.
Extending the the expected counterfactual error ECF by definition:
ECF-
()xx)) t)dxdt
(32a)
XXT
x (w' - - (xx)) = 0)dx (y - (x=) = 1)xx
(32b)
/ - (x) - (x)p(1 = 0)dx + / (3) - f (x)) (x))))=1((x)11 (x)p(( = 1)dx
(32c)
=p(t= = =0) Exx= (1y - fi (x))² + p(t = 1) Ex -
(32d)
Sp4=1 Ex /eS (x')] + 610.1) (x Ao +
1 ) ] + pi=0 Bx (x') + 6(2.0) (Nr (Ai+ + gmoLll
(32c)
(>++ ( 3 ( + pt=0 51.00 (di+ 1 3 ( =(rcF)s + M )0.11)
(32f)
Noticing that the counterfactual loss over the selected set S induced by AL: (ECF)s in (32f) can be further bounded by the IPM, e.g., Wasserstein distance, by adapting the proof of "Lemma 1" of the main text in (Shalit et al., 2017) to the subset
induced by AL. Let p(t = 1) yu be the marginal probability of treatment in S, thus p(t = 0) = 1 by noting that + p(t = 0) = 1, we bound the first term in (32f) as follows:
(fcF)s - [(1 - n1) (eF)s.. + u. (eF)s..o)
(33a)
[[(1 /1) (ccF) s._r) + u (coF)smul [11 p) (cF)s.m + f. (cF)s,=))
(33b)
=(1 p) |((c/)) Si_. - (cr)s.al μ [(ECF) - (cp)stal
(33c)
= (1 m) |==1 (p==0 (x) -= (x)) dxx + |t=0 (v'= -p- (x)) dx
(33d)
x
1 p==l (p' (x) - -² (x)) dx + +uc.
/ 1 1- (x) - (x)) dx
=(1 - p)
(33e)
x K
x
/ h(x)) (x) - -== (x)) dx
<(1 p) K sup
+ pu sus
h(xx) (x) - (x)) dx
(33f)
hEH
NCH
= . sup
-= h(x) (x) - (x)) dx = K (p'
(x))
(33g)
Thus, noting that (eF)s,., is indeed the expected factual loss over the subset Stnn induced by AL, thus simplying the notation to ES,, we conclude the proof by plugging (33g) back to (32f):
Σ P1-1 cs. + 8 (Nr + 3 (ALLt) + Ku.
1
CCF S
(34a)
(E(0,1}
where the constant Ky = K- (p' (x), is derived in the similar fashion in (Shalit et al., 2017).
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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Thus, we provide adjustable bound depending on the size of the covering radius, where the infimum of the upper bound value is by labeling all the pool set to make the covering radius the least, the gap originates from the labeling less data but with some covering radius. Note, that generalization bound by Shalit et al. (2017) quantifies well the risk upper bound given the fully labeled pool set. The work (Qin et al., 2021) builds upon (Shalit et al., 2017). but certain aspects remain underexplored, that is, it does not show the importance of the counterfactual covering radius. We fill this crucial theoretical gap by providing an informative and complete bound under the data-expanding context from scratch to derive a new bound tailored specifically to treatment effect estimation with AL. Furthermore, Lemma A.5 unveils the distinctive counterfactual covering radius which originates from the unique nature of the counterfactual prediction. We visualize the counterfactual covering radius (t) in Figure 1(b) and (d) to facilitate the conceptualization of Lemma A.5 in the treatment effect estimation setting under AL paradigm, where the full coverage on the counterfactual class is required.
Discussion:
A.5. Proof of Theorem 4.1
Definition A.6. Let (t.e) of size B(t,e) denote the optimal (OPT) subset for treatment group t. For each point vi E (t,a) let the cluster of be C. = (u' E Dr : d(u', v²) = minw'ss d(u', ,')}. As such, we have partitions C2; C......, ((s))
where each point u² E Dr placed in the closest Cr w.r.t. of E S((,))
Theorem 4.1. Under Assumption 2.3, the sum of the covering radii returned by Algorithm is upper-bounded by 2 x Engo,1 (O. OPT (())))
Proof of Theorem 4.1. Let the output of the Algorithm be S, specifically, the total budget B, additive arithmetically, is split into four parts with B B(1,1) o(,,0) Bio,0) + B(o, i.e., with each part to acquire designated point to reduce one of the four radius (1,1), 0(100) 0(o,0). and (0,1) at time. Thus, we have S S(1,1) U (1,0) U U ((o,1). and noting that S = S(111) U (1,,,)) and So = Si,, ( To bound each of the radius:
For u=1 DD to reduce (1,));
d d(u'= S1) s (('""),((,nn)) S x OPT
(35)
where the first incquality is because S(1,1) C St, and the second inequality is by Lemma A.8.
For === E Do to reduce 0(1,0)
d (u 1= Si) S d (u"=", (q.0o) S 22x OPT
(36)
where the first incquality is because S(1,0) C St, and the second inequality is by Lemma A.9.
For f= E Do to reduce 000.0) :
d (u²=0, Su) S dd du'=(, S(0,0) S 2 x OPT. O where the first inequality is because S(o.0) C Sp. and the second inequality is by Lemma A.8.
(37)
For f== E DD to reduce 60.11
d (1'=1, Sa) S du' S(o,1) 2 x OPT (Q..)
(38)
where the first inequality is because S(o,1) C Sy. and the second inequality is by Lemma A.9.
Since for all u the above holds, thus it follows that
max d d((u=1,,1) + max d
max d du'==0, + max d(u'=1,So) uEDo (0p
(39a)
NEDiNS,
DEDaYSo
MED, 1
<2x x OPT (1.1) +2 OPT.
01.00 (1.00 +2 x OPTo:
+2 OPT5
(39b)
b(o.0)
(39c)
= )111) + 01000) + 610.0) + 60,11 x2x Σ (OPT + OPT
(39d)
:(01))
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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
In the following, we define d(u, Q) := minvee d(u, v) where Q can be any set, e.g., the optimal solution S(.,)) also we denote the OPT as the minimal covering radius returned by the optimal set S", i.e. OPT mD minves- d(u, v).
Lemma A.7. Without loss of generality to Definition A.6, we have Vu, w C. for v Q, then d(u, w) S OPT
Proof of Lemma A.7.
d(uww) d(u, u) d(w,r) d(u, Q) + d( Q) S OPT + OPT = 2 OPT,
(40a)
(40b)
(40c)
where the inequality (40a) is by the triangular inequality, the equality (40b) is by Definition A.6, and we complete the proof with the inequality (40c) due to d(u, Q) min.eg d(u,
S max min.eg d(u, v) OPT.
Note, that the minimal covering radius 6(.-) under the optimal solution St,) is denoted as OPT Lemma A.8. With budgel Bi,,)) Subset S(e,)) for the trealment group I returned by Algorithm I is a 2 - OPTo for covering the set Dj of the treatment group t.
Proof of Lemma A.8. When covering the treatment group i, we dose not have any assumption for the data distribution, thus we split the proof into two scenarios in similar fashion by Dinitz (2019), i.e., Ve ((,1) S(. n C2 + Ø and Bv E (1,y) S(..)) ne; = Ø:
Ve E S (1.1) not * 2: Let w E St. net for Dr, u E C with S(.0) (noting that C2 C De), we have:
Su.n) S d(u, w) S OPT (c.
(41)
where the first inequality is because w E S(..e). and the second inequality is by Lemmma A.7 for u, w E C. Note, that Eq. (41) holds Vu E DD. as C c Dr, then ((.0) = max d(u, ((.,)) I 2 x OPT
Ee St (r.)) (c.o) = Ø: Since (St,,)] = | B(,,)), by the pigeonhole principle, Eu' St (1.)) (1..)......8 (t.)) (c,
2. Thus, assume that for z, m E SSu,)) ney and z is added to S(r.)) before m. Let S(..)) = S( \m, then for E D.,
we have:
d(u, S(,))) S d(u, St,e))
(42a)
< d(m, S((,))
(42b)
S d(m,2 2)
(42c)
< 2 x OPT
(42d)
where incquality in (42a) is because S/,,)) C S(1.)) (the radius decreases monotonically with larger set), the inequality in (42b) is because the greedy selection with larger distance with set S(,,) (m is selected before u), the inequality in (42c) by definition that d(m, d(...) = min d(m, v) and E St (t.e): the last inequality in (42d) by Lemma A.7.
Since for all E Dr, we have d(u, S(.s)) 2 OPT then it follows that max d(u, Sq..)) 2 OPTs conclude the proof for A.8 by enumerating all the scenarios.
Lemma A.9. With budget B()) Subset S for the treatment group t returned by Algorithm I a OPT ( for covering the set Di-t of the treatment group
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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Proof of Lemma A.9. Note, that the reduction of the counterfactual covering radius ò(c,1-e) requires the acquisition from group I, which is complex because the radius ott is calculated by ((1,1-1) = _ minjeS(,, d(x:- -f ,x)),
noting that the proxy collection - is from the treatment group 1 t, however, the acquisition targeting querying the sample from group to reduce the counterfactual covering radius ((,,--))
Thus, by the greedy nature of the Algorithm 1, when ((,1-e) is the one to be reduced, the mentality for the query step is to calculate the proxy point 01-s arg max A,,-: minjeSu.1 ") d(x!-* exx|", xx)) from group 1 t, then find the nearest point a E Dr to a1-t as the factual query to expand (t.1-t)- If there always exists d(at, a1-t) = O, the radius reduction can be real quick.
Definition A.10. Let cluster Fir = flu E Di dlu' ) = min d(u--t,'') ,')} } for of E ((aa-1)
The proof adopts the similar mentality as shown in Proof of Lemma A.8, For now we show the proof for the first scenario and the second scenarios follows.
tuu E - nF;; # Ø: Let SI- let E F; with E ((.1-1) (noting that Fø C Di-r), we have:
S
d(u' S(,11))) = ddu' ,(=' ,))
(43a)
S d(u'- _w w--1)
(43b)
S d(u'tt ,et) + d(u' - (0)
(43c)
$ d(u'-t,
1- ((,,,))))) + d(w -,,,)))))))))
(43d)
< OPT 6(0,1-0)
(43e)
where the equality (43a) is by the Assumption (Strong Ignorability) 2.3 (i.e., 0 < p(t)x) < 1), s.t., for Si returned by Algorithm 1, we can have identical set ((,,1-)) E Dj to the proxy collection S1-1, the inequality in (43b) is because ---t E Si the inequality (43c) is by the triangular inequality, the inequality (43d) is by Definition A.10, and inequality (43e) is by d(u d(u'-t,S; -)) min d(ul- (t)) I max,1--ED,, minees d(u' 1-t 2)) = OPT u e
A.6. Proof of Theorem 4.4
Assumption 4.3 Given the fixed covering radius (1,,)) and (t), there exists the optimal solution S S; C S* for treatment group I, such that A -ATTT =D
Definition A.1I. Let the pool set be D of size n, S₁ of size Be be the solution for group returned by Algorithm 2. family of sets U {Ul " where Vi, Ui C D. Note, that the subscript of Ut denotes a single selected point U in S S [em] due the fact that each 8; as the center with the fixed radius covers set of point, i.e., by definition U, = (1s (U:) Ace,1-e) (1)))) (w:). Let the uncovered set be Ω = D)) Uies Us up to iteration r, and assume that Algorithm 2 were to pick Ui U2 ..., Uy for which Ut is one of the set in U. Denote the set covered by optimal solution S as O
wi = 144, 0 11 and 1/2 = [0] 4j.
Theorem 4.4. Under Assumption 4.3, Algorithm is a (1 - 2) approximation for the full coverage constraint on the equally weighted graph and unscaled out-degree.
Proof of Theorem 4.4. We prove the (1 - ) approximation for the full coverage constraint by Algorithm 2 by extending the method by Dinitz (2019) for solving the conventional Max k-Cover Problem into our Factual and Counterfactual Coverage Maximization for the data-efficient treatment effect estimation problem. The objective is to query S = S] U that maximizes the mean coverage P(A).
As defined Section 4.2, AF= UxES, A(,))()) and A UxES, A(10)(x)), by further in Definition A.1 A.1 for the
22

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
factual (AF and counterfactual (A covered region by St, have:
AF
U ACF
U
uji
(44a)
ASB.
Σ Wj
(44b)
JEB.
|+] 1]B
(44c)
2 |0| |0| e 1 (1 |0),
(44d)
where the equality in (44a) is by definition of the covered region in Section 4.2 and the set U, in Definition A.1 the equality in (44b) and (44c) is by Definition A, and the inequaltiy in (44d) is by Lemma A.14.
Under Assumption 4.3, there exists optimal solution s.t. θ = D, which further implies:
(1 - (1-e-))p D l
AF UU 2 - e |
P(AF
U ACP)
(45)
D
Without loss of generality, the proof above applies for proving that
P(A- U A 2 1 1
(46)
To conclude, as we have:
P(A) 1
1 P(ACF) P(A + P(A + 1 P(A +
1
(47a)
P(AF) PIAFUA=TT ( 1 P(A A
(47b)
2
2 1-
(47c)
where the equality in (47a) by the definition for the mean coverage P(A) and note that our maximization goal in Eq. (10)
leave out the constant coefficient 1 144, which does not affect the ultimate goal. The equality in (47b) is by the independence between the factual covering and counterfactual covering, the inequality in (47c) is by conclusion in Eq. (45) and (46). O
Lemma A.12. For set P. Q and let
N J, we have P Z P -1
Proof of Lemma 12. P |
@ due to the fact that pnn |
Q for
P
Lemma A.13. 1). < @0UUs. 4/
Proof of Lemma A.13.
1/
|
Σ
-
< jey Uui
(48)
isi
JSi
The first and second equality is straightforward by definition and observation, and the inequality is by Lemma A.12 due to the fact that the optimal cover θ is the larger.
Lemma A.14. fj8 |0| -1
Proof of Lemma A.14. Construct the subtraction:
7/ 7/- =
[0]
Σ Wj
|0]
|0]
Σ w/ jsi-1
133)
-w
(49a)
7/ 7k-1 Wi
(49b)
23

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Lemma A.13 implies that optimal solution @ covers least 7/ uncovered samples (the uncovered samples are w.r.t. the covered set by S.) with Be sets, thus by the pigeonhole principle, when 7 filled into B, sets, each set assigned averagely Th/ Br, however, there are less than B, sets availables for the Algorithm 2 to query, thus there exists the set covers least n// B, uncovered samples. By the greedy nature of the Algorithm 2, we have wi+1 2 n; then it further implies that: B,
71-1 1T-1(1 B.
1
7) 7/-1 Wi 11-11
= B.
(50a)
76 1
(50b)
Tk-1 B,
For wi 0 implies that 1/0- = =00, and maximally with budget Br, we have T]B and then construct the following:
nB.
nB:
AB.11
1/2
7
(1
B,
(51)
7/o
1B,
1BB-2
B.
B. quantities ahove
Noting that:
lim (1
1
1
)
1
1)
1
(52)
B 700
B.
E
1/0
E
Thus, it can be concluded that 1/8, < 10
B. Related Work
Treatment effect estimation. Many early works in causal effect estimation (a.k.a,, the treatment effect estimation) focus on group-level estimation, e.g., conditional average treatment effect (CATE). The widely used inverse probability weighting method (Rosenbaum & Rubin, 1983; Imbens & Rubin, 2015) and the doubly robust model (Robins et al., 1994) designed to mitigate the selection bias in CATE estimation, but are not generalizable to the unseen individuals or groups without labels. So far, various methods (Shalit et al., 2017; Louizos et al., 2017; Alaa & Van Der Schaar, 2017; Yao et al., 2018; Yoon et al., 2018; Shi et al., 2019; Zhang et al., 2020; Kallus, 2020; Jesson et al., 2020; Wang et al., 2024) have been proposed due to the proliferation of deep learning (DL). These parametric models are good at modeling the individual- and are generalizable to unseen instances. Furthermore, the strong expressive power of such deep models can handle the high-dimensional data and relax the pivot assumptions in causal effect estimation, e.g., unconfoundedness assumption, by learning the deconfounded latent representations via neural mapping for treated and control groups. Additionally, there is another branch of work that investigate the treatment effect estimation under interference (c.g., the violation of the SUTVA assumptions) (Rakesh et al., 2018; Ma & Tresp, 2021; Ma et al., 2022; Lin et al., 2023; 2024; Chen et al., 2024; Lin et al., 2025), which is out-of-scope to the foucs of this paper.
causal effect
Active learning. The concept of active learning (AL) dates back over a century (Smith, 1918). Over time, has evolved into a prominent branch of machine learning research (Settles, 2009; Ren et al., 2021: Zhan al., 2022). The primary goal of is to optimize model performance in a cost-efficient manner, achieving low model risk while minimizing the number of labeled samples required. AL methods typically categorized into three main scenarios: query synthesis (Wang et al., 2015), strea (Fujii & Kashima, 2016), and pool-based approaches (Wu, 2018). This paper focuses on pool-based AL, particularly in regression problems, where key acquisition strategies include uncertainty-batt
sampling (Gal et al., 2017), density-based querying (Sener& & Savarese, 2018), and hybrid methods (Ash et al., 2019). For example, Bayesian Active Learning by Disagreement (BALD) (Gal et al., 2017) uses epistemic uncertainty to select unlabeled samples, while core-set (Sener& & Savarese, 2018) prioritizes samples based on their maximum distance to the nearest neighbor in the hidden space. ACS-FW (Pinsler et al., 2019) combines core-set and Bayesian approaches. balancing sample diversity and uncertainty in batch-mode acquisition. Although general AL methods are not specifically designed for CEE, benchmarking these methods can yield valuable insights.
Treatment effect estimation with active learning. Thus far, some progress has been made in this area of research. For instance, (Sundin et al., 2019) proposes a querying criterion based on the estimated S-type error rate-tth the probability that the model incorrectly infers the sign of the treatment effect. However, this work focuses on estimating the correct sign of the treatment effect, which differs from the risk metric used our study. For research aligned with the same risk metric,
24

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
Qin al. (2021) introduce theoretical framework that extends the upper-bound formulation from (Shalit et al., 2017) mainly by core-set approach (Tsang et al., 2005). Despite this, their proposed algorithm QHTE does not adequately address distribution alignment during data acquisition. To mitigate acquisition imbalance, Causal-BALD (Jesson et al., 2021b) adopts an information-theoneei perspective, introducing the μpBALD criterion. This criterion scales the acquisition metric inversely with counterfactual variance, encouraging the selection of samples that align with similar counterfactuals when certain treatments are underrepresented. This represents a notable improvement over its predecessor, μBALD, an uncertainty-
roftmax-BALD method (Kirsch et al., 2023). However, Causal- -BLLD depends heavily on accurate uncertainty quantification and computationally intensive training using complex estimators, such as deep kernel learning (Wilson et al., 2016). Recently, Wen al. (2025) introduced a straightforward yet effective algorithm, MACAL, which reduces distributional discrepancies while remaining model-independent Despite its advantages, MACAL requires querying data in pairs (one from the treated group and one from the control group), limiting its generalizability in scenarios where optimality can be achieved by querying from only one treatment group. Furthermore, while MACAL includes convergence analysis for auj-objectives, it lacks guarantees on overall risk upper-bound convergence. Additionally, some studies (Deng et al., 2011; I; Addanki et al., 2022; Connolly et al., 2023; Ghadiri et al., 2024) leverage AL for efficient experimental trial design, where treatment information is applied only after sample acquisition, rather than being included in the initial pool. This setup differs fundamentally from our focus, where treatment information is available from the start. Besides, the recent exploration of AL for treatment effect estimation the feature level (Piskorz al., 2025) differs from our individual-level data acquisition task.
Algorithm 1 Greedy Radius Reduction
1: Input: Pool set D D, U Do: random initialization Sinit Si U.S,, where S and So are the random initialization set for the treated and control group respectively: budget B; distance metric d(.,-)
2: S + Sinit, ,1 <- Ø, So
Ø
3: while [S] < [gini + B do
4: 8 - m (111)))(1.10))))..) (1.0) 60.00 (0.1)
5: 5 = (1.11 then
6: a - arg max,ep, minyes, d(x;=1,x])
(To reduce 6(1,)) by querying data point from group = -1}
7:
else if = (0,0) then
8:
a f arg ma minyes, d(x; --,0,
{To reduce 6(0,0) by querying data point from group 0}
9:
else if 5 = k(.0) then
10:
a' arg mp minjes, d(x!=0 x
(To reduce 6(1.0) by querying data point from group = 0}
11:
b- arg ep,
{Find the nearest point from = 1 to a' to reduce counterfactual radius 6(.0)
12:
b S then
13:
- b. S. 3 fa')
{Counterfactual radius d(1,0) be reduced, add b into the training set}
14:
else
15:
Repeat Line 4-25 by excluding 0(1,0) from the max function {Counterfactual radius 6(1,0) cannot be reduced, go back to other reducible covering radii from the largest one} end if
16:
17:
else if ő = 1).1) then
18:
a' < arg max :D,, V,, minjes. d(x!=1, x
[oo reduce 6(o.1) by querying data point from group = 1}
19:
b arg mi, d(x;=0, ,,))
{Find the nearest point from = 0 to a' to reduce counterfactual radius )(0,1)
20:
b & S. then
21:
- b. St < (a')
(Counterfactual radius 6(o.1) can be reduced, add into the training set}
22:
else
23:
Repeat Line 4-25 by excluding 1) from the max function {Counterfactual radius k(o11) cannot be reduced, go back to other reducible covering radii from the largest one}
24:
end if
25:
end if
26:
S 5 S (a}
27:
end while
28: Output: S
25

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
C. Additional Details
C.1. Algorithm
Proxy collection S1-1- To promote sample diversity the acquired data, the reduction of the counterfac- tual covering radius 6(c,1-t) queries the data from group t, however, such radius is calculated by d(t.1-t) -6 minjes, d(x; x5), where a direct acquisition from the treatment group - performed. But our actual acquisition targets the query from group to reduce the counterfactual covering radius ó(t,1-1) by covering the counterfactual samples. Thus, by the greedy nature the Algorithm I, when 6(1,1-1) is the one to be reduced, the mentality for the query step is to calculate the proxy point a' arg D minjes((.
d(x; 1t xxx) from group 1 - t, then find the nearest point a₂ Dr to a' as the eventual query to expand S,, and adding the proxy pint a' into the proxy collection $1 already marked position.
Rationale of querying factual samples: Note that querying the factual sample to reduce the counterfactual covering radius is not a necessity to help reduce the counterfactual covering radius, because a direct acquisition on the counterfactual samples can indeed reduce the counterfactual covering radius by the rigorous math definition. However, only acquiring counterfactual samples impedes the sample diversity in the counterfactual group. for example, querying the counterfactual sample fall under the neighborhood of an acquired counterfactual sample to reduce the counterfactual covering radius cause redundancy under limited budget.
C.2. Algorithm 2
The data acquisition performed in this algorithm is mainly repeating the two steps: 1). Pick the node with the highest scaled out-degree (which is recalculated each round for new point selection); 2). remove the incoming factual/out- going counterfactual edges to the picked node and its neighbors. Note that the scaled out-degree is by multiplying the out- degree with an coefficient e that is directly associated with the covering radii and the covered points to further balance the distribution discrepancy. That is, given the union of factual and counterfactual covering ball induced by center x: Acc,) (x) (t,,1-r) (x), three main possible scenarios are:
(a) (x) # -0AA #Ø (b) (x) 21A + (c) (x) + 2 2AA =
Figure 7. Visualization of the factual covering (dashed manifold) and the counterfactual covering (solid manifold) on the local neighbor- hood from the center Note that the zero-neighbor scenario is omitted it the least preferred to maximize the coverage.
Thus, the scaling coefficient e caleulated as:
(x),)) A (x) (a,,-1)(x)
c(x) = (1x) - ((x)),
E [0, 1].
(53)
where the minimum and maximum of the coefficient e is obtained respectively by ( 0 (Figure 7(b) and 7(c)) and <= (Figure 7(a)). Noting that the shape of the coefficient w.r.t. Ç over the interval [0,1] downward-opening parabola, which gives the maximum on the evenly divided scenario, with e declines when departing from J. Additionally, when
26

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
scenario 7(b) is preferred scenario 7(c) from overlapping perspective with higher number of counterfactual neighbors prioritized.
C.3. Dataset Details
Toy: The 2-dimensional dataset is generated by creating multiple clusters for each treatment group t. Let center of cluster
of (a 2/22) to be drawn randomly from the uniform distribution:
1 22,2 ~ Uniform([-9 Br, 9 B,]),
(54)
where the 3r is a offset to create larger distribution discrepancy between two treatment groups. Let the collection of the centers of size for group t be V {o/) the randomly added center of should satisfy:
dwevi_, (1 v') 1.5 x 0.9*,
(55)
where j counts from 0 with unit increment each time when Eq. (55) is not satisfied for the randomly generated center of over 100 times. That is, we reduce the minimum distance to a smaller one if the qualified sample cannot be found from the remaining pool set.
Given the centers, np we define the mean p²} = =V][1] and variance of and generating X'(i) E R2 for cluster i:
x²(1) ~ Nia ([(0)~N(( of), snsm ne = X X')..
(56)
Thus, set 31 = 2 and 30 = 22 number of clusters n1 = 50 and n² = 30, number of samples for each cluster ni = no- = 200, such that we generate covariate matrix X E I 10000 for treatment group f = and covariate matrix x 2e00xx for treatment group
See visualization main text Figure 6(a) for the generated data. Furthermore, let T E (x) be the treatment indicator, simulate the response curve y, we have:
yf sin (1.5 =f,1) cos (1.5 2 + 5t, A (x1, E (X'
(57)
IBM (Shimoni et al., 2018): This dataset is based on the real-world 177 covariates from cohort of 100,000 individuals, from the publicly available Linked Births and Infant Deaths Database. The generated response curve is based on the randomly selected 25,000 individuals out of the 100,000 base, and the potential outcomes have 10 different simulations according to (Shimoni et al., 2018).
CMNIST (Jesson et al., 2021a): This dataset contains 60,000 image samples (10 classes) of size 28x28, which are adapted from MINIST (LeCun, 1998) benchmark. CMNIST completely distinct from the previous tabular datasets by leveraging the image data for the treatment effect estimation. The potential outcomes are simulated 10 times and generated by projecting the digits into I I-dimensional latent manifold as described in (Jesson et al., 2021a).
C.4. Further Discussions
It is seen that in Section 5.3 for ablation study, the performance difference shown in Table indicates that the no measurable and significant performance gain on IBM and CMNIST dataset, respectively. Here, we qualitatively discuss the causes for this phenomenon from the data distribution perspective.
The entire data distribution for IBM is plot in Figure 8(a), where a well-blended distribution is seen. We further split the entire data distribution into sole plots on Figure 8(b) (1 = 1) and Figure 8(c) (t = 0). is observed that data distributions on different treatment groups are almost identical, which is aligned to the dataset description in the original paper that generates the IBM benchmark (Shimoni et al., 2018). Thus, the high-density region is built on the overlapping region since two groups are fully overlapped. Therefore, given all the samples on group = 0 known, when data acquisition starts on group 1, algorithm only needs to acquire the sample from the high-density regions as the overlapping condition is synchronously satisfied. Thus, FCCM and FCCM is indistinguishable on such data distribution. In the meanwhile, it is observed that the entire CMNIST data distribution (Figure 8(d)) embeds significantly less overlapping regions with the covariates extracted from the MNIST (LeCun, 1998) benchmark which has 10 classes of data (10 digits). Unlike IBM (Shimoni et al., 2018), the data generating process in (Jesson et al., 2021a) creates significant distribution discrepancy for two treatment groups, where
27

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
(a) IBM
(b) 1=1
(c) -= 0
(d) CMNIST
(e) 1=1
11 t 0
Figure High dimensional data projected into 2-dimensional space via t-SNE, with colors indicating overlapping, group and group = 0. Left: entire data distribution; Mid: distribution group = 1: Right: distribution group / 0.
the high-density regions for group t (Figure 8(c)) and group overlapping regions on the entire data distribution. Therefore, our proposed method FCCM prioritizes the data acquisition toward the high-density and overlapping region, while FCCM only focus high-density region where less overlapping seen, thus resulting in significant performance difference seen in Table 2.
(Figure 8(f) are vell-distinguished, leaving less
C.5. Practicability of the Assumptions
Strong ignorability: The validity of the SI can be approximated by carefully selecting sufficient relevant covariates and constructing more balanced dataset.
Lipschitz continuity: Theorem 3.3 assumes the Lipschitzness of the p' (y)x,, this can be practical assumption if the regression model f. e.g., a well-regularized neural network (NN), learns smooth mapping from x to which implies the Lipschitzness for p² x). Also, the NN is differentiable w.r.t. x, thus the squared loss I is also bounded and sufficiently differentiable w.r.t. x, further implying the Lipschitzness of 1y-
Constant k: The existence of the constant thoroughly discussed by Shalit et al. (2017) in Appendix A.3 and A.4.
Additionally, we discuss consequence for the proposed theorems when the above-mentioned assumptions not satisfied:
Lipschitz continuity for Theorem 3.4: If the Lipschitzness does not hold, the multiplicative constant will be unbounded, and thus the reduction of the radii may not help control the risk upper bound. Strong ignorability for Theorem 4.1: Strong ignorability provides an ideal scenario for acquiring the counterfactual samples, where a quick bound reduction by Algorithm 1 seen in Figure 2(a). If strong ignorability cannot be guaranteed in real-world data, e.g., CMNIST, the reduction of the bound will be significantly slower and less effective for Algorithm 1, as shown in Figure 2(b). Thus, motivates propose Algorithm that can handle compromised data distributions more effectively via slight trade-off on coverage.
28

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Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
C.6. Hyperparameters Tuning
We conduct all the experiments with 24GB NVIDIA RTX-3090 GPU on Ubuntu 22.04 LTS platform with the 12th Gen Intel 17- 17-12700K 12-Core 20-Thread CPU. As stated in the main text, for fair comparison, take the consistent hyperparameters tuned in (Jesson al., 2021b; Wen et al., 2025) for the estimators: DUE-DNN (Van Amersfoort et al., 2021) and DUE-CNN (Van Amersfoort et al., 2021) shown in Table 3. Additionally, we search the best hyperparameters, i.e., covering radius and edge weight for counterfactual linkage, for Algorithm 2 with the validation set shown in Table 4. Note, that in Section 4.3, we approximating a narrower range around 95% threshold to further determine the covering radius for the best performance.
Table Hyperparameters for Estimators
Table 4. Hyperparameters Algorithm
Hyperparameters
DNN
CNN
Hyperparameters
Search Space
Tuned
Kernel
RBF
Matern
6(111) for TOY
[0.11, 0.12, 0.13]
0.11
Inducing Points
100
100
8(1,0) for TOY
[0.11, 0.12, 0.13]
0.11
Hidden Neurons
200
200
8(11)) for IBM
[0.11, 0.13, 0.15]
0.11
Depth
2
((1.0) for IBM
[0.11, 0.13, 0.15]
0.11
Dropout Rate
0.1
0.05
(1,11) for CMNIST
[0.40, 0.45, 0.50]
0.50
Spectral Norm
0.95
3.0
(1,0) for CMNIST
[0.40, 0.45, 0.50]
0.40
Learning Rate
l
le
Edge weight a
[1.0. 2.5, 5.0]
2.5
29

---

Enhancing Treatment Effeet Estimation via Active Learning: A Counterfactual Covering Perspective
C.7. Main Results under Highest Resolution
032
048
25-
PATAL
035
25
125
1296
286
15%
1056
85
100%
(a) 2% increment TOY dataset
(b) 2% increment IBM dataset
(c) 2% increment CMNIST dataset
C.8. Sensitivity Study
Note that the acquisition on treatment sample is insensitive on o(0,0) and 6,,, in our setting, as all control samples (t = 0) are seen. For a, our setting = 2.5 has an overall lower error across different acquisition budgets.
10
40%
67%
380
385
400
100
85
40%
00%
200%
(d) Analysis for the weight TOY
(c) Analysis for the weight IBM
(1) Analysis for the weight CMNIST
6.98
3.0
Cansumpptoon Dadgee
Badger
(g) Analysis 5(1.1) on TOY
(h) Analysis 8(1.1) on IBM
(i) Analysis for 6(1.)) on CMNIST
E+
26
385
40%
30%
1005
205
4C%
30%
1005
400%
60%
105m
(i) Analysis for 61. TOY
(k) Analysis for 8,1.0) IBM
(1) Analysis for &(1,)) CMNIST
30


Reviews:
[{"review_id": "3H73lRxaKy", "reviewer": "Reviewer_8nJA", "summary": "This paper tackles the problem of active learning for treatment effect estimation, where the task is to label a treated or untreated unit given a fixed budget. The authors take the core set approach and extend it by introducing a counterfactual covering radius along with the factual covering radius. An algorithm FCCM is then introduced that chooses to label data points to minimises the expected factual and counterfactual coverings. Through quantitive experiments, the authors show that their method reduces the PEHE metric faster that other methods. Qualitatively, the authors also show that their method is better at choosing data points from the overlap of treatment groups, which is more informative of the treatment effect.\n\n## update after rebuttal\nI thank the authors for their response, I will keep my already positive 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": "**Q1:** $\\Gamma$ is a pseudo-operator defined in the \"Input'' section of the Algorithm 1, i.e., $\\Gamma=\\arg\\max\\min d(\\cdot,\\cdot)$. We use $\\Gamma$ as a shorthand for the distance-based radius defined underneath Eq. (8) (line 203 right column) to keep Algorithm 1 concise. Take line 6 of Algorithm 1 as an example, to reduce radius $\\delta_{(1,1)}$, by applying $\\Gamma$ as the selection criteria, the selected sample from $\\mathcal{D}_ {1} \\setminus\\mathcal{S}_ {1}$ is $a=\\arg\\max_{a'\\in\\mathcal{D}_ {1}\\setminus\\mathcal{S}_ {1}}\\min_{j\\in \\mathcal{S}_ {1}} d(\\mathbf{x}^{t=1}_ {a'},\\mathbf{x}^{t=1}_ {j})$, where $D_ {1}$ and $S_ {1}$ are respectively the unlabeled and labeled sets for $t=1$.\n\n**Q2:** As per line 273 (left column), the informative bound in Theorem 3.3 consisting of four covering radii is derived under the full coverage, i.e., all actively acquired samples should together cover 100\\% of the full data space (we will mathematically explain the full coverage in Q3). Thus, the straightforward solution in Algorithm 1 strictly maintains 100\\% coverage requirement using currently the smallest radii (in Definition 3.1 and 3.2) at every acquisition step, while progressively minimizing the four radii via newly acquired samples. However, due to this full coverage requirement, Algorithm 1 cannot work effectively on real-world datasets where inter-group distributions do not align well (Figure 2 and lines 255-269 left column). Thus, in Section 4.2, we further build Algorithm 2 (FCCM) upon Algorithm 1 as a solution. **Clarifying the statement:** Instead of computing the minimal radii in Algorithm 1, Algorithm 2 fixes the radii by a small constant when performing data acquisition. Though the 100\\% coverage requirement cannot be fully satisfied in early acquisition steps, as the acquisition expands, we can effectively increase the coverage of the data space. So as shown in Figure 3(b), FCCM can achieve up to 25\\% bound reduction compared to Algorithm 1 with only a negative $1\\%$ coverage gap (99\\% coverage). Thus, in line 238 (right column) we state ``given the relatively smaller fixed covering radius'', we can '' approximate (approaching) the full coverage''. We will add more explanations to those parts to better motivate Section 4.2 in an updated version.  \n\n**Q3:** For each treatment group $t$,$P(\\mathcal{A}^{t}_ {F})$ is the proportion of the data points from the unlabeled pool set $\\mathcal{D}_ {t}$ that has been covered by the training set $S_{t}$ in the covering ball $\\mathcal{A}^{t}_ {F}$, i.e., $P(\\mathcal{A}^{t}_ {F})=\\frac{|\\mathcal{A}^{t}_ {F}|}{|\\mathcal{D}_ {t}|}$. Analogously, $P(\\mathcal{A}^{t}_ {CF})=\\frac{|\\mathcal{A}^{t}_ {CF}|}{|\\mathcal{D}_ {1-t}|}$, then the mean coverage rate $P(\\mathcal{A})=\\frac{1}{4}(P(\\mathcal{A}^{t=1}_ {F})+P(\\mathcal{A}^{t=0}_ {F})+P(\\mathcal{A}^{t=1}_ {CF})+P(\\mathcal{A}^{t=0}_ {CF}))$. We will add this formal definition to complement our textual descriptions in lines 266-271 (right column) to make it easier to follow. A further note is that, for Algorithm 1, we have the full coverage, i.e., $P(\\mathcal{A})=1$, strictly satisfied at every acquisition step, while Algorithm 2 -- FCCM is to maximize the mean coverage rate $P(\\mathcal{A})$ to approximate the full coverage ($P(\\mathcal{A})=1$) given the four fixed small covering radii to constitute a lower bound value.", "claims_and_evidence": "The claims seem to be supported as their method does outperform previous active learning schemes. It is also clear why it is performing better (better acquisition in the overlap region).", "methods_and_evaluation_criteria": "The benchmarks are standard for these tasks and PEHE is a standard metric for ITE.", "theoretical_claims": "The proofs seem correct.", "experimental_designs_or_analyses": "The experiments seem sound.", "relation_to_broader_scientific_literature": "Although the idea of using covering sets for active learning is not new, using it for the counterfactual covering seems to be novel. It is also demonstrated that this leads to lear performance improvements.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- What is Gamma in L228? I don't believe this is defined anywhere.\n- I don't really understand what section 4.2 is trying to solve? It is shown that the covering depends on the data distribution in section 4.4 but statements such as \"approximating the full coverage given the relatively smaller fixed covering radius\" (L238 RHS) does not make sense to me. \n- This problem is compounded as I dont believe P(A^t_F) (and other variants) are properly defined, are these the normalised means over the corresponding covering radiuses?", "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": "v7UenlMaaN", "reviewer": "Reviewer_hAfw", "summary": "This paper proposes an active learning (AL) approach tailored to enhance the treatment effect estimation from observational data, when labeling outcomes is costly.  The authors introduce a theoretical formulation using the concepts of factual/counterfactual covering radii, to upper bound the (fundamentally incalculable) population-level risk --- expected precision in estimation of heterogeneous effect (PEHE), defined in Eq. (3) --- effectively by the sum of four factual/counterfactual covering radii, up to a multiplicative constant and lower-order terms, cf. Eq. (4) and Corollary 3.4. \n\nTwo distinct greedy algorithms to minimize the covering radii are introduced: (1) A direct radius-reduction algorithm (Algorithm 1 in Section 4.1) motivated by traditional core-set techniques, and (2) the factual and counterfactual coverage maximization (FCCM) method (Algorithm 2 in Section 4.2), which seeks a more balanced and effective radius reduction through maximizing coverage under given radii constraints. Empirical experiments conducted on synthetic, semi-synthetic, and image-based datasets demonstrate that the FCCM approach notably outperforms established baselines by systematically prioritizing samples from regions of high density and significant overlap between treatment groups.", "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": "**Potential Failure Modes:** FCCM is designed to better handle the partially overlapped data for a quicker bound reduction while maintaining high coverage (Figure 3). As such, in scenarios where the two treatment groups have no overlapping regions (e.g., biased treatment assignment), the data acquisition of FCCM will be challenged as there are no overlapping, counterfactual pairs to be identified. One possible remedy is to operate FCCM in the latent space where the inter-group distributions are aligned by methods like (Zhang et al., 2020; Wang et al., 2024), but it also offsets the model-independent advantage of FCCM. We will add this discussion in an updated version. \n\n**Practicality of Strong Ignorablity (SI):** It is noted that the focus of this paper is active label acquisition, it can then integrate with the selection bias-aware method to make predictions. The validity of the SI can be approximated by carefully selecting sufficient relevant covariates and constructing a more balanced dataset.\n\n**Practicality of Lipschitz Continuity:** Theorem 3.3 assumes the Lipschitzness of the $p^{t}(y|\\mathbf{x})$, this can be a practical assumption if the regression model $f$, e.g., a well-regularized neural network (NN), learns a smooth mapping from $\\mathbf{x}$ to $y$, which implies the Lipschitzness for $p^{t}(y|\\mathbf{x})$. Also, the NN $f$ is differentiable w.r.t. $\\mathbf{x}$, thus the squared loss $l_{f}$ is also bounded and sufficiently differentiable w.r.t. $\\mathbf{x}$, further implying the Lipschitzness of $l_{f}$.\n\n**Q1:** **Elaboration on The Sentence:** Despite that $\\mu\\rho$-BALD ($\\mu\\rho$) highlights searching for samples within the non-overlapping region, its acquisition criterion naturally prefers samples that lead to higher estimated uncertainty by its criterion. **However**, the criterion design of $\\mu\\rho$ is not density-aware. For example, in early acquisition steps, two unseen data points $a$ and $b$ can be of the same uncertainty, while $a$ is an outlier yet data $b$ belongs to a dense cluster (not seen). Then, prioritizing the acquisition of $b$ over $a$ can help generalize the estimator to more samples, which means training on $b$ can eliminate more estimation risk when data is limited. However, $\\mu\\rho$ does not differentiate the importance of $a$ and $b$, and thus \"account for less risk'' in such cases. **How FCCM addresses this challenge:** Pertaining to the previous example, the factual covering from FCCM can prioritize the acquisition of $b$ over $a$, as $b$ has more neighbors (i.e., higher out-degree) than $a$ in the graph constructed in Algorithm 2. To enhance the distribution alignment, FCCM makes use of the weight $\\alpha$ imposed on the counterfactual edges (line 308 left column). For example, assume $c$ is in the same treatment group as $b$, while $b$ and $c$ have similar numbers of neighbors in the graph. Then, if $c$ has more neighbors from the counterfactual group than $b$ does, then the priority of $c$ is higher than $b$ owing to the additional bonus (with $\\alpha>1$) on counterfactual edges when calculating the out-degree. For cases with significant non-overlap (almost no overlap), please refer to our response to potential failure modes. **Why $\\mu\\rho$ queries non-overlapping samples in Figure 6(c):** As stated in line 85, $\\mu\\rho$ relies heavily on the accuracy of estimated uncertainty (model-dependent), which can be erroneously high for certain samples outside the overlapping region, boosting the acquisition metric score. Thus, a few such samples are selected as per Figure 6(c). In contrast, FCCM is model-independent, and the acquisition score is robustly computed via the out-degree of nodes (Algorithm 2).\n\n**Q2:** Please see the anonymous link: https://anonymous.4open.science/r/To-Reviewer-hAfw for sensitivity analysis results on $\\delta$ and $\\alpha$ (click into folder \"To-Reviewer-hAfw''). Note that the acquisition on treatment sample $t=1$ is insensitive on $\\delta_{(0,0)}$ and $\\delta_{(0,1)}$ in our setting, as all control samples ($t=0$) are seen. For $\\alpha$, our setting of $\\alpha=2.5$ has an overall lower error across different acquisition budgets. We will include all results in an updated version.\n\n**Q3:** **Theorem 3.3 (Lipschitz continuity):** If the Lipschitzness does not hold, the multiplicative constant will be unbounded, and thus the reduction of the radii may not help control the risk upper bound. **Theorem 4.1 (SI):** SI provides an ideal scenario for acquiring the counterfactual samples, where a quick bound reduction by Algorithm 1 is seen in Figure 2(a). If it cannot be guaranteed in real-world data, e.g., CMNIST, the reduction of the bound will be significantly slower and less effective for Algorithm 1, as shown in Figure 2(b). Thus, it motivates us to propose Algorithm 2 which can handle compromised data distributions more effectively via a slight trade-off in coverage. \n\n**We will adopt all the constructive suggestions to update the paper.**", "claims_and_evidence": "The main claims---(1) a counterfactual covering perspective provides a tractable risk upper bound, and (2) the proposed FCCM enhances estimation performance and improves theoretical risk upper bounds---are well-supported by empirical evidence provided through experiments. The choice of datasets (TOY, IBM, and CMNIST) effectively demonstrates the general applicability and performance advantages of the FCCM approach. Nonetheless, a deeper discussion linking theoretical guarantees to empirical outcomes, especially considering variations in dataset properties such as the degree of overlap between treatment groups, would enhance the validity of these claims. While the empirical evidence is strong, a more explicit acknowledgment of potential failure modes or limitations of the proposed methods could strengthen the authors’ analysis and presentation.", "methods_and_evaluation_criteria": "The chosen methods and evaluation criteria---e.g., the PEHE metric---are appropriate and commonly accepted within the causal inference community. The diversity in datasets (synthetic, semi-synthetic, and image-based) effectively demonstrates methodological versatility. However, more explicit explanations and motivations for the chosen datasets in the main text would improve clarity. While Appendix C.2 provides some context, essential details such as dataset characteristics (e.g., types and ranges of covariantes and response variables) and preprocessing steps remain insufficiently clear.", "theoretical_claims": "I reviewed the main theoretical claim (Theorem 3.3) and associated lemmas (A.2, A.3, and A.5). While I did not verify every line in the proofs, the theoretical derivations appear generally sound, utilizing standard techniques. However, assumptions such as strong ignorability and Lipschitz continuity, though commonly used in the literature, deserve more explicit justification regarding their practicality. Moreover, clarifying and simplifying the presentation of assumptions and key theorem statements would significantly improve readability and comprehension.", "experimental_designs_or_analyses": "The experimental designs are sound, leveraging suitable baselines including QHTE, Causal-BALD, MACAL, LCMD, BADGE, and BAIT, effectively situating the proposed method within existing literature. Nevertheless, the robustness of the FCCM algorithm could be more comprehensively assessed through sensitivity analyses, particularly concerning hyperparameters such as the radii parameter $\\delta_{t,t'}$ (for $t, t' \\in \\{0,1\\}$) and the weight parameter $\\alpha$.", "relation_to_broader_scientific_literature": "The paper situates its contributions within the broader literature, linking clearly to existing AL methods such as BALD and core-set approaches, as well as contemporary causal inference methods like QHTE and MACAL.", "essential_references_not_discussed": "The discussion of related literature provided in Appendix B seems mostly sufficient.", "other_strengths_and_weaknesses": "* Strength: The theoretical framework based on factual and counterfactual covering radii is innovative and potentially valuable for future methodological development.\n\n* Weakness: The paper's presentation could be significantly improved to enhance clarity and readability, particularly in defining key concepts and motivating algorithmic choices.", "other_comments_or_suggestions": "The paper presents an interesting idea and a decent contribution; however, I believe its readability can be significantly enhanced.  Below I list some suggestions.\n\n* Definitions could be streamlined to improve clarity and reduce redundancy.  For example, Definitions 3.1 and 3.2 could be merged to define covering radius $\\delta_{t,t'}$ generically. which can then be specialized to define factual/counterfactual radii. The same comment also applies to Definitions 4.2 and 4.3\n\n* The statement of Theorem 3.3 is currently dense and challenging to follow. Explicitly separating and clearly stating assumptions outside the theorem (e.g., by defining \"Assumption\" environment) would significantly aid comprehension. I would also suggest the authors further discuss the practical implications and limitations of the theoretical assumptions explicitly.\n\n* The description of Section 4.2 is unclear and confusing.  I would suggest the authors revise description more precisely and compactly, e.g., by defining quantities and sets, e.g., $\\delta_{\\mathrm{sum}}$, $\\mathcal{A}$, $P(\\mathcal{A})$ explicitly using mathematical expressions instead of vaguely defining them in English. \n\n* Sentences throughout the manuscript often tend to be overly long and verbose. Breaking these into shorter, clearer sentences would enhance readability and improve interpretability. \n\n* Additional Minor comments\n  - Please consider revising the captions more informatively. For example: (1) please add \"by Algorithm 1\" in Figure 2; (2) I think the caption of Figure 3 should contain \"Algorithm 2\" instead of \"Algorithm 1\"; and (3) please add \"by Algorithm 2\" in Figure 4.\n  - I think Algorithm 2 is missing an input parameter $\\alpha$.", "questions_for_authors": "1. Could you elaborate on the following sentence you wrote in Section 5.1: \"For example, although $\\mu\\rho$BALD bias the data acquisition toward the overlapping region, it does not further embed the property to query from the high-density region, and thus accounts for less risk\" and explain the mechanism how FCCM specifically addresses distributional discrepancies compared to existing methods like $\\mu\\rho$BALD, particularly in scenarios with significant non-overlap? I am asking this question because (1) Figure 6-(c) seems to suggest that $\\mu\\rho$BALD indeed places some points outside the overlapping region, instead of \"biasing the data acquisition toward the overlapping region,\" and (2) I want to gain intuitive understanding on how FCCM (Algorithm 2) addresses the challenge.\n\n2. How sensitive is FCCM to the choice of hyperparameters, particularly the radii parameter $\\delta_{t,t'}$'s  and the weight parameter $\\alpha$? More details or experiments here would significantly enhance confidence in the robustness of your method.\n\n3. Can you discuss the implications on your theoretical guarantees and empirical results if key assumptions, such as strong ignorability and Lipschitz continuity, do not hold in practice?", "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": "35LVudrJmy", "reviewer": "Reviewer_FYYH", "summary": "This paper tackles causal effect estimation in active learning, where only features and treatments are known, but outcome labeling is costly and incomplete. To ensure balanced label expansion, it bounds generalization error using factual and counterfactual covering radius. Starting with a greedy approach, it develops the FCCM algorithm, optimizing coverage in real-world data. Experiments on benchmarks show state-of-the-art performance in causal effect estimation.", "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": "**Q1:** For instance, when performing A/B tests to compare two software versions, the service provider needs to assign two distinct user groups to different versions (i.e., treatments). To understand which software version offers a better experience for a specific user profile (i.e., features), it is necessary to collect users' feedback via customer surveys. Though the features and treatments associated with each user are known to the service provider, collecting all treatment outcomes -- users' satisfaction levels is time-consuming and economically unrealistic (e.g., paid surveys). In this case, active learning can be used to select a small subset of the most informative users, from whom the treatment outcomes can be obtained at a much lower cost.  \n\n**Q2+W1:** We provide the figures under the 2\\% increment evaluation scheme, which is the most fine-grained setting based on the 50 query steps in total in the anonymous link https://anonymous.4open.science/r/To-Reviewer-FYYH (click into folder ``To-Reviewer-FYYH''). Note that on the IBM dataset, all methods yield an impractically large prediction error at the initial acquisition steps (<20\\% budget), thus we show the results from 20\\% for better clarity.\n\n**Q3+W3:** Our main rationale is to avoid making the search space prohibitively large -- up to $\\mathcal{O}(n^4)$ if each radius has $n$ candidate settings. The key insight to reduce the search complexity is that, because $P(\\mathcal{A})=\\frac{1}{4}(P(\\mathcal{A}^{t=1}_ {F})+P(\\mathcal{A}^{t=0}_ {F})+P(\\mathcal{A}^{t=1}_ {CF})+P(\\mathcal{A}^{t=0}_ {CF}))$, each radius is **independent** and each of the sub-terms increases monotonically with the radius, then so does the mean coverage $P(\\mathcal{A})$. Thus, by the independence and monotonicity, the search space greatly reduces to $\\mathcal{O}(n)$ while the search of four radii stays in the same direction, thus we let the four radii share the same setting. **When will different radii help?** Though the shared value among four radii allows for effective and efficient hyperparameter tuning, it could potentially fail if the distribution discrepancy between the two treatment groups is very large. This is because the counterfactual covering radius can be $\\delta_{(t,1-t)}\\gg\\delta_{(t,t)}$ for the counterfactual coverage $P(\\mathcal{A}^{t}_{CF})$ to get close to full, lowering the utility of the uniform radii value. As such, if the distribution discrepancy between treatment groups is large, we can then identify a different value for each covering radius to maintain the high coverage under the linear complexity due to the independence and monotonicity. We will add this discussion to an updated version of our paper. \n\n**Q4+W4:** The reason for omitting 0\\% in IBM is that the estimation risk at the first few steps is too large to be practical for all methods. Thus, the reported results start from 20\\% for clarity. We will append a more explicit explanation on the figure caption to eliminate potential confusion. \n\n**Q5+W2:** Thanks for the suggestion, we will update the table to show the PEHE results for both FCCM and FCCM$^-$ in our paper. Please check the anonymous link https://anonymous.4open.science/r/To-Reviewer-FYYH (click into folder ``To-Reviewer-FYYH'') for the detailed results.\n\n**Probability Threshold $\\lambda$ ($\\gamma$ in Our Paper):** We will add a discussion of the magnitude of the probability threshold $\\gamma$. It is noted that higher confidence gives a larger upper bound due to the increase in the third term of the bound, but it remains a constant during our optimization. \n\n**SUTVA:** We will update it with the full spell, namely Stable Unit Treatment Value Assumption (SUTVA). \n\n**Typos:** Thanks for pointing them out, and we will correct them correspondingly. For ``by change of variable reverting y to x'', we intended to express that the integral over the domain $\\mathcal{Y}$ is changed to the domain $\\mathcal{X}$ by the change of variable. We will modify the description to clear the confusion.", "claims_and_evidence": "The claims are supported by the evidence.", "methods_and_evaluation_criteria": "The method is developed by extending the covering radius concept to the counterfactual sample setting, which in my view is a neat and intuitive solution to the problem. The evaluation adapts major benchmark datasets in causal effect estimation to the active learning setting, and covers a variety of strong baselines. Thus, the methodology and experiment sections are rigorous.", "theoretical_claims": "(1) In the proof of Theorem 3.3, it would be useful to explicitly describe the magnitude of $\\lambda$, such that the probability of $1-\\lambda$ is sufficiently large to be practical. \n\n(2) In the proof of Lemma A.3, the acronym SUTVA should be expanded as it is not introduced elsewhere.", "experimental_designs_or_analyses": "The experimental results show a promising level of performance gain from the proposed FCCM algorithm. The settings are rational and the radius hyperparameters are provided, where the algorithm is made available for reproducibility. On top of the quantitative results, I find the visualizations in Figure 6 also helpful for showing FCCM’s advantages.", "relation_to_broader_scientific_literature": "This paper resolves the issue of limited labeled data in the causal inference space, which aligns with real-world challenges when building data-driven models under data scarcity due to cost and/or privacy concerns. Hence, the relevance to the broader of causal inference is high.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths\n\n1 This paper clearly differentiates the problem it studies with another line of work in data-efficient causal inference, which is active experiment design. The practical scenarios where only the treatment effect (but not treatment) is unknown is well motivated and presents some important challenges unique to the problem.  \n\n2 The solution approaches treatment-aware data selection/labeling from an inter- and intra-group covering radius perspective, which is a technically sound and innovative solution. \n\n\n\nWeakness\n\n1 In terms of the experimental setting, as each dataset has 50 acquisition steps, it might help demonstrate the performance trend of FCCM if the results at more budget levels can be shown. For example, a figure similar to Figure 5 can be added to Appendix to showcase the prediction performance with a finer-grained increment (e.g., 10% or even 2% percent).\n\n2 The presentation of the ablation study results in Table 2 is somewhat untraditional. It would be more intuitive to directly report the PEHE values of both variants, with the calculated percentage difference as an additional reference. \n\n3 The active learning strategy needs to maximize a synergic objective with four covering radii (two inter-group and two intra-group) combined, where the authors state that all four radii share the same setting. It is worth discussing whether or not using a uniform hyperparameter setting for all four radii is the best possible practice. \n\n4 In Figure 5, the reported results for both TOY and CMNIST datasets start from 0% budget, while IBM starts from 20%. Is it because the error at zero budget is too large for IBM? Since the causal estimator is the same across all active learning methods, the results at 0% are in fact less informative and can be removed for TOY and CMNIST.", "other_comments_or_suggestions": "Typos/presentation issues: \n1 Line 380, “a fine granularity results” should be “fine-granularity results”\n2 Line 845 in Appendix, “distributions” is mis-spelled\n3 Line 957 in Appendix, the meaning of “by change of variable reverting y to x” is unclear", "questions_for_authors": "1 The Introduction mentions a real-world application for “collecting customer preferences from those who have already received different services”. The applicability of the studied problem to this example is not very clear to me; can you explain with more details?\n2 How does FCCM’s performance look like when using a finer-grained increment of the labeling budget?\n3 Any insights into sharing the same hyperparameter value among all four radii? In other words, in what scenarios will different radii values help?\n4 Is there a reason that results for IBM with 0% budget are omitted?\n5 Can the authors provide the PEHE results of FCCM and FCCM- from the ablation study?", "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": "DZyglAEOcQ", "reviewer": "Reviewer_CkNS", "summary": "This paper proposes a data-efficient method to construct a regression model for estimating the mean difference between the treatment and control responses. Specifically, the author upper-bounds the evaluation error and proposes two radius-based approaches to minimize the upper bound with a limited label budget.", "questions": "", "limitations": "", "rating": 2, "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": "**Active Learning vs. Coreset Learning:** The problem setting we described in lines 114-127 (left column) is an active learning loop that consists of 5 steps, where in Step 4 we identify a subset from the completely unlabeled pool set $\\mathcal{D}=\\{(\\mathbf{x}_ {i},t_ {i})\\}^{N}_ {i=1}$ for the oracle to obtain the labels (i.e., treatment outcome $y_i$). The newly labeled samples are then used to refine the estimation model in Step 5. As such, our setting differs from coreset learning that operates with all samples' labels $y_i$. As our labels are actively acquired, Algorithm 1 corresponds to Step 4 (unlabeled subset identification from $\\mathcal{D}$) of our problem setting. In Eq.(8), though it is formulated using the labeled training sets $\\mathcal{S}_ {0}$ and $\\mathcal{S}_ {1}$, the calculation of Eq.(8) only requires covariate $\\mathbf{x}_i$ and its corresponding treatment assignment $t_i$ (lines 202-203 right column), hence are applicable to all samples in the unlabeled pool set $\\mathcal{D}$. In both Eq.(8) and Algorithm 1, we slightly abused the notation $\\mathcal{S}$ to make the notations less cluttered, as their calculations do not actually rely on the access to treatment outcome labels $y_i$. To avoid potential confusion, we will update the description and notation in corresponding parts to highlight the active learning setting.\n\n**$\\mathcal{R}$, and Class $\\mathcal{H}$:** Thanks for pointing out the typo \"$\\mathcal{R}$'', we will correct it to the feature space \"$\\mathcal{X}$''. The family of functions $\\mathcal{H}$={$h \\mid h: \\mathcal{X} \\rightarrow \\mathbb{R}$} does contain the regression model $f$ as part of the function $h$. As per Theorem 3.3, we have $h_{f}(\\mathbf{x},t):=\\frac{1}{\\kappa}l_{f}(\\mathbf{x},t)\\in \\mathcal{H}$ ($t$ is a constant for any given $\\mathbf{x}$), which contains the loss function $l_{f}$ to evaluate regression model $f$. **The rationale for defining $\\mathcal{H}$ is that** the constant $\\kappa_{\\mathcal{H}}$ in our derived bound is dependent on $\\mathcal{H}$. As per line 966, the term in Eq.(32c) is further upper-bounded by Integral Probability Metric (IPM) under the premise that $h_{f}(\\mathbf{x},t)$ is in $\\mathcal{H}$, which is a useful bounding technique also seen in (Shalit et al., 2017).\n\n**Corollary 3.4 and Noisy Response $y$:** Thanks for improving the completeness of the Corollary, which should be claimed with the probability of least $1-\\gamma$ since it is a further simplification of Theorem 3.3. Its main idea is to reveal what optimizable terms are closely related to risk convergence. For the gap $\\Delta$, we follow the common practice in (Sener \\& Savarese, 2018) by assuming zero training loss and noiseless observation for the theoretical analysis purpose. Under the active learning setting, it is reasonable that the labels obtained from the oracle are sufficiently reliable, i.e., $y_i$ has minimal noise.\n\n**How We Query in Algorithm 1:** As stated in lines 215-219 (right column), the intuition behind the query process is to greedily prioritize the reduction of the largest radius, which is a commonly used intuition also seen in (Tsang et al., 2005; Sener \\& Savarese, 2018) for the $k$-Center problem. As per Eq.(8), each of the four radii is defined in lines 202-204 (right column), thus, we just need to query the point (no label) that constitutes the radius $\\delta$, and the query is explicitly defined in line 6 of the full Algorithm 1 (Appendix C) if we need to reduce $\\delta_{(1,1)}$. The visualizations in Figure 1 are a graphical expression of Algorithm 1's intuition, i.e., the point at the other end of the radius is the next acquisition goal. Notably, Algorithm 1 is a naive instantiation of the covering radius-based active learning and comes with a strong (yet impractical) assumption on fully overlapping data distributions (lines 232-234 right column). Thus, we made its description relatively brief to pave the way for our proposed main algorithm FCCM (Algorithm 2), which relaxes the full coverage constraint to achieve stronger radius reduction under the same labeling budget. We will add this explanation when introducing Algorithm 1 to provide more intuitions.  \n\n**Random Variable $T$:** Thanks for pointing out the typo, we will correct $t$ to $T$ where needed.", "claims_and_evidence": "My primary concern regarding the paper is that the proposed approaches bear a stronger resemblance to coreset learning rather than active learning. In active learning, users interact with the real world, acquire responses (i.e., labels), and utilize these responses to refine the learning process. However, as seen in Algorithm 1 and Eq. (8), the core of the proposed approach focuses on finding a coreset that covers the samples in the treatment and control groups. In other words, there is no interactive learning in which the treatment/control responses are contributory.", "methods_and_evaluation_criteria": "The authors evaluate the performance of the proposed algorithm across various label budgets and compare them with multiple baseline. This makes sense.", "theoretical_claims": "I did not check the correctness of proofs. However, I have a couple of concerns regarding the theory.\\\n(1) \\mathbf{Theorem 3.3}: What is $\\mathcal{R}$? What is the hypothesis class $H$ comprised of? Does $H$ contain regression functions which are used to model among $x,t$ and $y^t$?\\\n(2) Corollary 3.4: Shouldn’t this be a $(1-\\gamma)$ probability-type result? I am thinking the response $y$ is noisy and there is chance even $\\gamma$ is small, $\\Delta$ could also be big.", "experimental_designs_or_analyses": "I have a bare-minimum check on the experimental results, as the methodology part raises significant concerns.", "relation_to_broader_scientific_literature": "The paper states that the proposed method can be applied to evaluate the difference between treatment and control responses in a label-efficient manner.", "essential_references_not_discussed": "I am not aware of.", "other_strengths_and_weaknesses": "I think that lines 6 and 7 in Algorithm 1, which involve searching for points, are critical. However, the authors have included these parts in the appendix, and it seems they did not provide the intuition behind these queries in the main paper. It would be helpful to include an explanation of what data should be selected to help reduce the maximum radius.", "other_comments_or_suggestions": "In assumption 2.3, should use $T$ to indicate a random variable and and use $t$ to indicate the realization., e.g., $p(Y=1\\mid x)$.", "questions_for_authors": "Please see the Theoretical Claims.", "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: This paper proposes an active learning approach tailored to enhance the treatment effect estimation from observational data, when labeling outcomes is costly. The reviewers found the paper to be novel, with both the theoretical analysis and experimental results being sound. Most of the reviewers' concerns were satisfactorily addressed in the rebuttal. However, the authors are encouraged to incorporate the reviewers' suggestions in the final version to enhance the clarity of this paper.
Decision: Accept (poster)