Debiased Front-Door Learners for Heterogeneous Effects
Authors: Yonghan Jung
Venue: ICLR 2026 Poster (2026)
Keywords: Front-door adjustment
Heterogeneous treatment effects
Debiased learning
Quasi-oracle rates
DR-Learner
R-Learner
Primary area: causal reasoning
TL;DR: We propose FD-DR and FD-R learners for heterogeneous front-door effects that remain debiased and achieve quasi-oracle rates even with slow nuisance convergence , and we validate them in synthetic and real-world studies.

Abstract: In observational settings where treatment and outcome are confounded by unobserved factors but an observed mediator satisfies front-door conditions, estimating heterogeneous treatment effects remains underdeveloped. We introduce two debiased learners for heterogeneous front-door effects: FD-DR-Learner and FD-R-Learner. Both methods are constructed to be robust to nuisance estimation error, and we show they achieve fast quasi-oracle rates even when nuisance functions converge as slowly as $n^{-1/4}$. We provide error analyses that clarify their behavior under overlap and nuisance misspecification. In synthetic experiments varying sample size, nuisance noise, and overlap severity, both learners consistently outperform a plug-in baseline, with FD-R showing stronger stability under weak overlap. In a real-world case study using FARS data on primary seat-belt laws, the methods deliver reliable personalized effect estimates and interpretable heterogeneity patterns. Overall, the proposed learners offer practical and sample-efficient tools for heterogeneous causal estimation under front-door identification.

Published as a conference paper at ICLR 2026
DEBIASED FRONT-DOOR LEARNERS FOR HETERO- GENEOUS EFFECTS
Yonghan Jung
School of Information Sciences
University of Mllinois Urhana-Champaign mughan@11linoos.
ABSTRACT
In observational settings where treatment and outcome share unmeasured confounders but an observed mediator remains unconfounded, the front- door (FD) adjustment identifies causal effects through the mediator. We study the heterogeneous treatment effect (HTE) under FD identification and introduce two debiased learners: FD-DR-Learner and FD-R-Learneh Both attain fast, quasi-oracle rates (i.e., performance comparable to an oracle that knows the nuisances) even when nuisance functions converge slowly n-144 We provide error analyses establishing debiasedness and demonstrate robust empirical performance in synthetic studies and a real-world case study of primary seat-belt laws using Fatality Analysis Reporting System (FARS) datasct. Together, these results indicate that the proposed learners deliver reliable and samplo-offeiintt HTE estimates in FD scenarios. The implementation is available at https: /tpp::///ithan tigb........ com/ yonghanjung/FD-CATT
Keywords: Front-door adjustment, Heterogeneous treatment effects, De- biased learning, Quasi-oracle rates, DR-Learner, R-Learner
INTRODUCTION
Estimating causal effects from observational data is central to disciplines such as public policy. A crucial challenge is unmeasured confounding, where the treatment a unit receives influenced by unobscrved variables that also affect the outcome. The front-door (FD) criterion (Pearl, 1995) addresses this by using an observed mediator that transmits the treatment's 's influence to the outcome and is plausibly explained by observed covariates.
A concrete instance (which we also use in our empirical study in $ 5.2) is the effect of adopting
a primary seat-belt law (X) on occupant fatality (Y), with observed seat-belt use (Z) as the mediator, and observed covariate C. This scenario is depicted in Fig. la. Because of unmeasured confounding between X and Y, the naive contrast E|Y | X 1]]]E[]FY X 0] biased cstimate of the causal effect. When belt use responds strongly and the observed covariates plausibly explain belt use, the setting is consistent with FD structure, enabling identification and estimation of the causal effect via FD adjustment. (Pearl, 1995).
law adoption
Although robust FD estimation has advanced recently (refer & 1.1), most methods target population averages (the average treatment effect). In practice, platforms require person- alized effects i.e., the conditional front-door effect r(C') to support decision-making at the user context level. We address this gap by adapting hetcrogencous treatment effect ostimators developed for the standard ignorability (Rubin, 1974) (or back-door adjustment (Pearl, 1995)), such as the DR-Learner (Kennedy, 2023) and the R-learner (Nie & Wager, 2021), to the front-door setting. Concretely,
FD-DR-Learner. We construct a pseudo-outcome whose conditional mean equals r(C). Regressing this pseudo-outcome on C. yields a debiased estimator that achieves quasi-

---

Published as a conference paper at ICLR 2026
HTE Debiasedness
Nie Wager (2021)
X
Kennedy (2023)
Xu & Gretton (2022)
X
Xu et al. (2024)
Fulcher
(2019)
Guo et al. (2023)
Wen et al. (2024)
x
Jung et al. (2024)
(a)
(b)
Chen el al. (2025)
/
Ours
/
(c)
Figure 1: (a) Causal diagram illustrating the front-door (FD) structure; (b) back-door (BD) structure; (c) comparison table indicating whether methods estimate FD effects, het- erogencous treatment effects (HTE), and are debiased. Our proposed learners satisfy all three.
oracle rates (the rate achievable with true muisances) whenever the nuisances converge al n m114A-rates.
FD-R-Learner. We develop a three-stage procedure: (1) estimate how X changes Z across C; (2) estimate how changes in Z shift Y given (X, C); (3) compose these estimates to obtain TCC) This estimator achieves quasi-oracle rates whenever nuisances converge at n 1//4-raates
3. Theory and Practice. We provide error analyses of the proposed estimators, show- ing debiasedness under slow muisance convergence. We demonstrate our findings with synthetic and real-world data analysis.
This paper is organized as follows. $3-4 presents the FD-DR and FD-R learners. $5 reports simulations and case study. All proofs are deferred to Section
1.1 RELATED WORKS
For back-door adjustment (g-formula), large literature establishes debiased and sample- efficient estimation. Classical approaches include augmented inverse probability weighting (Robins et al., 1994; Bang & Robins, 2005), as well as targeted maximum likelihood es- timation (TMLE) (Van Der Laan & Rubin, 2006; van der Laan & Gruber, 2012). More recently, estimation frameworks leveraging machine learning for heterogencous treatment effect have produced flexible estimators with finite-sample guarantees, notably the DR- Learner (Kennedy, 2023), the R-learner (Nie & Wager, 2021), the EP-learner (van der Laan el. al., 2024), orthogonal statistical learning (Foster & Syrgkanis, 2023), and double/debiasee machine learning (Chernozhukov et al., 2018).
Beyond back-door adjustment, research on developing debiased estimators for the front-door (FD) model has grown steadily. Fulcher et al. (2019) develop a doubly robust estimator for the FD average treatment effect estimator. Guo al. (2023) propose one step and TMLE estimators for FD adjustment. Jung et al. (2024) introduce unified covariate-adjustnen formulation that enables robust and sample-effiiien FD estimation. On different thread, modern deep learning- based FD estimators have been developed for scalable estimation (Xu Gretton, 2022; Xu et al., 2024), but these methods lack the debiasedness property.
Beyond average effects, work on heterogencous or conditional FD has also emerged. Chen et al. (2025) study conditional FDD and introduce Lobster:eet, a multi-task neural estimator for the conditional FD effect. However, their estimator lacks debiasedness. In this work, we propose debiased Icarners for heterogencous FD treatment effects, which allow any machine learning method to be used off the shelf while guaranteeing fast convergence to the target even when muisance parameters converge slowly. A summary comparing existing works with ours in Table 1c
2

---

Published as a conference paper at ICLR 2026
2 PROBLEM SETUP
We observe iid. samples of V= (C,X, where is observed covariates, X E {0, 1] binary treatment, Z E {0, 1} binary mediator, and Y is the outcome. Let U represent an unobserved variable that jointly influences (C, XY.) We define the muisance functions
m(zxe) Y Z 2, X x,C cj.
(1)
e(x c) Pr(X-
= c) with (i) - e(1 C), and eo(C) a e(0 C).
(2)
<(= xc) Pr(Z = 2 X =
(3)
We assume positivity: 0 c(x c), {(= xc) < 1 for x, E (0,1) and almost every We also assume Var(Y) < 0o.
The data-gennrating process is depicted in the causal diagram G in Fig. la. Specifically, the structure salisfies the conditional front-door crilerion (Pearl, 1995; Fulcher et al., 2019):
1. Every directed path from X to Y is mediated by Z in & (no unmediated direct effect).
2. Every spurious path between X and Z is blocked (d-soparated) by C' in g.
3. Every spurious path between Z and is blocked (d-separated) by (X, C) in g.
Let (C) 2 E[ do(X = =)), c], where do(X = σ) denotes an intervention that fixes X to E. Under the graph in Fig. 1a, the conditional treatment effect r(C) is identified by
T(C) - (C) ro(C) = Σ (a(z |10) q(= [0]]ee(C)m((z²C, 2
(4)
2.1 PRELIMINARIES: R-LEARNER FOR BACK-DooR ADJUSTMENT MBD-R-LEARNN
In this subscction, we review the standard R-learner of (Nie & Wager, 2021), key building block for developing our FD-R-Learner. We refer this standard R-learner as BD-R- Learner (shortly, BDR), since it is developed under the back-door (BD) criterion (Pearl, 1995), as depicted in Fig. 1b.
To review the BD-R-Learner, define the nuisance functions vj A mex,myh, where
ex(C) f e(1 C) and my A E[y C).
(5)
Let T (C( 2 E| do(X = .), C7 in the BD setting. is identified
(C) A (C) -0 (C) 1,
(6)
In the BD graph in Fig. 1b, the data generating process be expressed the following partial linear model (Robinson, 1988):
X eC) + E[ex CJ 0; and
Y ==a(C) XX(C) +ey, E(ey =
(8)
where a(C') = T0 (C' and BBD)=) (C).
The BD-R-learner's ' loss function, equipped with an arbitrary nuisance îj = {ēx, my) is defined
BDR (Y,X,Y = my (X-xx((())
(9)
Population and empirical risk functions for samples D(Y,X,C) A ((Y, X,, 12) are give by:
L LEDR (T.
Ey
[ADD ((Y,X + +
(10)
BDR (T
Σ BDR
(J
(11)
D
EVED
where À is a hyperparameter and J is a regularizer. BD-R-Lcarner defined as follows:
3

---

Published as a conference paper at ICLR 2026
Definition 1 (BD-R-Learner (Nie & Wager, 2021; Foster & Syrgkanis, 2023)).
The BD-R-Tearner fBDR is learned from the following procedure:
1. Split the i.i.d dataset D(Y,X, {(Y, X,, Ci) into D1 and D2.
2. Learn 7 A (ēx, } using D1.
3. Find @BDR mrg LRDR XY
One may repeat steps (2 with the partitions swapped and average the estimates, The debiasedness property of the BD-R-learrer is captured by the following result:
Proposition 1 (Error Analysis of BD-R-Learner (Nie & Wager, 2021; Foster Syrgkanis, 2023)). Let || denote the Lp(P) norm. Let ęBDR denote the BD- R-Learner estimate from Def. 1. Let z b denote S b up to constant factor. Suppose there cxists a function R.DR (c, 7, such that, with probability -,
(T, 77) min I (T, (T. & R (c,
(12)
TET
Then, with probability 1-
1 BDR s REDR (c, PBDR m + l|xx exlla + I - my l1llex -ex 113
(13)
This result exhibits the property debiasedness: BDR converges at a quasi-oracle rate RBDR even when εx converges only at n- 1114
3 FD-DR-LEARNER
We first introduce the LD-DR-Learner, a learner for the conditional FD effect T(C) that exhibits double robusiness properly. In addition the nuisance functions m(ZXC),
e(X C)) and d(Z XC) in Eq. (1), (2), (3), define the following weights:
q(Z IC)
(ZXC)
TxC
(14)
< XC)
e(X C)
We also define the following functionals of nuisances (m, e),
Tmo (ZC) &
Σ m(ZxC)e(z) | C): zE(0.1)
(15)
V Umeg Coneg(CC) (XC)
Σ (=e) [C),,
(16)
sE(0.1)
Smg, (XC)
Σ m(xXC)q(2 (*)).
(17)
=E{0,1)
These induced functionals sufficient express the front-door estimand.
Lemma (Expressiveness of nuisances). For E {0, 1}
E[{((ZXC) = (XC)(()) = E[smgs
(18)
Based on these nuisances, we define the pseudo-outcome for the FD-DR-Learner.
Definition 2 (Front-door Pseudo-Outccome (FDPO)). For F E (0, 1), let n (m.e,q). The front-door pseudo-outcooee (FDPO) at denoted as ( (V;n):
(XC) (ZC)- Umeg (0mem( Smag (XC). (19)

---

Published as a conference paper at ICLR 2026
FDPO enjoys the following identification and robustness properties.
Lemma 2 (FDPO Property). For {0, 1), let v e,
Consistency: 7=(C) (V)))(V:)) C]: and
Double Robustness: Let n be an estimate of 17. Let ex - e(X | [) and 9=s - g(= |
(V: - 9a (V:n)]
(20)
Blfm - ( + ff - Hyne mge!!
(21)
+ Σ Hlm -()- x0)H((< C)
C)H- |))
(22)
We learn TC) by regressing the difference between the FDPO at æ = and I = 0 on C. Specifically, define the population and empirical risk functions for the FD-DR-Learner
CF - (pp (V; -oo (V;(V; +(C)))++A(())
(23)
XF ( 1
Σ (V) -oV - =(C)) +X](F)) VED
(24)
where i denotes any nuisance set, is any candidate function, Sa is the FDPO for a e {0,1}, A is regularization parameter, and J penalizes model complexity.
Using the risk functions Eqs. (23) and (24), we define the FD-DR-Learner as follows:
Definition 3 (FD-DR-Learner). Let D1, D2 be two disjoint (independent) splits
of size of V; (C,, Zi Yi). Let T be the model class for т, and let Xn be a regularization level.
1. Fit f) - {m, i é) on DT.
2. Compute ÂDR arg min CDR FET A,, (7, using D2.
3. (Optional) Swap the roles of D and Dd (cross-fitting to obtain a second estimate
H', and return the avcrage FOR (f+
Finally, the following result formalizes the debiasedness of the learner:
Theorem Analsi-Analyis of FD-DR-Learner). Let be the model class for (, such that,
any f. Suppose there exists quasi-oracle rate function R (E, with probability at least -,
LR (+, .) min DR (f, i) S (c, F. for all + TT. FET
(25)
Let îDR denote the FD-DR-Learner With probability at least 1 -
||DR - 71 S RDR(C (c, fD,, 1)
Σ 2e(0,1)
|| 1--mm1lll - Eall2
(26)
Σ
||mma
Vinng 11 nan 1
(27)
(0.1)
Σ
||e Ex 12211120
Iea 122
(28)
(..
In words, the error ÂDR is controlled by the quasi-oracle rate and second-order products of muisance errors, reflecting the method's debiasedness. For example, if all muisances converge at the n 1/4 rate, then the FD-DR-Learne îOR converges at the quasi-oracle rate (behaving as if the true muisances were known).
5

---

Published as a conference paper at ICLR 2026
4 FD-R-LEARNER
In this section, we introduce the FD-R-Learner, a learner for the conditional FD effect rCC) that exhibits robustness property such that the estimator converges fast even when nuisances converge slowly. To derive the FD-R-Learner, we re-express the data-generating process as the following partial linear model:
Proposition 2 (Partial Linear Model for FD). Suppose the FD criterion holds (i.e., Fig. la).
X = ex((C) +ex, E/ex |C| = 0, Z =a(C)= + Xb(C)+ tEz E[ez XC] 0,
(29)
(30)
Y f(XC)+ + Zg(XC) tey,
Elsy [XC = 0,
(31)
where ex(C E[[ C and
a(C) _ E do X 0), C). b(C) A R|Z do(X -c()), S(XC) Y do = XC), 9(XC) - E do = -)
Since C' satisfies the BD criterion relative to (X, 7) (by the definition of the FD-criterion $2), we can learn b(C)) using the BD-R-Learner as follows:
8 E arg min 7BR BEB (2,,,,
mz)),
(32)
where the population and empirical R-risk functions are defined in Eq. (10) and (11), (Di, D2) is a partition of the sample D, and ēx, my estimated nuisances for ex
Pr(X- C) and mz - E[Z C), learned from D1. error analysis of the BD-R-Learner implics that b converges to converge as slowly as the n 11/4 rate.
the quasi-oracle rate even when the estimated muisances
Similarly, 9 can be estimated using the BD-R-Learner, since (X,C) satisfies the BD criterion relative to (2,Y) Y) (by the definition of the FD-criterion), and Z and Y admit the Robinson (1988)'s partial linear model as in Eq. (7) as follows:
2 = ex(XC) + e = f(CC) + Zg(XC)++ + E(cy ZXC| = 0,
Blez XC] = 0,
(33)
(34)
where ex(XC) A aCC) + Xb(C) from Eq. (30), so that ex(XC) E| XC because E[Z do(X C) E ã, Cj. Therefore (XCC) can be estimated using LD-R-Le as follows:
g arg min BDR gEe As 2xx To A (ēz, my
(35)
where (Dig, D2g) is a partition of the sample D, and ēz my are estimated nuisances for ez - EiZ XC] and my _ E|Y XC), learned from Dig. The error analysis of the BD-R- Learner implies that ÿ converges to g at the quasi-oracle rate even when estimated nuisances converge slowly.
Such debiasedness propertics of b(C) and g(XC) is crucial for the estimation of heterogo- neous FD causal effect r(C',, since it can be expressed as functional of b and g:
Theorem (Heterogencous Treatment Effect via Partial Linear Equation).
Let b and g denote the functionals in Prop. Define 7,(C) A Elg(XC) CJ. Then,
r(C) = (C))
(36)
A remaining challenge to estimate ex e (x-ex (1xxx()))(( in a sample- efficient manner. One straightforward approach is to construct a plug- in intestmmaor by sub- stituting the estimated nuisances:
(Cllu(C) (C)G((( + (1 2x (C))
(37)
6

---

Published as a conference paper at ICLR 2026
However, even if b and 9 converge quickly (Prop. 1), golugg may still converge slowly when mussance estimates converge poorly, because its error depends directly on the accuracy of ēx (C). Specifically,
+xx]))))xxxxxxxxxxxxxe (C) - -g((C) = {g(1C) - -0010000000-00000000000000------
(38)
+ +11 (01=
- g(0C))
(39)
Thus, the convergence of (C) is bottlenecked by the accuracy of εx, and be slow when muisance estimates converge poorly.
To address this challenge, define the pseudo-g function, which serves as a pseudo-outcome for (XC), as follows:
Definition 4 (Pseudo-g). The pseudo-g Sū, (XC) with 7. - (Ex(C),,(XC)) is:
S _ +
(40)
The pseudo-g function enjoys the following consistency and robustness properties:
Lemma 3 (Property of pseudo-g). Let Si (XC) denote the pseudo-g function.
Consistency: ElG,, (XC) CI = (C) with ne - (ex(C),g(XC))-
2. Error Correction: For any estimated εx and g,
(41)
The error analysis Lemma implics that the pscudo-g's error no longer depends on the error of εx; depends only on the error of g, which be learned sample-efficietlly via the BD-R-Learner.
We now formally define the FD-R-Learner follows:
Definition 5 (FD-R-Learner), Let (D,, D2, Dz) be disjoint (independent) splits of size n of V; = (C,, X,, Z,, Yi). Let B, Q, denote the function classes for b, 9; 7, respectively. Let \b,n, Agmn d7n be regularization levels.
(Nuisance Fit) Fit f7 A fēx, m2) and ng - (ez,my) using D1. (BD-R-Learner for b) Learn d using D2 with î from Eq. (32).
3. (BD-R-Learner for g) Learn ÿ using D2 with n from Eq. (35).
4. (Pscudo-g) Evaluate the pscudo-g function C (XC) using Dz.
5. (Learn ?) Find
7 E arg min gET [aal
1
Σ
(G (X.C.) - >(,)) + (c))
(42)
:.V.Ds
6. Return in((C) A ((C)(C))
One may repeat steps (1-5) with the partitions swapped and average the estimates. Finally, the following result formalizes the robustness and sample-effcciency of the learner:
Theorem 3 (Error-Analyiis of FD-R-Learner). Let (,,,, EKC. (XC) 7(C)}² + (T be population risk for 7. Suppose there exists quasi-oracle rale

---

Published as a conference paper at ICLR 2026
(a)
(b)
(c)
(0)
3.35
0.5
0.23
0.40
0.55
0.30
013
0.55
0.25
RMSE
012
10
0.0
10000 21003
32002
42002 55000
12003 20002 30002
40000 10020
0.10
0.2
04
Sample size
Nuisance noise level
Weak-overlep sewerity
Figure 2: Synthetic study: RMSE (mean + 95% CI) of FD-PI (plug-in), FD-DR, and FD-R estimators: (a-b) varying sample size n, where (a) no structural nuisance noises is added, and (b) muisances converge at the n n-1/4 rate; (c) varying the level p of nuisance noises pe, W//11/4, 1/4,n1/// n1/);; and (d) varying weak-overlap severity.
functions Ro(e, 76), Role, ).) and Rr(e, 7, me) such that, with probability
| (6, mT))
min (bb,,, - Ra(c, b,(c,,,,7u b,7))
(43)
WEB
I (ă. mg)
min Lx(g', 7,1 - Re(c, mg),
(44)
d'eQ
(7) 7.)
min yeer
(7 77)) - Rr 7, 7.))
(45)
Let FR denote the FD-R-Learner in Def. 5. With probability 1-6,
प -1. S Ry(e, T, mh + (e 7. mgy) + Rr 7, 77)) + 1 - |1 + +|1xx - ex m2 m2 m2 1
(46)
(47)
+ pス - <z 11 + || 11 - my
(48)
In words, the error of fि. is controlled by the quasi-oracle rates, together with second-order products and fourth-order terms of nuisance errors, reflecting the method's debiasedness. For example, if all muisances achieve the n 114 rate, then fR converges at the quasi-oracle rate.
4.1 COMPARISON BETWEEN ED-DDR-LAARE AND F-D-RLLEARNER
FD-DR-Learner. FD-DR enjoys a double robustness property (Thm. 1), where its error depends the quasi-oracle rate plus nuisance errors represented second-order products. Consequently, quasi-oracle rates are attained if either (i) is accurate, or (ii) (m, é) are accurate, even when the other blocks are misspecified or converge slowly. This makes FD- DR a default when q or (m, é) can be fit well. A caveat is variance under weak overlap (positivity): its construction uses inverse weights and density ratios (o.g., (xC and & (ZXX))), which can inflate finite-sample variance when e or approach 0 or 1.
FD-R-Learner. FD-R avoids the density ratios required by FD-DR, making more variance-friendie under near-violations of overlap. This makes FD-R strong choice when overlap weak. In addition, FD-R provides interpretability by decomposing the estimation into two BD-R-learner subproblems for the pathway components b(C") (effect of X + Z) and g(XC) (effect
+ Y). These intermediates (b and g) are useful for diagnostics. However, FD-R requires more nuisance fits (ex, mz, ez, my plus b, compared to the FD- DR-Learner.
Practitioner Guidance. Prefer FD-DR-Learner when or (m, e) can be accurately timated. Prefer FD-R-Learner when overlap is weak (to avoid density ratio estimation) or when interpretability of the X - Z and Z + Y pathways is valuable.

---

Published as a conference paper at ICLR 2026
(
(b)
(C)
54
Fb-R
mean 7 cf top-ez
0.03
Counts
41
-9.33
-0.22
-4.33
-2.4
04 -2200 2310 180-0 011---212-000 8:60 t5n cáo
6.4
-5.66 -2.06
-0.03
0.00
0.02
?
top-at Ot fraction (largest ?)
[SHAP]
Figure 3: Belt Laws and Fatalitics: (a) histograms for FDR (C,) and (C;) (b) concen- tration curve showing the mean values of &DR and FR within the top-a fraction (largest f), where both learners exhibit downward trend; (c) SHAP feature importance highlighting age, time of date, and driver status as the most influential features for both learners.
5 EXPERIMENTS
In this section, demonstrate the debiascdness of the proposed estimators. In all ex- periments, nuisance functions are learned with XGBoost (Chen & Guestrin, 2016). We compare the proposed FD-DR and FD-R-learners with plug-in (PI) estimator Ppr of the larget estimand:
PC)
& Σt (0lz |C)) - q( [C)))(( = x C)m(zzC).
(49)
22
Details of simulations described in Sec. C. The implementation is available https:
//github. wawyyynngggggn/DDD.
5.1 SYNTHETIC DATA ANALYSIS
We assess the proposed estimators synthetic scenarios where the true heterogeneous FD effect r(C) is known. Fig. reports the root mean squared error (RMSE; mean 95% CI) of the plug-in bascline (FD-PI, Frı), the FD-DR learner, and the FD-R learner across four regimes.
Panels (a-b) vary the sample size n. Both FD-DR and FD-R consistently dominate FD-PI as grows. When no additional structural noise is introduced (panel (a)), all estimators converge expected, but the proposed FD-DR and FD-R learners achieve substantially lower егтоr. When the nuisance functions are restricted to converge at the n -1/4 rate (panel (b)), FD-DR and FD-R still exhibit reliable convergence, demonstrating robustness to imperfect nuisance estimation, whereas the plug-in in estimator converges much more slowly.
To further probe robustness under noisy nuisances, we inject controlled noise of the form pe into the muisance functionals, where e ~ NNnn 1/1411/4,, 1/1.111,n111// -1/4) and (0.0, 0.2, 0.4, 0.6, 0.8, 1.0). Panel (c) shows that the RMSE of îPi deteriorates rapidly as p increases, while FD-DR and FD-R maintain stable performance. FD-R achieves lower RMSE across higher noise levels, reflecting its reduced reliance on nuisance accuracy.
Finally, panel (d) examines weak-overlap scenarios by pushing e(X = 1 and q(Z = 1 X,C) toward zero. In this regime, the plug-in estimator exhibits severe degradation, and FD-DR suffers noticeable variance inflation due to its use of inverse weights. FD-R, by contrast, remains stable and consistently outperforms both FD-PI and FD-DR. These patterns mirror analytical comparison in 4.1, confirming that FD-DR excels when mussance are accurately estimated and overlap is sufficient, while FD-R is particularly advantageous under weak overlap or noisy nuisance models.
9

---

Published as a conference paper at ICLR 2026
5.2 CASE STUDY: STATE SEAT-BELT LAWS AND FATALITIES (FARS)
We use a state-year panel constructed from National Highway Traffic Safety Administration (NHTSA) sources (National Highway Traflic Safety Administration, 2000) using the Fatality Analysis Reporting System (FARS). In our data, the treatment is whether a Seale-ear has primary seal-bell law (enforcing seat-belt use) (X), the mediator is the bell-use (Z) from NHTSA surveys, the outcome is the occupant fatality Y, and C' collects covariates affecting (X, Y). The front-door structure is plausible here because the effect from X to Y operates through increased belt use Z; rich set of covariates C helps mitigate confounding bias between (X, Z) and between (Z, Y). Also, positivity holds in this datasct, because belt use is neither zero nor universal across law regimes; L.e, some occupants do not fasten seat belts even under primary law.
Fig. 3 summarizes the analysis. Panel (a) shows the distributions of (C,) and (C:),,, in- dicating lower fatality rates under primary-law regimes. Panel (b) reports the concentration curve (mean 7 within the top-a fraction ranked by f). Only small portion of units exhibit increases fatality under the primary law regime (X=1), whereas over 95% of units show decreases, consistent with a preventive effect of primary-law adoption. Panel (c) presents SHAP feature importance, highlighting age, time of day, and driver status as dominant factors explaining heterogeneity.
Together, our results indicate that primary seat-belt laws reduce occupant fatality rates for the majority of units, illustrating the practical utility of our approach for estimating causal effects from real-world datasets that fit the FD setting.
6 CONCLUSION AND DISCUSSION
Summary. We developed two heterogeneous FD treatment effects estimators: FD-DR- Learner and FDD-RLLerner. Both attain quast-oracle rates under 1/4-rate nuisance con- vergence (Thm. 1, 3). A comparison
the two estimators for practitioners is given in $ 4.1. In synthetic stress tests (varying n, slow muisances, weak overlap) they dominate plug-in baseline, and in our FARS seal-belt case study they deliver reliable personalized FD estimates.
Limitations & future work. (i) Positivily. Our guarantees assume overlap for e(X C) and q(Z X X);;; near-violations inflate variance (especially for FD-DR). We recommend overlap diagnostics, ratio stabilization, and overlap-aware uncertainty, and plan adaptive routing toward FD-R under weak overlap. (ii) Binary mediator. Our theory uses bi- nary Z, whereas many practical settings feature continuous or multidimensional mediators. Extending FD-DR and FD-R learners to handle such scttings is promising direction.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Heejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference models. Biometrics, 61 (4):96622-9) 973,, 2005.
Tianqi Chen and Carlos Guestrin. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Dala Mining, pp. 785 -94, 2016.
Winston Chen, Trenton Chang, and Jenna Wiens. Conditional front-door adjustment for helerogeneous mreatment assignment effect estimation under non-compliance. In Confer- ence on Health, Inference, and Learning, pp. 194 230. PMLR, 2025.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duñlo, Christian Hansen, Double/debiased
Whitney Newey, and James Robins. Double debiased machine learning for treatment and structural parameters: Pouble/debiasee machine learning. The Econometrics Journal, 21 (1), 2018.
Dylan Foster and Vasilis Syrgkanis. Orthogonal statistical learning. The Anmals of Slatis- lies, (3):8877 908, 2023.
Isabel R Fulcher, Lya Shpitser, Stella Marcalle, and Eric Tchetgen Tchetgen. Robust in- ference on population indirect causal elfects: the generalized front door criterion. Journal of the Royal Statistical Sociely: Series B (Stalislical Melhodology), 2019.
Anna Guo, David Benkeser, and Razich Nabi. Targeted machine learning for average causal effect estimation using the front-door functional. ar preprint orXiv:2312. 10234, 2023.
Yonghan Jung, Jin Tian, and Elias Barcinboim. Unified covariate adjustment for causal inference. Advances in Neural Information Processing Systems, 37, 2024.
Edward H Kennedy, Towards optimal doubly robust estimation of heterogeneous causal effects. Electronic Journal of Statistics, 17(2):3008-331 3049,, 2023.
National Highway Traffic Safety Administration. Fatality analysis reporting system (fars) 2000 data files. Technical report, U.S. Department of Transportation, 2000. URL https: //www. nhtsa. www.sa.oooffllaa... 09-24.
wwwpmmlosdsppnns//taaaaa /bo.a//ss//bbmmoass////////00// Accessed: 2025-
Xinkun Nie and Stefan Wager. Quasi-oracle estimation of heterogeneous treatment effects. Biometrika, 108(2):299 -1929993999-1 2021.
J. Pearl. Causal diagrams for empirical research. Biometrika, 82 6264))669-710, 1995.
James M Robins, Andrea Rotnitzky, and Lue Ping Zhao. Estimation regression coeffi- cients when some regressors are not always observed. Journal of the American statistical Association, (0(427):846-88) 866, 1994.
Peter M Robinson. Root-n-consistent semiparametric regression. Econometrica: journal of the Econometric Sociely pp. 931 954, 1988.
Donald Rubin. Estimating causal effects of treatments randomized and nonrandomized studies. Journal of educational Psychology, 66(5))688, 1974.
Lars van der Laan, Marco Carone, and Alex Luedtke. Combining t-learning and dr- learning: A framework for oracle-efficient estimation of causal contrasts. arXiv preprint ar Xii:2442200197. 2024.
Mark van der Laan and Susan Gruber. Targeted minimum loss based estimation of causal effects of multiple time point interventions. The international journal of biostatistics, 8 (1), 2012.
Mark J Van Der Laan and Daniel Rubin. Targeted maximum likelihood learning. The International Journal of Biostatistics, 2(1), 2006.
11

---

Published as a conference paper at ICLR 2026
Lan Wen; Aaron Sarvet, and Mats Stensrud. Causal effects of intervening variables in settings with unmeasured confounding. Journal of Machine Learning Research, 25(345): 1-54, 2024.
Liyuan Xu and Arthur Gretton. A neural mean embedding approach for back-door and front- door adjustment. In The Eleventh International Conference on Learning Represenlations, 2022.
Ziqi Xu, Debo Cheng, Jiuyong Li, Jixue Liu, Lin Liu, and Kui Yu. Causal inference with conditional front-door adjustment and identifiable variational autoencoder. In The Twelfth International Conference on Learning Representations, 2024.
12

---

Published as a conference paper at ICLR 2026
Supplement to " "Debiased Front-Door Learners for Heterogeneous Effects"
CONTENTS
Introduction
1.1 Related works
Problem Setup
2.1 Preliminaries: R-Learner for Back-Door Adjustment (BD-R-Learner)
FD-DR-Learner
4
FD-R-Learner
4.1 Comparison between FD-DR-Learner and FD-R-Learner
5 Experiments
5.1 Synthetic Data Analysis
5.2 Case Study: State Seat-Belt Laws and Fatalities (FARS)
10
6 Conclusion and Discussion
10
A Preliminaries: R-Learner and DR-Learner Analysis using Orthogonal Sta- tistical Learning
15
A.1 Examples (R- and DR-learners)
15
A.1.1 R-Learner
15
A.1.2 DR-Learner
16
A.2
Assumptions
16
A.3
Assumption checks for the examples
16
A.3.1 R-Learner: assumptions hold
16
A.3.2
DR-Learner: assumptions hold
17
A.4 Main Result .eaul.........
17
A.4.1 Nuisance Leakage: R-lerrner
18
A.4.2 Nuisance Leakage: DR-learner
19
Proofs
21
B.1 Proof of Lemma
21
B.2 Proof of Lemma
B.3 Proof of Theorem
22
B.4 Proof of Proposition
23
B.5 Proof of Theorem
24
B.6 Proof of Lemina 3
24
B.7 Proof of Theorem.
25
C Simulation Details
27
13

---

Published as a conference paper at ICLR 2026
C.1 Synthetic simulation
27
C.2 Real-world study: State seat-bel. laws and fatalities (FARS)
28
14

---

Published as a conference paper at ICLR 2026
A PRELIMINARIES: R-LEARNER AND DR-LEARNER ANALYSIS USING ORTHOGONAL STATISTICAL LEARNING
We study a population risk L(T, n), where the target T E T and the nuisance n E H live in normed spaces (T, I|T) and (H, 1llx), respectively. Throughout, to denotes the true muisance. We define the (possibly non-unique) oracle minimizer
TO arg min Lo), TET
(A.1)
which we assume nonempty.
Directional derivatives. For functional F and direction h, the (Gâteaux) derivative with respect to variable at æo is
F( + th) F(xo)
V F(ro)
A lim +00
(A.2)
t
and second derivatives (2 he] are defined analogously; mixed derivatives such as VLL will be used for orthogonalily.
Sample splitting and plug-in. We assume a two-way split into independent folds of approximately equal size: one to learn á (using data D,), and one to learn by minimizing L(T, f)) over T, i.e.,
arg L(T, f)
so that
To
T
This separation prevents overfitting-induced bias when we later linearize around (To. no).
Target-class statistical term. Let (T(T;n.e) 2 0 be data-dependent rate function such that, with probability at least 1 -
L(T,n) M(T < RT(Tine)..
(A.3)
You may instantiate Ri via localized complexity (e.g,, critical radius) or algorithm-specific bounds; we keep it abstract to highlight how nuisance error propagates into target error.
Goal and norms. Our goal is to upper bound the target error ||7 rol.. When write || we mean the (P) norm with respect to the underlying distribution.
A.1 EXAMPLES (R- AND DR-LEARNERS)
We use standard notation: T (0,1) (treatment), X (covariates), Y (outeome). The estimand is the CATE
Tx(X) A ( Y()) X].
(A.4)
under the usual positivity (c S (X) < 1 a.s.) and i.i.d. sampling. We assume
IL X ( x B|y 1, x], We (0,1).
(A.5)
A.1.1 R-LEARNER
The Robinson decomposition posits
= fo(( T To(X) cy, E(cy T 0,
(A.6)
T' = mx() + tex, E X =
(A.7)
15

---

Published as a conference paper at ICLR 2026
and with mu(X) A E X] we have (X) fo(X) + To xo())ro(X)) Hence
Y mo(X) (T-=t(()) To (X) cy
(A.8)
Thus, viewing 70 OLS-type coefficient. in residualized regression, we define
(T, 710 (mo, mo)) - B B mo(X) (T - -xo(x))((x))-}
(A.9)
so that To E arg min, (T, 70).
A.1.2 DR-LEARNER
We define following nuisances:
wT( - 2 -1
(o - B|Y T, X],
(A.10)
P(T X)
Define the psoudo-outcome
(V; vo - (uo, mo}) - Wo X) mo(T, mo(1, X) po(0,
(A.11)
and the squared-loss objective
(DR(F,n) & [[(v(V:n)=(x)))
(A.12)
This loss is centered at the CATE in virtue of (l)((:no) X = (X(X).
A.2 ASSUMPTIONS
We now state structural conditions that yield fast rates. The exposition follows the orthogonalsstatiistiallleeee
(OSL) template: first-order optimality at truth, curvature in and orthogonality to damp the impact of nuisance error.
Assumption A.1 (First-order optimality in T). Moving away from To cannot reduce the population risk at the true nuisance:
L,T, [h) for all feasible directions h, from To
(A.13)
Assumption A.2 (Strong convexity (quadratic growth) in T). There exist constants X > 0, x 2 0, and r 10,1) such that, for any 7 on the line segment between and
- To, vol A||T To Tol
x|1n
noll
I+T H
(A.14)
Rationale: The risk function L(T, m) is in bowl-shape over т. The к term allows mild curvature deterioration when A no; the exponent 4/(1 (1)r) is chosen to balance mixed terms via Young's inequality later.
A.3 ASSUMPTION CHECKS FOR THE EXAMPLES
We verify that the R- and DR-losses satisfy the above, elarifying how positivity yields curvature and how residualization/
/DR DR construction yields orthogonality.
A.3.1 R-LEARNER: ASSUMPTIONS HOLD
First-order optimality. With ý 2 Y mo (X), 7 2 T (X(),,
V, LR no)[kr] =E TT) T h (X) -2 E T XT h (X)] =0. (A.15)
16

---

Published as a conference paper at ICLR 2026
Hence Assumption A.1 holds.
Strong convexity. We have
VEL(T, T- rol -2)((T(X) - -o(x)) [T
(A.16)
where
- - (X))) | Var(T) | Xx
+ (To(X)-T(x)} 2 (1(X)(1-molll (A.17)
Therefore,
V.L(F, To. Tol -
(A.18)
([(( (Xa)) X)]
(A.19)
[E((r(X)
(A.20)
2 2E 20 [T((( - =o(x))2 c|
(A.21)
= 2c(1 - -c)ll -
(A.22)
Hence, Assumption A.2 holds with À = 2c(1 c) and x = 0 (taking || = -12))
A.3.2 DR-LEARNER: ASSUMPTIONS HOLD
First-order optimality.
since E[p(V;n)) No) [h+] -2
V (To
=0,
(A.23)
Strong convexity. We first note that
VTLDR(TO. To) -2E((p((Vn)) -(x))(((X -x))
(A.24)
This gives
V LDR (0 -
2/|7 To Tol12
(A.25)
which shows that x and X 2 (taking || ||7 || 112))
A.4 MAIN RESULT
Theorem A.1 (Fast Rate Convergence). Suppose Assumption A.1 and A.2 hold.
Then,
||T Toll? RT (V,
Tol ((T. m
(A.26)
+ -ln noll
I+r
(A.27)
Proof of Thm. A.1. By applying the Taylor's expansion and rearranging, we have L(F, 1 LF I)((F = L(F - L(To. V- f)lf 7o),
where 7 is the line segment between f and To
Using Assumption A.2, we have
4
2/ Toli < L(F,n) L( n) V L(T, f)(F Tol 11
no H
1+r
2
(F:(.))))
17

---

Published as a conference paper at ICLR 2026
Since V, (To, 7o] by Assumption A.1, have
X
2 Tul ~ (T + (V,L(Tu. nol - V,L( f)lf Tu]}
(A.28)
1 | Vol
+
(A.29)
The middle difference
[ [] Tol
(A.30)
the reuisance leakage of the first-order optimality condition. is the main channel through which nuisance error affects the target. Under Neyman orthogonality, the leakage is higher than first order in |oll (typically quadratic or a product of nuisance errors), s0 f inherits only a higher-order remainder rather than linear bias. In particular, for the DR-learner factors into product of nuisance errors (yielding double robustness), whereas for the R- learner it enables fast rates once the nuisances are sufficiently accurate. We quantify these forms below for each loss.
A.4.1 NUISANCE LEAKAGE: R-LEARNER
Theorem A.2 (Error Analysis: R-learner). Suppose Assumption A.1 and A.2 hold with || || ||2- Let a å I and X a 2c(1 c), where is constant satisfying 5 (X( 11-6 - Then, with probability e;
32a
32 x² - -olll +
- 1 z ス c) +
xa I mollā molla. (A.31)
Proof of Thm. A.2. Let h, (X) & +(X) (XX). Let 8m( & (m(() mo(X)) and (X) & (x(X) - (x))))
Note, the first-order risk function is the following:
V,LR(T, -EE m(X) -(X)(T
xX))) - - - (x)],,
(A.32)
We note that VRLR[TO, (h+) O, shown in the first-order optimality condition analysis. To analyze the leakage, we rewrite a few terms here:
- TT T) = mg To mo) 5m + 70O
(A.33)
T To 8
(A.34)
Then, we can rewrite the first-order risk as follows:
VnIn(To (To, (/-)))
-2E((cr 6m T (T- To - 5z) hr)
(A.35)
2 (..5,)11)
(A.36)
S 2| + 2| 2
(A.37)
S 21100= ||T0|lx || ll + 211 1|4 1|0x114 1| ll2
(A.38)
2||h, 12 (ll 1| [1,.1aa).
(A.39)
18

---

Published as a conference paper at ICLR 2026
Then, for any 0, Young's inequality (with p = q = 2) gives
L7 no)[f ro] (T, f) mol
(A.40)
< 2114-112 (11 16. 6m114 (|6
(A.41)
s <allh, 11 + (11tol130 1 + ||óm 1| - -1,.
(A.42)
2
= a
Tol 80 1 l1 + 6m 1 112
(A.43)
Choose a = X/4. Let R _ RT f/ e). Then, by Thm. A.1, we have
2
2
16a
16
|| 12 -
RT +
A 11 +
1 +
X fl 11 12
(A.44)
X
1 1 11
2
16a
16
RT
1
6m 113
(A.45)
X2
x2
which completes the proof.
A.4.2 NUISANCE LEAKAGE: DR-LEARNER
Theorem A.3 (Error Analysis: DR-learner). Suppose Assumption A.1 and A.2
hold with || || || [|2. Then, with probability 1 1-c,
|| Toll (f; 8|| wo w0ll2|1140 mll.- (A.46)
Proof of Thm. A.3. Let (x) & f(X) - To(X). Let (x(2)) & (m(XZ) ()( and (x) - (w(x) - wo()). Note
VOLDR(T,M(h/ = -E - hh,l < <2 - -hrl]
(A.47)
We note 7TL: = O, as shown in the first-order optimality condition analysis. Also,
EHp(V;n)) - He-ll
(A.48)
-
(A.49)
(A.50)
(A.51)
||/-||a|(w~-ap) -1)112
(A.52)
< |//||22| wollalluo - -Alla-
(A.53)
Then, for any > 0, Young's incquality (with p = q = 2) gives
2 2|E[(((V hh
(A.54)
s sa 11 + a
- - 1ll..
(A.55)
Choose a = 2/4. Let R 2 R (f; Then, by Thm. A.I, we have
2
16
| - Rr +
A4 1 +
x2 w|w wollālluo - m1l
(A.56)
2
16
= 1 112 < Rr
+
||W woll2llpo - -1112.
(A.57)
19

---

Published as a conference paper at ICLR 2026
which completes the proof.
20

---

Published as a conference paper at ICLR 2026
B PRoors
B.1 PROOF LEMMA
First,
B[rx(C)] Σ m(zxc)q(= ze)e(( c)p(c).
(B.1)
Then,
((= Tc)
E [
Y|
Σ E|
2xd
((xxe)
P(
Σ m(zze)g(2 Ic)e(x ee)(()).
(B.2)
q(2 xc)
Also,
E [T]]me] Σ Ia(x')
Σ m(zre)e(x c))(= x'c)e(a' c)p(()
(B.3)
c(x'' (c)
22'c
= Σ m(zxc)e(z c)a(= (c)p(c).
(B.4)
200
Finally,
B[Sing (XC) Σ Smg (xc)e(x) c)p(c)
(B.5)
m(zxc)g(2 ze)e( c)p(c).
(B.6)
B.2
PROOF LEMMA
First,
(B0080000-012200000--22/20200000 mm(xC)]] =
(B.7)
A Also,
T (XC))rme - V/meg (XC)) E EXC (ZC) XC| - Umeg Unmeg (XC)HI 0,
(B.8)
since
E[rmo (2C)|X(]
ΣΣ m(zaC)e(x (()(( XC) Umngg Sine((C (XC).
(B.9)
"mme(=C
Finally,
E[smas (XC) CC Σ m(xxO)g(2 [C)e( [C) (C).
(B.10)
Therefore, Elpp(VV:)) C] E(smqa (XC) ([C)|C]=1 TxC)
Now, we prove the doubly robustness. Let
(V; (x(X))) m((X()) + 2 (XC) (7C) (XC)) Unnge (TC)
T2(V;n) Eul - 6(x' ())sma
21

---

Published as a conference paper at ICLR 2026
We note that
(2(V; Sin((C)) Σ E(' (C))
(B.11)
=Vmge(Z()
Therefore,
Ti(V:A) + (V(:
(B.12)
=EXXXX) - mmZXC)) + (xC)))rmeeZZZ umage (XC))
(B.13)
+ (mge(TCC) + (x)) Umige (TC)
(B.14)
(V).
(B.15)
Now, from [[]] = EL (ZXC)m(ZX)),
- -s()
=V(Ti(V:() (v( [
+
)
-[(rx(C))
+ (XC)(((())) - Pmnge (XC) TTa (XC) (XC)
E[[Tz (XC))())) (XC)
Ef(ernae (xC)
Eliinge (xC) - Ez (xXC)))(2X()) + sing (XC) Vinge (xC)).
The first line can be handled as follow:
(Y m] + εzm {mm] = B((E= Ex (m m.).
(B.16)
The second line can be handled follow:
E (rie Pmgee TzVigg TzVingel E[[ Te HVinge Umggel }}-
(B.17)
The third line is handled as follow:
Ei (((ZXC)((ZXX) + Smg Vmage (IC))
(B.18)
B[m(zaC))(( (C))) C) c(x C)}
(B.19)
Blm(===)((21 [[)] C*) é(x C)H
(B.20)
22
Hl)( FC) - ((= (C))((<>) C) - e( C)}]-
(B.21)
This decomposition completes the proof.
B.3 PROOF THEOREM 1
By Thm. A.1, we have
pPDR - +| <
7)(f - 7] V.I.D. (T, ff +]},
(B.22)
since V.I.R (T, m)) - VZJ(T) (i.e., the strong convexity holds).
22

---

Published as a conference paper at ICLR 2026
Define hC) a F(C) T(C). First,
V
DR (T, [h+]] (5a (V:n) + - + ro)
= - (V:n) -
Σ El(pa(V:n) - -
=E(U,1)
Σ (c(B([s=(V:n)] -01=(()))AA(C))
#c(0,))
0.
Next,
(T,
Σ
1E/0,17
Σ =e(0.1}
Σ E
(5(0.1}
+
Σ
E (XCC)((C)
T (XC) mage (2XC) - Ungge (2XC))4,(C)
(E(0,1)
Σ
Ee [m(zz()) (a(z x0) 0(= [0){((x (C) - e(x (C),
BE(U,1)
|| l2
Σ
Im ml
122
| Vinge |1a| Tz 4|
(0,1)
Re (0
B
+ 1 Σ Σ
|x(= EC) a(> [C)laa|(((> C) e(x C
2E(0,1} zu
sa + a
(A+B+C), Va 0.
Choosing = X/4 completes the proof.
B.4 PROOF OF PROPOSITION 2
Define
ex 2 X - e(C)
(B.23)
ez 2 Z aCC) - Xb(C)
(B.24)
ey 2 Y S(XC) Zg(XC).
(B.25)
Then, E[E | [C]
0. Next, consider ez. For (0,1),
E(cz C = Z C aCC) Bb(C).
(B.26)
We note that E[ez 1C 0 since Z X = 1, aCC) + b(C). Also, Elez
OC|
since F[Z| X 0, C] (C) Therefore, E XC] 0.
23

---

Published as a conference paper at ICLR 2026
Next, consider ey.
B Z= 0, E|y -
(B.27)
=m(Z -, m
(B.28)
= 0,
(B.29)
and
Elcy 2 = 1, = E|y 1 - f(XC) g(C))
(B.30)
m(Z =
X, m(2 Z XC) m(Z 0, XC)
(B.31)
0.
(B.32)
B.5 PROOF OF THEOREM 2
We first note that m(XXC)- f(XC) Zg(XC), Define f(C) 2 Ex e(x C)f(xC) and
O(C) a Ezet c(x C)g((C) Then,
= E Σ c(x C)m(Zx() TC
(C)
(B.33)
= Σ c(x ([)(((() + Zg(xC)} &C
(B.34)
[C() Zg(C) EC].
(B.35)
From B[2 5C| u(C) + Rb(
(C) = (EI(() + ZO(C) [C]
(B.36)
DC
(B.37)
= +((0))
(B.38)
This implies that
= ((C)))((X)))) CJ.
(B.39)
B.6 PROOF OF LEMMA
We first prove the consistency result:
B (XC) [1]=[[=x()))](0C)) +ex(C)g(1C) (C).
(B.40)
Now we prove the error-correction property. Define (C) & ε (C), (C) 2 2C), (C) A g(IC), and (C(C) A g(OC). Also, (c(C) A ex(C), (C) 1 ex (C), (C) 2 g(IC), and so(C) e g(OC).
Also, 61 = 1 eo and ē = čo. Then,
ElG- - (XC) C - (C)
ex
= =1 + Eogo + {ar - Hg - ]o) E191 - -090
=1
=0
= -i (ū) ]1} - 9gg + c0 - 9n]
cnāo + ق + Cog0 + {er - ē11 - -
=-C
D-D
=1
=B
"i (ē) )1) + go
-0 60) +er
31 + eo
fer ē1 -
24

---

Published as a conference paper at ICLR 2026
Here,
0 -1) 5o -o)
= = -1 g0fei =1
{e e1 - 9o).
Therefore,
Elsax . 7g(()
= = (ū: - -gi + co -
(B.41)
B.7 PROOF OF THEOREM. 3
We first note that
F - - = 107 by
< 1ō(5 %i. + Ti (6- b)112
$ |l 7/112 + +16 bl12
By Thm. A.1,
||3 - 7 S Rr(C + (V, - v,,2(,,92),,
where, for hy A ||8 - 11.3
(,, = - (XC) -(0))_(()),
This gives that (,) 0. Therefore,
||. - 7.%1 = )t) v,
where
-2[((S, (XC)))(C)))
- -00)))
= +000000000000000H000000000000000
S {allk,(C)11 (C) ((]lbb(0))(((((((()] Ileo ((C)) (0(C) g(0C)
I aa)] (II(1C) + 1(0)))))
Set a A 1 Then,
23-
[C]] 23 (13(( (I5(10) I0(OC) -0
which implics that
1
1 < Rr(e, 23 (17(1)) ((((C)) -( II((CC) - g (())
2
or
|| - - $ Rr(s, [(C)) ((C)) + (0()C) 13
By Error Analysis of LD-R-Learner,
10(XC)- - & Role, + 1ē ezllà + Iyy 11llez ez
25

---

Published as a conference paper at ICLR 2026
We note that
I0(XC) o
(B.42)
= -
(B.43)
E BI[((CC) =(((=)))-(())@(xx==) =z]. xE(0,1)
(B.44)
which implics that
[H 9(aC)) = x) s Re(e, + 1|ēz -zzl + |y mrliillz ezllă:
which implies that, for {0.1),
-((=))=((=O]1)) & Re(c, + 16z ezlli + Il my - - c2llă.
Therefore,
11 19-7%1 7 5 R, + [@(1C) - + l0((0)) - g
$ Rr 7, Re,,,, 71.) + 1| - <zllt + Il my llillz ez113
Finally, by Error Analysis BD-R-Learner,
-(C))) s Ro(e, b, m.) + l|ēx exl1 + fl mzllilex Ex11?
Combining,
|| - -r112
$ 9 - 1/112 + | - bll2
Rr ?. fi=)) Re(c...m. mg) + Re(c, b, 76)
+ ||ēz czlli + Il my @la - ezll3 3 + 1θxx - exlli+1mm - mzllāllx - ex11a:
which completes the proof.
26

---

Published as a conference paper at ICLR 2026
C
SIMULATION DETAILS
The full information on implementation available at https:///githhi httpa////gtthub..oo/oogggggge yongggh ung/ FD-CATE.
C.1 SYNTHETIC SIMULATION
Data-generating process (DGP). For given sample size and dimension d,
C~ N(0, Ia);
U0,10.
Pr(X = C,U) + C,C),
+
Pr(Z = 1 - = a(ao +
Y 0, 0 C 8.Z 0.U tE,
c~,(0,1).
We choose "mooderate-positivity""" coefficients to avoid extreme propensities:
Pg 0.1, 3. 0.7 wx
B. 0.7,
0g 0.1,
0.7 wz, 02 = 1.2,
0g = O, 0₂ = 0.7 Wy,
02 = 1.4, 0₂ = -244
where wx, 12, wy ~ ((0, Ia) are ez-ormalized. The ground-truth heterogeneous effect
Ttrue 0z (00 + a. ax) 0 + a. c))
Learning algorithms and cross-fitting. All nuisances XGBoost (Chen & Guestrin, 2016) with the same configuration: n eestimattorssss = 50, max _ere__dephhhhh = 3, learning rate = 0.1, subsample 0.9, colsample,,ytree = 0.9, l2 penalty = 1.0, histogram tree method. We employ two-fold cross-fitting for FD-PI/FD-DDR. FD-R uses a three-way way split: nuisances on D1, (b, g) via BD-R D2, and the final Yg regression on D3, then swap/ swap/average. average. Final regressions on pseudo-outcoomes (FD-DR) or on the pseudo-g larget (FD-R) use ridge OLS with 10-6, To control variance, only denominators appearing in inverse reights/densig ratios are floored at 0.05; numerators never clipped.
Structural nuisance noise.
To stress robustness, we inject rale-level misspecification al
the n 11/ scale:
p E (e,g): p + / E {my,m): n u
e~ Mnn 1/1 n 1/11)
sweeping the knob 8 E 00, 04,,0,,04,, 1.0). Denominators used in weights frozen before noise and floored.
Weak-overlap stress test. We steepen the treatment logit via
Pr(X = 1| C = a (No + Ke (8. C + BrU),
Ke E (2,4,6,8110)
keeping the mediator regression moderate. This inflates inverse-eeight variance (allecting FD-PT/FD-DR) while FD-R remains variance-friendly (no density ratios).
Design grid and reporting. Unless noted, d 10. (1,000, 2,500,5,000, 10,000, 20,000, 50,000} and use R = 100 Monte Carlo replications. We report RMSE.
We vary
RMSE (I[(F(C) Turee (onue(C)) (C))31) 1/2
with mean + 95% normal CIs across replications.
27

---

Published as a conference paper at ICLR 2026
C.2 REAL-WORLD STUDY: STATE SEAT-BELT LAWS AND FATALITIES (FARS)
Setting and estimand. We study the effect of adopting primary seat-belt law on motor- vehicle occupant fatality rates using a state year panel constructed from the National High- way Traffic Safety Administration's 's Fatality Analysis Reporting System (FARS) and com- panion NHTSA survey tables (National Highway Traffic Safety Administration, 2000). Let C' denote observed state year covariates, X E {0, } indicate whether a primary law is in force, E {0,1} 1} be the observed seat-belt use, and Y the occupant fatality. Our target is the conditional
effect T(C).
The FD assumptions are plausible here because (i) the causal pathway from X to Y operates via increased belt use; and (ii) rich may be sufficient explain spurious paths between X and Z; and Z and Y; and (iii) posilivily holds empirically since belt use neither nor universal in either law regime.
Our result is described in Fig. 3.
farrree_frr___eee fars__r___a_ serrs_2000000 _e___2000_______e py) provided in supplements, we build balanced state year panel and construct (C, X, Y) as follows.
Data and preprocessing. Following the analysis script (analyze_
Treatment X: indicator that a primary seat-belt law active in a given state and year.
Mediator Z: bell-use from NHTSA surveys ({0, 13).
Outcome Y: occupant folatily.
Covariates C: state and year fixed effects and policy-relevanl controls compiled at the
state-year level: coarse weather severity, and road-class mix and drivers' status.
Estimators and learning protocol. We fit the plug-in FD baseline (FD-PI), FD-DR- Learner, and FD-R-Learner exactly as in the synthetic study: all muisances are learned with XGBoost (XGBoost; 50 trees, depth 3, learning rate 0.1, l2 penalty >=1); FD-PI/FD-DR, use two-fold cross-fitting, and FD-R uses three-way split (nuisances + (b,g) via BD-R 79 via pseudo-g). followed by swap-averaging, The final regression on pseudo-outcomes (FD-DR) or on pseudo-g targets (FD-R) uses ridge OLS. To stabilize fimite-ample variance we floor only denominators that appear inverse weights/densite ratios at 0.05; numerators never clipped.
0.9,
28


Reviews:
[{"review_id": "TB8IT3f53w", "reviewer": "Reviewer_YM1c", "summary": "This work presents estimators for the CATE under a front-door adjustment formula. This contrasts with existing works, which focus on a back-door formula. The authors derive variants of the DR- and R-leaners for this setting and provide an error analysis that reveals double robustness.", "questions": "Minor comment on references on line 41-42:\n\n* R-learner was first proposed in Corollary~9.1 of\nRobins, James M. \"Optimal structural nested models for optimal sequential decisions.\" Proceedings of the Second Seattle Symposium in Biostatistics: analysis of correlated data. New York, NY: Springer New York, 2004.\n* DR-learner was proposed in Section 3.1 of \nvan der Laan, Mark J. \"Targeted Learning of an Optimal Dynamic Treatment, and Statistical Inference for its Mean Outcome.\" (2013).", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "To my knowledge, this the first estimator of the CATE under front-door adjustment. This is important since it makes it possible to estimate the CATE in a wider variety of settings than was previously possible.\n\nThe theoretical arguments appear to be sound.", "weaknesses": "The theoretical analysis ends up being somewhat similar to that of existing backdoor methods, but that's no fault of the authors - that analysis strategy works well.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lRE8IA7ix5", "reviewer": "Reviewer_sZgx", "summary": "This paper tackles the challenge of estimating heterogeneous treatment effects (HTEs) under unmeasured confounding using the front-door (FD) criterion. It introduces two novel debiased estimators—FD-DR-Learner and FD-R-Learner—that achieve quasi-oracle convergence rates even when nuisance functions converge slowly at $n^{-1/4}$. FD-DR leverages doubly robust pseudo-outcomes, while FD-R decomposes effects into interpretable causal pathways. Both methods demonstrate superior performance and robustness in synthetic and real-world experiments, providing accurate and stable HTE estimates where traditional plug-in estimators fail. Overall, the study advances causal inference by enabling reliable, personalized effect estimation under front-door settings with theoretical and empirical validation.", "questions": "1. While theoretically appealing, the front-door identification relies on strong and often unverifiable assumptions—such as the absence of unmeasured confounders between mediators and outcomes. As noted by Imbens and others [1], these conditions are rarely satisfied in practice, raising concerns about the empirical credibility and real-world applicability of the proposed methods.\n2.  Figure 3(a)’s histogram suggests that the sampling distribution of the estimator deviates noticeably from normality.\n\n\n\n[1] Potential Outcome and Directed Acyclic Graph Approaches to Causality: Relevance for Empirical Practice in Economics", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper is among the first to tackle HTEs under FD.\n2. The proposed method is supported by rigorous theoretical analysis.\n3. Experimental results demonstrate that the proposed approach.", "weaknesses": "1. The proposed methods and theory are restricted to binary mediators.\n2. The paper does not include empirical comparisons with existing HTE-FD methods (e.g., Chen et al., 2025, LobsterNet).\n3. Beyond double robustness, a key advantage of debiased estimators lies in their asymptotic normality, which enables valid statistical inference. However, the paper does not provide any inference results or variance estimation, leaving the uncertainty quantification of the proposed estimators unexplored.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8YMcpj9BwB", "reviewer": "Reviewer_KeNg", "summary": "This paper considers the front-door criterion to identify heterogeneous treatment effect where there exists unobserved confounding. The authors proposed two estimator for this regime: FD-DR-learner and FD-R-learner. The FD-DR-learner follows the DR-learner framework that constructs a doubly robust pseudo-outcome leveraging FD identification of the treatment effect first, then regresses the covariates on the pseudo-outcome. The FD-R-learner first uses the (backdoor) R-learner to fit some of the nuisances, then constructs a \"pseudo-outcome/function\" from the learned nuisances to obtain the final estimator. Moreover, the author also provides an error analysis for the proposed estimator.", "questions": "- This paper only considers binary mediators, can the estimators be extended to higher dimensional or continuous mediators?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Proposed two novel estimators with discussion on their performance, and when one is preferred over the other. \n\n- Provided error analysis that demonstrated the robustness of the proposed estimators.", "weaknesses": "- Proposed estimators are mainly built on top of ideas from exiting estimators.\n\n- The FD-R-learner requires the estimation of many nuisance functions (has to split data into 3 folds). This could suffer when the sample size is small. \n\n- The paper discussed cases where one estimator is preferred over the other, but did not include experiments to support the claims.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "csGI8e7lXI", "reviewer": "Reviewer_mEQU", "summary": "The authors propose an adaptation of the R-learner for CATE estimation using a front-door structure with a binary mediator (instead of a back-door adjustment which the R-learner was developped for). They fill the gap with prior work that had been focusing on targeted averages, while their method targets conditional effects $\\tau(.)$. To do this, they adapt two learners (DR and R-learners) to the FD setting. They reach quasi-oracle", "questions": "Could you consider a simulation with controlled FD violation to show the behavior of the estiamtor under misspecification?\n\nCan you give more details on density ratios?\n\nWhat happens in small sample sizes? The sizes considered in the synthetic study are very big and FD-PI seems to have better RMSE in lower sample sizes, can you comment on this?\n\nCan your method provide confidence intervals for $\\tau(.)$?\n\nOther baselines could be considered in the experiments such as the oracle estimator and the mentionned Chen et al. 2025 method.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The theoretical properties are clear and the setting too. The subject is interesting and well positioned. The empirical evaluation design seems sound.\n\nI especially like the apparent soundness of the results", "weaknesses": "I found the paper's notation quite extensive and hard to read. Especially equations 1, 2 and 3 need to be described, and more importantly there is a lack of explanation in the. main text of the nuisance functions defined in Eq. 15, 16 and 17. The authors need to explicitly state what $\\bar x$ refers to too.\n\nWhy did you not consider comparing your performances with Chen et al. 2025? \n\nI am not sure whether the environments for definitions, propositions and theorems are in accordance to the conference's template", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)