Efficient and Sharp Off-Policy Learning under Unobserved Confounding
Authors: Konstantin Hess
Dennis Frauen
Valentyn Melnychuk
Stefan Feuerriegel
Venue: ICLR 2026 Poster (2026)
Keywords: policy learning
causal inference
unobserved confounding
partial identification
Primary area: causal reasoning
TL;DR: We develop novel bounds on the policy value under unobserved confounding using the marginal sensitivity model, and derive a semiparametrically efficient estimator.

Abstract: We develop a novel method for personalized off-policy learning in scenarios with unobserved confounding. Thereby, we address a key limitation of standard policy learning: standard policy learning assumes unconfoundedness, meaning that no unobserved factors influence both treatment assignment and outcomes. However, this assumption is often violated, because of which standard policy learning produces biased estimates and thus leads to policies that can be harmful. To address this limitation, we employ causal sensitivity analysis and derive a semi-parametrically efficient estimator for a sharp bound on the value function under unobserved confounding. Our estimator has three advantages: (1) Unlike existing works, our estimator avoids unstable minimax optimization based on inverse propensity weighted outcomes. (2) Our estimator is semi-parametrically efficient. (3) We prove that our estimator leads to the optimal confounding-robust policy. Finally, we extend our theory to the related task of policy improvement under unobserved confounding, i.e., when a baseline policy such as the standard of care is available. We show in experiments with synthetic and real-world data that our method outperforms simple plug-in approaches and existing baselines. Our method is highly relevant for decision-making where unobserved confounding can be problematic, such as in healthcare and public policy.

Published as a conference paper at ICLR 2026
EFFICIENT AND SHARP OFF-POLICY LEARNING UNDER UNOBSERVED CONFOUNDING
Konstantin Hess 1.2," Dennis Frauen 12, Valentyn Melnychuk!2, 12, Feuerriegel 122
'MMU Munich Munich Center for Machine Learning
"Corresponding author: k. hees@ mm..de
ABSTRACT
We develop novel method for personalized off-policy learning in scenarios with unobserved confounding. Thereby, we address key limitation of standard policy learning: standard policy learning assumes unconfoundedness, meaning that no un- observed factors influence both treatment assignment and outcomes. However, this assumption is often violated, because of which standard policy learning produces biased estimates and thus leads to policies that can be harmful. To address this limitation, we employ causal sensitivity analysis and derive prramttrrayy efficient estimator for a sharp hound on the value function under unobserved confounding. Our estimator has three advantages: (1) Unlike existing works, our estimator avoids unstable minimax optimization based on inverse propensity weighted outcomes. (2) Our estimator is semi camammmetttriilllyyy efficient. (3) We prove that our estimator leads to the optimal confounding-robee we extend our theory to the related task of policy improvement under unobserved confounding, i.e., when a baseline policy such as the standard of care is available. We show in experiments with synthetic and real-world data that our method out- performs simple plug-in approaches and existing baselines. Our method is highly relevant for decision decision-making where unobserved confounding can be problematic, such as in healthcare and public policy.
policy. Finally,
INTRODUCTION
Policy learning is crucial in many areas such as healthcare (Feuerriegel et al. 2024 Kraus et al.
2023), education (Chan 2023), and public policy (Ladi & Tsarouhas 22220). However, collecting data through randomized experiments is often either infeasible or unethical. Instead, methods needed that use observational data to inform decision-making Here, we focus on off-policy learning optimize decision policies from observational data (Athey Wager 202I).
The reliability of standard off-policy learning is compromised when unobserved confounding is present (Kallus al., 2019). Unobserved confounding arises when factors affect both treatment choices and outcomes but are not recorded (Pearl 2009). For example, the race of a patient may unfortunately affect the access to treatments (Obermeyer al. 2019). yet race is typically not recorded in patient records. Hence, standard off- off-policy learning that relies on the assumption of no unobserved confounding will lead biased estimates and may even generate harmful policies.
As remedy, confounding -obuut policy learning aims to find the optimal policy under worst-case unobserved confounding. This is typically achieved using the marginal sensitivity model (MSM) (Tan) 2006), a framework from causal sensitivity analysis that bounds the effect of unobserved confounding. However, the existing method for confonnding-robur policy learning under the MSM (see (Kallus) & Zhou 2018a for the conference paper and (Kallus & Zhou 2021) for the journal version) has notable shortcomings. First, it must numerically optimize the worst-case effect on the regret function due to unobserved confounding. Such minimax optimization is based inverse propensity weighted outcomes and hence unstable. Second, this method is statistically suboptimal: it lacks the property of semi-pad cemmmeeraettrcc efficiency and thus suffers from suboptimal variance properties.
In this paper, we address the above shortcomings by developing a novel semi semi- parametrically efficient and sharp estimator for personalized off-policy learning under unobserved confounding. Here,

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thereby Published as a conference paper at ICLR 2026
semi-parametto efficiency means the unbiased estimator with the lowest possible variance. Our key novelties are the following: (i) We derive a ftosed-form expression for a sharp bound on the _ As a result, directly minimize our closed-form bound and, unlike existing works, avoid an unstable minimax optimization based on inverse propensity weighted outcomes. (ii) We propose an estimator that is cemi-parametrically efficient. Hence, our estimator is the first to achieve the lowest variance among all unbiased estimators for our task.
value function of candidate policy under unobserved confounding
Methodologically. we proceed as follows. We first derive a sharp bound on the value function for scenarios with unobserved confounding and, hence, avoid the unstable minimax optimization as in other methods. We then propose novel one-step bias-corrected estimator to achieve semi- -arammerti efficiency and thus guarantee that our estimator has the lowest variance among all unbiased estimators. For this, we derive the corresponding efficient influence function of the sharp bound on the value function. We finally provide theoretical guarantees that minimizing our estimated sharp bound on the value function ensures that our method yields the optimal confounding-roke are particularly crucial in high-stakes applications such as medicine or public policy, where unreliable policies lead to harmful consequences.
policy. Such guarantees
Our work makes the following contributions (i) We propose a novel efficient estimator for our sharp bound the value function. (ii) We derive an estimator for our bounds that is semi- parametrically efficient. (iii) We generalize our theoretical findings to the related task of confounding- robust policy improvement. (iv) Through extensive experiments using synthetic and real-world datasets, we show that our method consistently outperforms simple plug-in estimators and existing baselines.
2 RELATED WORK
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We provide an overview of three literature streams particularly relevant our work, namely, stan- dard off-policy learning (i) with and (ii) without unobserved confounding well as (iii) causal sen- sitivity analysis. We provide an extended related work in Appendix A (where we distinguish our work from other streams such e.g., unobserved confounding in reinforcement learning).
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Table 1: Overview of related methods. Oprescu ct al. (2023) is designed for a different task, and standard policy learners ignore the issue of un- observed confounding. Only Kallus & Zhou 22118|| 202I) deals with our setting, but pro- vides neither sharp bounds nor efficient esti- mator. Only our work can deal with unobserved confounding, discrete treatments, provides sharp bounds, and an efficient estimator.
Off-policy learning under unconfoundedness: Off-policy learning aims to optimize the policy
value, which needs to be estimated from data. For this, there are three major approaches: (i) the direct method (DM) (Qian & Murphy| 201 I) leverages estimates of the response functions; (ii) inverse propensity weighting (IPW) (Swaminathan & Joachims 2015) re-weights the data such that in order to resemble samples under the evaluation policy; and (in) the doubly robust method (DR) (Athey & Wager 202 I Dudik al. 2011). The latter is based on the efficient influence function of the policy value (Robins et al. 1994) and is efficient (Chernozhukov et al. 2018 van der Vaart 1998).
Several works aim at improving the finite sample performance of these methods, for instance, via re-weighting (Kallus) 2018 [021] or TMLEE (Bibaut et al.] 2019). Further, several methods have been proposed for off-policy learning specific settings involving, for example, distributional robustness (Kallus et al. (2022), fairness (Frauen et al. 2024b)) interpretability (Tschernutter et al. 2022).. and continuous treatments (Kallus & Zhou 201 &b) Schweisthal et al. 2023). However, all of the works assume unconfoundedness and, therefore, do not account for unobserved confounding.
Off-policy learning under unobserved confounding: In scenarios with unobserved confounding. standard approaches for off-policy learning are biased (Kallus & Zhou, 201 Sa 2021), which can lead to harmful decisions. The reason that, under unobserved confounding, the policy value cannot be
We use the term "sharp" in earlier work from causal sensitivity analysis (Frauen al. 2023 valid upper (lower) bound of a causal quantity is sharp if there does not exist another valid bound that strictly smaller (larger).
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Published as a conference paper at ICLR 2026
identified from observational data. As remedy, previous works leverage causal sensitivity analysis or related methods to obtain bounds on the unidentified policy value (Bellot & Chiappa 2024| Guerdan
et al. 2024 Huang & Wu 2024 Joshi et al. 2024: Namkoong et al. 2020: Zhang & Barcinboim 2024), which can then be used to learn an optimal worst-case policy. Optimizing such bounds is often
termed "confounding-rohue robust policy learning". However, these works do not consider sharp bounds
under unobserved confounding and do not provide semi- aamii-aammttrillllyy efficient estimators.
Closest to our work is (Kailus & Zhou, 2018a) with extended version published in (Kallus & Zhou 202T). Therein, the authors propose a method for confounding-robuse policy improvement, yet with two notable shortcomings: (i) it is not based on closed-form solutions for the bounds, and (ii) is nor based on semi- paramiaattttrillllyy efficient estimator of these bounds. Therefore, Kallus & Zhou (2018a) 202T) require solving a minimax optimization problem that is relies in inverse propensity weighted outcomes, which is wnstable. Further, their estimator is suboptimal because it fails to achieve parameeria efficiency, meaning it does not achieve the lowest-possible variance among all unbiased estimators. We provide a thorough discussion on the work by Kallus & Zhou (2018) Sa 202T and a detailed comparison to our method in Supplement B
Causal sensitivity analysis: Causal sensitivity analysis (Cornfield et al. 1959) allows practitioners
to account for unobserved confounding by using so-called sensitivity models (Jin et al. 2022] Rosenbaum 1987), which incorporate domain knowledge on the strength of unobserved confounding. As a result, the sensitivity model allows to obtain bounds on a causal quantity of interest, which, if the sensitivity model is correctly specified, can then be used for consequential decision-making
(Jesson et al. 2021).
prominent sensitivity model is the MSM (Tan 2006). The MSM gained popularity in recent years and, for instance, was used to obtain bounds on the conditional average treatment effect (CATE) through machine learning (Jesson al.] 2021 Kallus et al. 2019 Yin et al., 2022). Works like Aronow & Lee (2013): Miratrix & Wager (2018); Zhao et al. (2019) target mean effects under the MSM. rely on Hajek-style estimators which have fimite-sample bias, and are not designed for policy learning. Yadlowsky et al. (2022) study ATE bounds, but their framework is limited to binary treatments. Finally, BBruns-Smith & Zhou consider dynamic policies in Markov decision processes and must estimate ratios of transition probabilities, which yields a fundamentally different estimator that is not sharp for action spaces with more than two actions. Only recently, sharp bounds on the
CATE have been derived (Bonvini et al. 2022; Dorn & Guo 2022 Frauen et al. [222aa] 2023 Jin
ct al. 2023). Other works have considered the estimation of such bounds (Dom et al. 2024 Oprescu al.
2025). instruments for partial identification [Schweisthal al. (2025). However, these works
only consider causal sensitivity analysis for CATE but not policy learning. Further, their works are limited to binary treatments. In contrast, method can handle discrete treatments, and, therefore, requires entirely different influence functions.
Research gap: To the best of our knowledge, we are the first derive a semi-uuramttiically efficient estimator for a sharp bound on the value function using the MSM. Thereby, we enable optimal confounding-robut off- policy learning.
3 PROBLEM SETUP
learning
Data: Let Y E e C R be our outcome of interest, such as the health condition of patient. We follow the convention that w.l.o.g. lower values correspond to better outcomes (Kallus & Zhou 8a). Further, let X X' C Rdd. denote covariates that contain additional information, such as age, gender, or disease-related information. Finally, let A E AA= (0. 1,.., du - 1} be the assigned treatment (or action). Note that we do not restrict our setting to binary treatments but allow for arbitrary, discrete treatments. For the product space, we use D = y x X x A. In the following, we assume that we have access to an observational dataset D,, = ((Y,, X,, A that consists of n i..dd copies of YYX,AA) E DD
robusi golky
laaring
Figure I: We can only block backdoor paths for observed confounders X. Hence, under unobserved confounding U. we can- not point-identify V(T)..
Policy value: Policy learning aims find the best policy for assigning treatments, given covariates. Formally, a policy :T( x) is a conditional probability mass function π : AA x X + [0, 1] with Eaes T(a x) 1, corresponding to the prohability of receiving treatment A a, given covariates
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= z. The value VTT) policy is defined as V(T) = E
= EeEs a(a
X
where Y(a]
denotes the potential outcome for Y when intervening treatment A (Neyman 1923 Rubin 1978). Hence, the policy value (Tr) is the expected average potential outcome when adhering to the policy
Standard off-policy learning: Off-policy learning aims to find a policy æ that has the best policy value among all LI for some policy class 1I. Of note, it is standard in the literature (Frauen et al.) 2024b; Hatt et al., 2022; Kallus & Zhou 2018a) to restrict the analysis to policy classes II with finite complexity such as neural networks.
The value function is identifiable under the following three assumptions (Rubin 1978): (i) Con- sistency: [[A] Y; (ii) Strong overlap: There exists e E (0. such that e S p(A X a)) & 1 v a E A,x E X; (iii) Unconfoundedness:: Y[a) 11 A X A a E A. Then, the policy
value is identified from the observational data via V(T) = E Σne a(a | x)Q(a, x)]. where
Q(a, x) = E|| X = x, A = a] is the conditional average potential outcome function.
The optimal policy can then be learned via
staandard arg min (IT), =Enl
where ((x) is an estimator of the identified policy value. Recall that we follow the convention in (Kallus & Zhou 2018a 2021) that lower Y are better, aim to minimize the value function.
Allowing for unobserved confounding: The assumption of (iii) unconfoundednerss is problematic and often unrealistic (Hemkens et al. 2018): Unconfoundedness requires that the observed covariates X capture all factors that affect both treatment choice and outcome. In this work, we do not rely on the unconfoundedness assumption, which is restrictive and oftentimes unrealistic. Instead, we allow for unobserved confounding, which we denote by a random variable U E U C IR (see Figure 2
Importantly, under unobserved confounding, Instead, we aim to partially identify the value function V(x) by leveraging causal sensitivity analysis. Specifically, adopt the MSM (Tan 2006) bound the ratio between the nominal propensity score
cannot point-identify the value function (mm).
e(a, x) = p(A = a X = x),
(2)
which can be estimated from data D, and the true propensity score
e(a, xr, u) = p(A = X = x, U = u), which is fundamentally unobserved. Formally, the MSM assumes
(3)
e(a, x) e(a, x,
< 1 - e(a, e(a, v.)
<
(4)
for some I 2 1 that can be chosen by domain domain knowledge (Frauen et al. 2023 Kallus et al.
2019) or data-driven heuristics (Hatt et al. 2022) (see Supplement
Intuitively, I close implies that the impact of unobserved variables U on the treatment decision is small, whereas a large means that observed variables X do not contain sufficient information to fully capture the treatment decision. In particular, r 1 implies that the true propensity score coincides with the nominal propensity score. Hence, there is no unobserved confounding and our scenario simplifies the naive unconfoundedness setting. Conversely, if we let I > 1, the true and the nominal propensity scores differ, and, therefore, we account for additional unobserved confounding.
Formally, the marginal sensitivity model gives rise to set of distributions P(T) over D x l that are compatible with the constraints in Equation [ This set is defined as
P(T) - { DE p ((DxU)) / p(d, u) du p(d)vd E D, r
E(A 1 1 (-E(AXX,U) E(A E(A,X,U)
<T
(5)
where P(D x() is the set of all possible joint distributions of the observables and the unobserved con- founders, and where the nominal propensity score ē(A, and the true propensity score ē(A, U) result from & as in Equation 2 and Equation 3 respectively.
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Different from standard off- off-policy learning under the unconfoundedness assumption, can point-identify the value function, and, hence, optimizing the objective in Equation T is biased. Instead, need to account for the worst-case scenario that occur under unobserved confounding. That is, we are interested in: Which policy yields the optimal value under the worst-case unobserved confounding, given our sensitivity constraints?
Objective: Formally, the optimal confounding-robust policy n" is the solution to the minimax problem
π = arg min sup tlll
(TT). SEP(T)
(6)
However, the existing method (Kallus & Zhou 20 2021 for our task has key limitations: (i) It requires directly solving the minimax optimization problem, which can be unstable due to inverse propensity weighting. Instead, we later derive a closed-form expression for the inner supremum (i.e., an upper bound), which reduces Equation to simple minimization task. (ii) This method is not semi-pan cami--prametrically efficient, thus leading to suboptimal sinite-sample performance. As remedy, we later derive an estimator that is ceri-paramtrrically efficient.
4 SHARP BOUNDS AND EFFICIENT ESTIMATION
In this section, we introduce our estimator for sharp bounds of the value function under unobserved confounding. For this, we first derive a closed-form solution for the sharp bounds of the value function (Section which directly solves the inner maximization in Equation 6 Then, we present our estimator for these bounds (Section 4.2), which is based on non-trivial derivations of the efficient influence function offer aani-paramttric efficiency. Further, we provide learning guarantees when optimizing the bounds of the value function (Section 4.3). Finally, we propose an extension of our method for scenarios where the aim is to optimize the relative improvement of policy over a given baseline policy such the standard of care in medicine (Section 4.4
4.1 SHARP BOUNDS FOR THE VALUE FUNCTION
We now derive our sharp bound for the value function under unobserved confounding, given sensitivity constraints P(T) in Equation Recall that our aim is minimize the value function V(=), and, hence, we are interested in an upper bound for V(r). That is, we seek to find the value function in the worst-case confounding scenario under the MSM, which is given by V+ (x) = SUP (IT)..
Jesker sulos
2007
RL
Figure 2: Under unobserved confounding. the value function is unidentifiable, and the ground-truth optimal policy is unknown. Ig- noring unobserved confounding can lead to policy with large regret, and may introduce harm. Further, optimizing w.r.t. subopti- mal bound can lead to an overconservative policy. Instead, we seek to find the optimal confounding-robus policy by minimizing a sharp bound on the worst-case effect of unob- served confounding.
By definition, closed-form solution to this maxi- mization problem ensures that (i) the bound is valid, i.., V++*(=) V(T) for all ÿ E P(T), and that (ii) the bound is sharp, i.c., there does not exist a valid upper bound (+(((T) such that (++(
V+*((T) (x)).
In order to derive V+*((T,, we first introduce the conditional average potential outcome function
Q(a, x) [a] X xr)
which is the expected potential outcome for treatment A a, given covariate information X = T. Importantly, because do not make Assumption (iii) of unconfoundedness, the quantity Q(a, x)) is not point-identified.
We now state first theorem, which provides sharp upper bound of the value function under sensitivity constraints P(T). Further, we also provide the sharp lower bound V inffee((r) V(=).
which we later need for extensions in Section 4.4
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Proposition 4.1. Let Q+ suppep(r) Q(a, 2) and Q=,((,))) inf (rP Q(a, 2) be the sharp upper and lower bound for the conditional average potential outcome, respectively, given our sensitivity constraints P(T). Then, the sharp upper bound suppeP(r) V(x) = V+ (m) and the
(x) ( for the value function (Tz are given by sharp lower bound inf pEP(T) (TT) = =V-
V V=*(()= xx -0 (a, x)T(( xx) dp(r).
(8)
Proof. See Supplement
F.1
Our above derivation of the closed-form solution has a crucial advantage over existing works (Kallus & Zhou, 2018a 2021): we avoid unstable minimax optimization that is based on inverse propensity weighted outcomes, and, instead, we can directly work with ((T), which simplifies Equation 6 to ' arg min V (xT). As result, we have reduced the original minimax problem to much simpler minimization task.
4.2 SEMI-PARAMETRICALLY EFFICIENT ESTIMATOR FOR THE SHARP UPPER BOUND
In this section, derive paramerrmettriclllyy efficient estimator for sharp upper bound V+* (xx) of the value function V semi cammipaaaattrilllly efficient estimators are desirable because they achieve the lowest possible variance among all unbiased estimators (Hines et al.) 2022 Kennedy 2022).
In order derive such estimator of V++*(T) (7r), we first need decompose the estimand Q**(aa
in Proposition 4.1
Frauen et al. 2023 ). Sharp bounds x²) of the conditional average potential outcome (1, function given by
Definition 4.2 (Don & Guo 2022
Q (a. x) = = (a. x) (x) (a, ((..)))=((a))))) xa,
(9)
where we let
(a, = b² c(a, + + b² =1 (
(10)
and
(a. = E (Y, (Y, x = A= ((a (Y, A,
X= =xx
(11)
with
A (v. 0, x) (NSP
A A a, = lv2 (
(12)
where a+ -T/(1+ +) and 1/(1+ and where F (g) the conditional quantile function
(g) inf E Ev: p(V Sy X = x, A = a) 2 q)-
(13)
In order to achieve semi-parametric efficiency for the sharp upper bound / (++((T), we need to carefully take into account the nuisance functions in Equation 88 and Equation respectively. That is, the key difficulty lies in that V+ (TT) depends on several nuisance functions
9
(14) If we followed a naive plug-in approach (i.e., if we estimated i) from data D, and plugged them into Equation 8 and thus Equation 9b. our final estimator (++ would suffer from first-order bias due to estimation errors in the nuisance functions. As a remedy, we present a one-step bias-corrected estimator. That is, we estimate the first-order bias and subtract it from our plug-in estimate (Hess)
n = {e(a, x), F "ontt, (a, x)), ()),
al. 2026 Frauen et al. 2025a Kennedy 2022 van der Vaart 1998).
oc and ply x, a) admit a continuous density bounded away from zero in neighborhood of (a This holds, for example, the density p(y xx a) is strictly positive and continuous. Then, an estimator for the sharp upper bound of the value function is given by
Theorem 4.3. Let Y|
(+)*(T)= = P.. Σ
Taxx [0::x - -e..(( Bexx (bxxx +b* +b*,txx)
TAX (b (b 2xx + b² xxxx)
TAXX
(5xx (5ix- -5xx-xxxx (xxiaaa*)(tt - aa))
(15)
êx
+ [ (YDYAX- - 2xx) + t (Y - )]}
If. top, [[[[[[] co, the above estimator is semi-parametrically efficient.

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Proof. See Supplement F2
We now have semi- parametrrrilillyyyy efficient estimator for the sharp upper bound of the value function under unobserved confounding. Algorithm [ presents flexible procedure learn confounding- robust policies for parametric policy classes He (e.g.. neural networks). For simplicity, Algorithm | only uses a single spit, but we emphasize that cross-fitting can be used increase sample efficiency.
4.3 LEARNING GUARANTEES
Algorithm 1 Confounding-robut policy learning.
In this section, we provide asymptotic learning guarantees in the form of generalization bounds when learning the confounding robust policy via our []gorithm[] Of note, is nor obvious that minimizing the estimated sharp up- per bound (++**(T)) provides a meaningful, confounding- robust policy œ². Hence, provide learning guarantees where we show that, with high probability, minimizing our estimated sharp upper bound yields the optimal policy.
Input: Data D,, {(Y,, (.)) sen- sitivity parameter I' 2 1, sample split (0, 1), learning rate X. parametric policy class He, training iterations K
Output: V++*(())
I: Perform sample split D (Employ cross-fitting to improve sample- efficiency.)
[pn] ((a-p)n)
For this, we show that minimizing the estimated sharp up- per bound (=z) with respect to indeed minimizes the true, unknown value function ((T) on population level.
Estimate nuisance functions D [pm]
V 3: Evaluate v on [(1-p)n]
Fortunately, our method only requires one additional as- sumption, namely, boundedness of the outcome |Y| S Cy. This is very mild restriction and reasonable in practice.
Initialize policy m
(0) E IIg
for l: = 0 to K - 1 do
6: Estimate V++*(= (+)
in Equa-
tion 38 (using evaluated n)
We express the flexibility our policy class
terms of the Rademacher complexity R,, (T). which is a common choice in the literature (Athey & Wager, 2021 Frauen
Update policy parameters:
g(++1) gb)) AV VVVV-T(T (mT (1))
end for
et al., 2024b; Hatt et al., [2022] Kallus & Zhou 20 2018a).
10: Return: Robust policy " + 7
For example, the class of neural networks with bounded
size and smoothness, e.g., with fixed depth, bounded width, and weights controlled by simple norm constraints (so the networks are Lipschitz and outputs bounded), has Rademacher complexity (II) E On 1/2).
Theorem 4.4. Assume (Y| S Cy. Let Cp == 2C,, (1 (1+1-1 + P) and let R₂ (IT) denote the empirical Rademacher complexity the policy class IT. Then, with probability at least 8,
sup TEll
s 2C, (TI +
22
2n
log
(16)
Equivalently, on the same high- crebb-rrbbbbiiityy event, simultancously for all क e
((T) S (++((T)) + +0,, (11) + 3/ / log
(17)
Proof. See Supplement
The above Theorem 4.4 has the following implication: given our sensitivity constraints P(T), estimated sharp upper bound (++.*(m) correctly bounds the true, unknown value function V(T) on population level with high probability. Therefore, given sufficient data D,,, minimizing (++.*(=) with respect to also minimizes V ((T) and, hence, yields the optimal n².
In sum, we have derived (i) novel sharp upper bound of the value function, which circumvents unstable minimax optimization based on inverse propensity weighted outcomes. Further, we have proposed (ii) an estimator for this bound that is estimator with the lowest possible variance. Finally, have derived (ii) learning guarantees, which show that minimizing our estimated bound via Algorithm T inddeed optimizes the true, unknown population value.
efficient, i.e., an unbiased
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4.4 EXTENSION TO POLICY IMPROVEMENT
Our main results from above focus optimizing the value function (TT),, which is common in practice (c.g.. Dudik et al. 2011: Hatt et al. 2022). However, in some scenarios, an established baseline policy To may be available; then, one may aim make small relative improvement yet with certain guarantees. This setting is commonly termed as policy improvement (Kallus & Zhou
2018a 2021 Laroche al. 2019 Thomas et al. 2015).
Hence, we no longer aim to minimize bounds on the value function V(T) but, instead, bounds on the regret of candidate policy against baseline policy (Kallus & Zhou 2018a). Specifically,
can define the regret of policy over baseline as Rm, (T) <(<=)
(To). Hence. a negative
regret implies that policy improves upon To. Importantly, the optimal confounding- policy
x in Equation can also be defined as the policy that achieves the best relative improvement over
baseline mg in the worst-case scenario, that is,
arg min sup TEll PEP(T)
Rmo (x)).
(18)
This definition is equivalent to Equation o Nevertheless, the above objective may be preferred in practice when aiming at policy improvement.
We now show in the following three corollaries that our results directly generalize to policy improve- ment. First, provide closed-form solution for upper bound of the regret function R (T), given our sensitivity constraints P(T).
Corollary 4.5. An upper bound for the regret function Rao (T) is given by (x)
fx Ea (a, x))(a x) (a, x)) dp(x).
Proof. See Supplement F.4
Next, we derive semi- pemmmerramtrrically efficient, one-step bias-corrected estimator, which is based on the efficient influence function.
Corollary 4.6. A semi-parametrically efficient estimator for the upper regret bound is given by
Ret (T) = Σ tPn ΣΣ
Tax Ót, - Ea,x 1T, + b (2,x)
+E(-,+}
+ xxx
+ ग -
6 fi, + +
[ -ct
(Fx (A -a
εA.
+ Exx
- - + ,x,xx (exx(rr8 ( -
where we let + = n and π No for readability.
Proof. See Supplement E.5
Finally, we provide improvement guarantees: given a baseline policy (c.g., the standard of care), if the empirical estimator R,, (iT)* is negative, which we can check by evaluating it, we are guaranteed that æ improves upon яo and introduces no harm.
Corollary 4.7. Under the same assumption as in Theorem 4.4 for any policy π E II and baseline pol-
icy mo E II, it holds, with probability 1 155, that RI, (T) I Rt (T) (IT)+ 2/ 2n log ( )3))
Proof. See Supplement F.6
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10.1
-005 1
0.01
-0.03 0.01
Smadard 13R estimater
=000
-0.71
-0.18
0007
0.01
-007
0.01
0006
0.01
=0.1k=001
Kallus Zhou 2018a 2021
-0.70
-1.22
0.00
-0.16
00.1
0.02
0.01
-0.08
Ellicient + shorp estimater (ours)
-1.12:20008
1.00
1: 0.08
0.89
0.13
0.60
0.64
0.14
0.58
0.17
0.50
0.20
0..00 0.22
Absolute improvement
+0.19
-0.20
-0.49
0.44
0.48
-0.46
-0.40
-0.22
Table 2: Varying confounding strength. We vary the confounding parameter I in the DGP along with the sensitivity parameter T in both our efficient estimator and the baseline (Kallus & Zhou 2018a| 202T). Then, we report the regret over a randomized policy (lower values are better). As confounding increases, our estimator is the only method that is robust and thus performs best.
5 EXPERIMENTS
In the following, we evaluate the performance of our method against: (1) the minimax optimization approach by Kallus & Zhou| (2018a) 2021) and standard methods for policy learning, namely, (2) the IPW estimator (Swaminathan & Joachims 2013) and (3) the DR estimator (Athey & Wager 2021 Dudik et al. 2011D. Importantly, the approach by Kallus & Zhou (2018a 2021) is the only baseline that can deal with confounding our task. To ensure a fair comparison, we use the same neural instantiations for all models in terms of (i) the nuisance functions and (ii) the policy no (see Supp. G All results are averaged over 10 seeds.
policy learning with the MSM and thus the only baseline for
Synthetic Data: As is standard in causal inference liter-
ature (Hess & Feuerriegel 2025] Hess et al., 20244] Kallus evaluate our method on synthetic data
Rogge ove aod 20ky
et al. 2019), in order to have access to ground-truth counterfactuals. Here, we use established data-generating process from
the literature (Kallus et al. 2019): First, we simulate ob-
served confounders X Unif 22, 2] and unobserved con-
& 2011 2001)
founders U ~ Ber 1). The potential outcomes [[a] are then given by Y[a] = (2a- 1) (2a-1)- (2n X) 22U (1 + 0.5X) + where -O(0, 1) random noise. Further, we assume binary treatment, i... de 2. For this, we first fix ground-truie trudd-ttthhhht 1 Then, we let the true propensity score be given by e(1, u) p(a:1yrr> p[a;1=5. where p(z; 7) = 1+ (1/e(1, 1)7, and e(1, x) = (0.75x + 0.5) is the nominal propensity score.
Figure 3: Robustness analysis. We set
I 7 in the data-generating process but use mis-specified sensitivity param- eters I' in our estimator (i.c., I' = is correctly specified, while * 7 is mis- mieeiiie)f..dd. We report the regret over a randomized policy (lower values are better). Our estimator significantly im- proves upon the standard DR estimator, even for completely mis-specified r.
Varying confounding strength: First, we demonstrate the performance of our method for increasing levels of unobserved confounding. For this, we increase the con- founding parameter I* in the data-genn process. We compare the regret of each method over randomized baseline policy. In our method and (Kallus & Zhou 2018a 2021). we set the sensitivity parameter I equal to D*
Our results are shown in Table 2 (i) As expected, the standard methods for off-policy learning (i.e., standard IPW estimator and standard DR estimator) perform well for zero to very low levels of confounding. However, the standard methods are biased and thus become ineffective for I > 1. (ii) The method by Kallus & Zhou (2018a 2021) performs well under low levels of confounding. Yet, the performance quickly deteriorates. (ii) Our proposed method performs clearly best for increasing I Here, our method achieves a relative performance gain by up to a factor of 4.
Robustness analysis: Next. we show that our method is robust to mis- mis-specification of the sensitivity parameter Г. We thus fix the confounding strength to r' (which corresponds to unconfoundedness) up to 100 (which mirrors almost assumption-free bounds). We again compute the regret of our learned policy over a randomized baseline policy to showcase the improvement, Figure 3 shows that our approach yields robust results even for mis-specified I' (i.e., when r + 7). In contrast, the method by Kallus & Zhou (2018a) 2021) improves for mis- specifeee values I < I*, but performance quickly deteriorates for larger I, which renders it unreliable. This behavior is expected, as their method only minimizes a bound on the regret over a bascline policy, and quickly reverts to the baseline when the specified I' increases (see Supplement B for detailed
in the DGP. We increase T from
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discussion). Further, even under the (almost) assumptions constraint of I 100, method provides significant improvements over the biased DR estimator.
pema--rraalllicallly efficient estimation: Finally, we show the benefits of our efficient estimation strategy over simple plug-in estimators. A semi- pammi--aaattrillllly effi- cient estimator is the unbiased estimator with the lowest variance (Kennedy 2022 van der Vaart (998). Hence, policies based on efficiently estimated bounds are learned better in low sample settings than those based on plug-in approaches. Therefore, we report the performance when we vary the number of training samples Dn, Here, we compare our method against a naïve plug-in estimator of our sharp bounds based on Proposition 4.1 We again report the regret over randomized baseline policy. Fig- u shows that our method performs better in low sample settings, and achieves larger performance gains when in- creasing the sample size.
Repeet rur dartazd palay
ohere Eficart extmmate
Namber of sampls fortmininn
Property of semi- parametrically efficient estimation. We compare our efficient estimator with a simple plug-in estimator of our sharp upper bound from Proposition 4.1 We report the regret over randomized policy (lower values are better). Our efficient estimator leads to a lower regret and benefits from increasing sample size duc to its optimal estimation properties. Importantly, this means that there are four different treatments available. The advantage of an RCT over observational data is that we can estimate the ground truth value function without bias, as the true propensity score is known. Our aim to find the optimal treatment strategy based on patient covariates in order to prolong the fime time-to-death (TD) outcome variable (in days).
Figure 4:
Real-world medical data; We evaluate method a real-world medical case study. For this, we use data from the International Stroke Trial (Sandercock al., 2011), which is a randomized control trial (RCT) that examines the outcomes for early administration of aspirin, heparin, a combination of both, or none on acute ischaemic stroke.
For this, we artificially introduce unobserved confounding as follows: In the training dataset, we randomly drop 60% of the untreated patients whose diastolic blood pressure is larger than the average, well 60% of the patients who received aspirin and whose blood pressure is lower than
Ingoveeenea TD RCT paba)
900
800
average. Then, remove the diastolic blood pressure variable. Thereby, we introduce unobserved confounding in the training dataset.
200
Kollaa 7htu 2001)
Results: We report the estimated improvement of the TD outcome of all methods over the randomized policy in Figure We report regret relative to a fully randomized (RCT) policy because, under unobserved confounding, it is the only baseline whose value remains identified and therefore provides neutral, assumption-free reference point for evaluating policy improvement. Here, we vary the sensitivity parameter for both the method by Kallus] & Zhou (2018 2021 and our method. Our method per- forms best at 24. Further, our method has the overall best treatment strategy. Only for very small I, the performance of our method deteriorates this is expected, it does not guard against unobserved confounding and is therefore biased, and in contrast to standard methods, requires estima- tion of more complex nuisance functions. The baseline performance is generally poor. Specifically, we observe a similar pattern for the method by Kallus & Zhou (2018a] (021) as the robustness analysis; it only optimizes against baseline policy, and quickly reverts as I' increases (see Supplement B). In contrast, our method is highly stable for appropriate parameterizations of I', which confirms the effectiveness of our method, and shows its applicability medical scenarios.
(Picked ActIrvatTTT
Burolivyy paramitrr
Figure 5: Real-world medical data. We compare our efficient estimator against the previous baselines based data from the International Stroke Trial. Our method yields the best treatment policy and is robust over different l'.
We develop novel semi parametrically efficient estimator for sharp bounds on the value function under unobserved confounding. Our results provide principled way for reliable decision- making from observational data, and show that robust policy learning can improve decision decision-making in sensitive applications such as healthcare.
Conclusion:
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ACKNOWLEDGMENTS
This paper is supported by the DAAD program "Konrad Zuse Schools of Excellence in Artificial Intelligence", sponsored by the Federal Ministry of Education and Research.
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A EXTENDED RELATED WORK
Offline reinforcement learning under unobserved confounding: Offline reinforcement learning deals with the problem of learning the optimal policy when the reward (value) function is defined over an infinite time horizon (Javurek et al., (026). Therefore, these works rely upon techniques that are different from ours.
Some works focus on off-policy evaluation under unobserved confounding (Bennett et al. 202
Kallus & Zhou/ 2020) and even propose computationally efficient algorithms for this task (Kausik et 2024). However, these methods primarily focus on the identification of policy value bounds without semi- sami-paamettrcalll efficient estimation procedures. In contrast, |ennett al, (2024) propose an efficient estimator for offline reinforcement learning. Different from our work, however, they require estimation of density ratios in order to evaluate the policy value. Further, Pace et al. (2024) propose a heuristic approach that learns representations of the unobserved confounders but does not provide theoretical guarantees for efficiency or unbiasedness. [hh et al. (2022) propose an approach that involves the approximation of bridge functions in partially observable Markov decision processes (POMDPs). Additionally, Shi et al. (2024) use mediators as auxiliary variables to construct confidence intervals for policy evaluation under unobserved confounding. Finally, Waang al. (202T) improve sample efficiency in offline reinforcement learning under both observed and unobserved confounding.
Frauen al. 2025b) develops retargeting strategy for the conditional average treatment effect to balance estimation error and decision performance.
Further, Rambachan al. (0255) study slightly related but different problem setting: a binary selection decision 1) E {0. with binary selectively observed outcome Y*, and they derive sharp bounds on prediction and fairness metrics (e.g. risk, ROC, precision- recall) under Mean Outcome Sensitivity Model (MOSM). The MOSM directly bounds the outcome gap
8(z) = P(Y* = 1| D = 0, = x) - P(Y* = 1 D =11x=
(19)
and their sharpness guarantees hold relative to this MOSM.
Our work concerns offline policy evaluation and learning with (possibly) multi-valued actions A E A and general outcomes Y. under a Marginal Sensitivity Model (MSM) on treatment assignment. We target the policy value (Tr) and derive closed-form, sharp MSM bounds together with semipara- metrically efficient based estimator.
Importantly, RRambachan et al. (2025) show that propensity-based models such the MSM imply particular MOSM (via outcome bounds derived from Bayes' rule), but the resulting MOSM "may not exhaust all implications" of the original MSM. Thus MOSM bounds are generally valid but not sharp for the MSM. In contrast, our analysis works directly at the level of the MSM and characterizes the sharp MSM-identifel set for V(x).
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B COMPARISON WITH KALLUS & ZHOU
(2018A) 2021
B. TECHNICAL SUMMARY OF KALLUS & ZHOu (2018A 2021
Kallus & Zhou (2018a 202T work under the same marginal sensitivity model as we do, specified via an odds-ratio bound I between the nominal propensity e(A X) and the true propensity e(A X, U). For a fixed policy and baseline policy To, they define IPW/Hájek-type estimator for the advantage of π over no. say (TT, Tn) for samples E (1....,n) n and consider the set of feasible weights (W1,- (w.....; Wx) implied by the marginal sensitivity model:
Ur = {w: : w. (T) I S T. (T), 1
Σ
W. 11.
(20)
1=1
Given Ur, they define the worst-case regret of π relative to To as
E W/ A2 To)
(T( mo) = sup
(21)
wEur
2i=1
and then select the policy that minimizes this worst-case regret. In the bintry-action implementation, the inner optimization is solved by sorting the (T mo) and exploiting the fact that the linear- fractional program over Ur attains its optimum at an extreme point: there exists single index k* {1,... (1..... n such that units on one side of k* are assigned their upper feasible weight and units on the other side their lower feasible weight. Hence the inner supremum reduces to a one- mie-dimensional
search over to find h=*, which Kallus & Zhou 201 8a 2021 implement via ternary search.
B.2 OUR APPROACH AND KEY DIFFERENCES
Our work builds on the same marginal sensitivity model but differs in several technical aspects:
Objective: regrel vs. value bounds. Kallus & Zhou (01 Sa 202T) optimize worst-case regret relative to a baseline policy, which is sharp for that regret quantity but does not provide closed- form sharp bounds on the value V(T) itself. In contrast, we derive explicit closed-forn sharp upper and lower bounds on V(r) under the marginal sensitivity model and directly optimize these bounds over the policy class.
Propagation of IPW instability. In Kallus & Zhou (2018a: 2021). the worst-case weights are determined by single threshold k. the sorted IPW/Hajek Wa--aeae--bsedd scores X; Because k- is global, small estimation errors in the inverse propensities (and hence in the X;) can change which constraints bind in the inner problem and thus alter k*. This then leads to a global change in the worst-case weights assigned to all units. Hence, local IPW noise can directly propagate into large changes in the estimated worst-case regret.
Closed-form sharp bounds and local ervor propagation. Our estimator avoids this discrete minimax search and global thresholding. We derive sharp value bounds in closed form and estimate them with semi- paammerrrattrrilllyy efficient, one-step bias-corrected estimator. That is, nuisance estimation errors then do not enter to the final bound estimate as first order bias, but only as lower order biases. Further, there is single threshold index k* in our method whose mis-estimation can flip the worst-case assignment for the entire sample. This yields substantially stable behavior in practice.
Sharpness and efficiency. While Kallus & Zhou 20188 2021 show sharpness for the regret interval relative to a baseline, our results provide the globally sharp identified set for V(x)) itself. Moreover, one-step estimator based on the EIF is cemi-parametrically efficient and Neyman-orthogonal with respect to the nuisances, whereas the Hajek//PW-type estimators used in Kallus & Zhou 2018a 2021) do not achieve this efficiency bound.
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Sharp regret does imply sharp value. The Kallus Zhou method yields the sharp identified set for the regret R(P) (T;; P) P) over P E P(T), but this does not imply sharp bounds on (T)) itself. The distributions that maximize regret and those that maximize the baseline value generally do not coincide. Consequently, the implied value upper bound satisfies only the loose inequality
sup ((T; < sup (V(T;P)- (TTO; P)) PEP(T)
sup (TTO: P), PEP(T)
(22)
PEP(T)
and this inequality is typically strict. Hence, even a sharp regret interval does not translate into sharp identified set for (TT). In contrast, our Theorem 4.1 directly characterizes the closed-form sharp bounds for (xx under the MSM.
Overall, we move from numerical minimax regret based on unstable IPW weights to closed-form sharp value bounds with efficient, sineenceffnnctionbare estimator that is both statistically and numerically more robust.
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C ADDITIONAL RESULTS: COMPLEX PROPENSITY SCORES
In this section, we provide additional experimen- tal results that illustrate the mechanism by which our efficient and sharp estimator improves over the method by Kallus & Zhou 82 2021 In particular, we show how estimation emrors in increasingly complex propensity score models
Rogret randombzdd polcy
propagate differently in the two approaches.
Experimental setup:
We extend the synthetic
Kalus (2018; 2021)
data-generating process from Section 5 beyond binary actions and allow for
Efficient sherp exlimwior (ours)
10
Dimension of treatment space
{0, do 1).
(23)
Figure 6: Performance comparison between our efficient and sharp method and Kallus & Zhou (2018a 202I) under increasing dimension da of the action-space.
We fix the unobserved confounding r
and run both methods with the same correctly specified sensitivity parameter I 5. Increas- ing du substantially complicates propensity esti- mation: estimating e(a X) becomes multiclass classification problem, and the resulting inverse- propensity terms become noisier as d, grows. This provides an ideal setting to study how each method handles nuisance complexity.
Results: Figure summarizes the results. The findings align with theoretical discussion in
Supplement
Our method remains stable. Our estimator maintains tight value bounds and stable performance d, increases. The one-step correction based on the efficient influence function ensures that first-order sensitivity to errors in the estimated nuisances is eliminated. Consequently. even as the propensity scores become harder estimate, the resulting bounds remain well behaved and lead an increasingly better, Confounding-robug policy.
Kallus & Zhou (0018a] 2021) become increasingly conservative. In contrast, the approach of Kallus & Zhou| 2018a] 2021) again reverts to the baseline policy. Their method computes worst-case regret relative to To via a minimax optimization over feasible weights Ur:
Sill w;Ar(
R (T(T)))) = sup
Σ U/
wEUr
where the AI(TT, mo) IPW/Hajek-type advantage scores. The optimizer selects single global threshold k* (via ternary search) after sorting the X; to determine which observations receive lower versus upper feasible weights.
When d, grows, the estimated A; become highly variable due to propensity estimation noise. This causes the inner maximization to lock onto overly pessimistic weights in which many non-baseline units receive their lower feasible weight. As a result, nearly all non- baseline policies appear unsafe, and the learned policy collapses back to no. even though the true confounding level is correctly specified.
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D IHDP DATASET
AEAL -022 033
-422
-6224-11
-022
-12
Repormmntt
Table 3: Regret over randomized policy for varying sensitivity parameter I. Lower values indicate better performance. Absolute improvement is computed against the best-performing baseline.
The IHDP dataset is widely used semi-synthetic benchmark based on the Infant Health and Development Program (IHDP) randomized trial. Here, the effect of home visits on cognitive test scores for infants are measured. Covariates are taken from the real RCT, and potential outcomes are simulated following Hill (2011).
Dataset:
Similar to our real-world data study in Section S because the original study is unconfounded, we can introduce a known amount of unobserved confounding a controlled way: here, we select covariates birth weight (indicator of neonatal health), head circumference at birth (developmental marker), and weeks of gestation (prematurity measure), and drop 60% of observations whose value for any of the three covariates is below average and has received home visits, as well as 60% of the observations whose value is below the average and has received no home visits. Then, we drop the three covariates and, thereby, introduce unobserved confounding. This mimics realistic setting where key health indicators are absent from the recorded data but nevertheless influence treatment decisions and the outcome variable.
We evaluate the regret of all methods relative to a randomized policy because, under unobserved confounding, is the only baseline whose value remains identified, whereas any non- randomized policy would implicitly rely on violated unconfoundedness assumptions and thus provide a misleading comparison.
Results:
The results in Table closely mirror the patterns in our previous real-world experiment in Section 5 As expected, the standard IPW and DR baselines are biased and completely insensitive the choice of T, since they assume unconfoundedness regardless of the true data-generating process.
The method Kallus & Zhou (2018a 202T) again behaves overly conservatively. In particular, effectively reverts back toward its baseline policy and produces regret over the RCT baseline that cluster near zero. This, again, shows that the worst-case optimization collapses under even moderate confounding.
contrast, our efficient and sharp estimator consistently achieves the lowest regret across all values of r. This demonstrates that method computes meaningful bounds on the unidentified value function and, on top, finds optimizes toward the global best performing, confounding-robee robust policy.
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E CHOOSING THE SENSITIVITY PARAMETER IN THE MSM
In this work, we adopt the MSM Tan 2006 in order to bound the ratio between the nominal propensity score
e(a,z) p(A= =a x
(24)
and the true propensity score
e(a, 2. = = -
(25)
Here, the nominal propensity score can be estimated from data, whereas the true propensity score fundamentally unobservable. In particular, the MSM is given by
c(a,x) e(a, x, u)
- 1- e(a, x) e(a, x, u)
T
(26)
for some sensitivity parameter r 2 1.
Typically, the sensitivity constraints I' are chosen via domain knowledge (Frauen et al. 2023 Kallus ict al., (2019) or data-driven heuristics (Hatt et al. 2022). For example, in practical applications, typically has benchmark variable (e.g.. hours with sunlight) that is known cause of the outcome (e.g.. vitamin deficiency), and one then wants to study how strong a confounder (e.g., other ecological activities such nutrition) must be explain away the effect of the benchmark variables. (Cinelli & Hazlett 2020) term this the robustness value, which quantifies the strength of unobserved confounding needed change conclusions.
Hence, to achieve this, a commonly used strategy for selecting I' is the following: We can search for the smallest such that the partially identified interval for the causal quantity of interest includes Then, we can interpret I as a measure of the minimal deviation from unconfoundedness required to invalidate the effect of intervention (Jesson al. 2021 Tin et 2023).
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F PROOFS
F.1 SHARP BOUND OF THE VALUE FUNCTION
Proposition F.1. Let Qh. (a, x)) be the sharp upper/lower bound the conditional average potential outcome, given our sensitivity constraints P(T). Then, sharp bounds for the value function (xx) are given by
V+)(()= / E (a, xx
v( (0,
dp()).
(27)
Proof. We provide the proof for the sharp upper bound V+* (Tr). The lower bound follows completely analogously by swapping the signs and replacing the supremum with an infimum.
We start by noting that the upper bound the value function depends on the set of admissible distributions P(T) induced by the sensitivity model, that is,
V+* (TT) = V (T, P(D)).
(28)
Hence, we can write
(++((T)
(29)
-V+.((T,(IT))
(30)
sup DEP(T)
V(T,ğ)
(31)
sup PEP(L) x
Qua, x, p) <aa d5(a)
(32)
sup (EP(T) x
Σ
Q 2t x. () <aa x) dp(x),
(33)
where Equation 33
follows from the equality p(D) = ((D) for all j E P(T).
Clearly, by definition of the optimal bounds Q (a, 22), we have that
Q x. P) < sup
Q(a, x)
P(T))
(34)
PEP(T')
for all ý E PPTT), and since Q+* E T. P), we know by dominated convergence (Fraeen ct al. 2023) that
sup PEP(T)
Σ Ola, T(a d)) dp(x)
1 a(a (x) dp(x).
(35)
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F.2 EFFICIENT ESTIMATOR OF THE SHARP BOUND OF THE VALUE FUNCTION
On the construction of paam-earamttricllly efficient estimators. Let (PP) V+* P)
denote our target functional in the nonparametric model, and let T(P) be the corresponding tangent space. For any regular parametric submodel {P') through P with score (2 = 0g log Ps (2) the pathwise derivative of y at P in direction E T(P) is
d (P:)
$ r((s = de
1-=0
The canonical gradient g* E (P) is defined as the unique element such that
r(() = P [g (Z) (2)] for all E T(P):
(36)
i... 9" is the Riesz representer of the pathwise derivative functional (see, e.g., van der Vaart 1998).
Chapter 25).
In the following. derive the influence function of V++*((T) via the chain rule for pathwise differentiability. Specifically, we decompose V+*((() into building blocks (conditional expectations, conditional distributions), plug in the known canonical gradients for each of these primitives, and then apply the chain rule as in Kennedy (2022) and Lemma S3 Luedike 2024). By construction, the resulting function gp(2 (Z) satisfies
d de
(P.)
= (B)(p(Z)s for every regular submodel {Pe}.
(37)
Hence yp is exactly the canonical gradient 9", and therefore the efficient influence function in the nonparametric model. By standard semiparametric theory (e.g.. van der Vaart (1998)) the one-step estimator based on this yp is semiparametrically efficient for V++'(T).
Theorem F22. Let E B||y]] sc and ply x. a) admit continuous density bounded away from zero in neighborhood of Fzza(at) This holds, for example, if the density ply Iz a) is strictly positive and continuous. Then, estimator for the sharp upper bound of the value function is given by
Pa,x ++)*(())= = Pn Σ Ta.x 10txx - Ea.x -eaxx (bpax + +b+ pxxxxx,)) )]
+ + + (b-At,x + +
+ TA.X
[5 - -exx)
- -at)
(38)
+ + (vÂt.A.- (ixxxx-Exxxx
(tx) +
txxx
- i - )1x
If. on top, B < oc, the above estimator is semi-parametrically efficient.
Proof.
The sharp upper bound of the value function is given by
*=*((T) :
/ E (o,t"(a,,))itt (), dp(x).
(39)
In the following, we derive the efficient estimator for this quantity. Therein, we make use of the chain rule for deriving efficient influence function (Kennedy) 2022). A proof of the validity of the chain rule for deriving efficient inftuence functions is provided by Luedtke (2024) (Lemma S3).
In order to avoid notational overload and for the sake of clarity, do not additional variables such as b², c², A n=t etc. until the final steps, such that the derivation becomes casier to follow. More- over, make the dependency nuisance functions 7/ S x),a, F-3 (a+)) a² (a, (),,), at (a, explicit by writing, for example, ++*((1;)) for V+* (T))
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The influence function of V+*** (I:: is given by
IF (T:
(40)
E @t (
x)) dp(x))
(41)
Σ
/ x(a x) UI (PD))) de
(42)
Σ x
/ x(a ((aa=))) x) IF (t
(a, + a(a x))(() IF (a, de
(43)
Σ / x(a I |x) -1x((x))--(())
p(x)) Q (a) dx + Σ aaa x)) p(x) IF Qt (a, x; m) dx
x
(44)
Σ x(a
- (=;)) + E/ M(a p))( IF (Qt dæ (45)
Hence, in Equation 45 we are left to compute the efficient influence function of Q+ the sharp upper bound of the CAPO. With at = r11(1 + r'), the sharp upper bound Q+* (a, x) is given by
(a, x). that is,
Q - Q+*(aa,;; x3,))
(46)
r-1 E Y IIs |X x, 2, A q a) + (1--))((a, x) + +T
(1---
(47)
E y1 (2eestoo) |x= ==
(48)
Hence, the influence function is given by
IF (at""(a,:;) m)
(49)
III (1 - r )e(a, ((ra,))+(=))) E ( XX= x, A-
(50)
(4)
+ (1- )e(a,,))))) + - IIF E vI
(Y
x== x, A=A==a) a)
(51)
(6)
+ IF (1- - (1-]))((,))) + r) R. Y1
(12F2200))
xxxAA== =
(52)
+ (1-a)a x) + +T) IFF E Y1
(53)
(d)
where we can apply the chain rule to the conditional expectation of the truncated first moments of Y since Y is integrable by assumption. We start with (a) and (b). For (a), we obtain
IF (1-1 ()),,)))++1 (-))
(54)
=(1 -11-1-1 r |IF (e(a,z))
(55)
=(1 -(1-r-1 1 ((x=x) (((A=o)e((a,x)))
(56)
p(x)
and, similarly for (b), we yield
IF (1 - ())a +1'
(57)
=(1 ---r) r) 1 (x==] (I(A=a) e(a,z))
(58)
p(r)
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Next, we compute the influence function of (c) via
LY xx= x. A 2,A=a) a))
IF
E
YI
(59)
-I
1 (USFF uply I,, a) de)
(60)
IF
(41ssPadtatn) uply a. + 1 (us 3IF
(p(s ppl z. a.)) dy
(61)
IF
(1 (1((FFEd (n uply dy 1 + /
"(uSFiddaa)) y IF (ply I x, dy-
(62)
(c)
(02)
For (c1), we first note that
IFF Faalat))
(63)
III
(P(Y xx=
x, a))
(64)
-TIF
(R(1
X x==,A=10 ai)
(65)
(x=xA==))
(L(rso)- - (r |x= 2, 2.A=
(66)
x(a,x)
(X=e,A==)
((YS5)) - Fna(v))
(67)
p(a,x)
(+)
Then, we can simplify (e1) via
/ IFF
[ (LusprrA(at)) yply x,
(68)
/ 8 {V Padd(at) IF (izllat) yoly x, dy
(69)
IF (Falltt) / 6( Ped(at)) uplu I dy
(70)
MF FZzd (2")) / (v- - Fad((a)) uply x dy
(71)
(g) d
IF (Faulor)) / 6/v- - Fnd(at)) yplv x. dy
(72)
dg
IF (Paal(")) / 8(y - Fad(at)) yply xr, a,
(73)
Fy. FFd(Fnn
1
IF (FealV")) / ((v- Faddla*)) uplu d.
(14)
x, ply"
1
1(x=nd
((rsw) Faalv") / 5(y y')u(l x. ay
(75)
p(u"1
x,
p(a,x)
1[x=x,==))
(1orsw)-o (a00)-aa)) -at) / 8(y - uuuply x,
(76)
plu"
x(a,2)
1
1(x=2,==)
(L - at v² plu" 22,a)
(77)
plu"1 Ixaa)
p(a,2)
1[x=x,A=al F..d
(a ({1 (L -aa at),
(78)
p(a,x)
for some 3² E y such that Fa,a = at, where 5(-) is the Dirac-delta function, and where use in Equation 73 that p(y|0, a) is continuous density strictly bounded away from zero in
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neighborhood of (a which guarantees that the conditional quantile is unique, and where Equation 74 exists by positivity of the conditional density.
Next, we simplify (c2) via
/
1 (vS ((SFF yIF (ply 2, a)) dy
(79)
y
1 (VSFFdd(att) y IF (( Xx= xx= A al) dy
(80)
(x==AA=))
Jy (USFrid(at))
(( - ply x, a dy
(81)
p(a,z)
1 (X=u,AA=a) x(a,x)
(Y (M(Sddda)))) R[y 1 xx= x, A ==)= al)
(82)
Then, combining (cz) and (c2), get
IF
IE (I(ISEcchaa)) 1x= x==xA=aa)
(83)
(x==,A==))
(MIssddan)EEY EY py |X= x, A= a) + Fz (rgF.
da -a= a²))
p(a,x)
(84)
Finally, we compute the influence function of (d) analogously to (c) via
IF E Y1 (V2 X = x, A A- a))
(85)
IF (1(1102 uply x, dy+ I, 1l 1122 y II (ply xr a)) dy
(86)
(d2)
Again, we start with (dz) via
/ IF (A(22Fdd(a)))) yply x, a,
(87)
IF (11 uply dy
(88)
/ IF (I (1(0SPEttiat)) uply dy
(89)
(X=z,A=a) (- F-d((ta)))))) (1 at).
(90)
2(a,2)
using the result for (c1) in Equation
78 Further, for (dz), we get that
J 1 (12 ( y IF p(v x,a)) dy
(91)
(X==nA==) p(a,2)
(I(y2matatt) E|Y (12 |x= -x=AA= =x,A= al).
(92)
Combining (d1) and (dz), we obtain that
IF
E Y X x, ==x,A=d))
(93)
(x==nA==)
(Y
E|
(Y2
|X= x, aj - (a) ([
- -a+))
p(a,z)
(94)
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Finally, state the influence function of Q+*((a,z) by combining (a)-(d) in Equation 56
Equation 58 Equation 84. and Equation 94 We get
IF (tt"((a;;;))))
(95)
( (x==) (1 1-1 (1r()(((A))) - x() B Y1 (YSFy -x= x. a
(96)
p(x)
{[x=xAA=u (1-I~))((.x)+
+ p(a,z)
(97)
(YSPada))))) - -E (VsPx 1 = =x=x -- al F (1 (1
- -a+)
(98)
+ (X=x) (1 -
- (((a)) B Y1 (128
| xx= x, A=
(99)
p(x)
1 (X=zA==)) (1-T)c(a.z)+T)
(100)
p(a,z)
Y1 YL (11
(22 X -x= -x=xA=- == - (( - -a*)
(101)
In order to simplify the above lengthy equation as in our main paper, we introduce the following variables:
g = (1-
(a, = ((te(a,)2)) + n=1)
A+,,,,;;,) 1 (USFF)d(at))
(+ 1 1v28 (a+)))
F (+,,,;)) = E| (Y, =x= = A A= = d)
x (a,x,))) = E (x |x= =
Then, Equation 101
simplifies to
IF
Q (a,2;n)
(102)
(1d==a--((o (0, (0,2;
(103)
p(x)
+ p(z)e(a,2)
c
- (a,,,))) (
(A (Y,
a,,,, a+))
(104)
1x==s)
b+ - -(ax))))(
(105)
p(a)
1x==a==a)
(1 (V - a+))
+
p(z)c(a,x)
(106)
(a, (0 = + (
[(x=)) {{
(107)
p(x)
+ [
(c(aaz)) (F (Y,a, (1.
(108)
e(a,z)
+c +o =a (Y.
(a... pt
(109)
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Finally, we can combine Equation 45 and Equation 109 V+** (अ) is given by
That is, the efficient influence function of
II (T;:)))))))))))))))){)))))))((
(110)
Σ taa -a.)
(111)
+ Σ x(a x) (A=a) - c(a,, x)) 1a, + t pi (a,
(112)
+ Σ x(a x)
[
x(a, (a (a, (a. (tt_ (F=))
(113)
(a - ( (x(a..::))(x(XX,...
(114)
= =- (T) + T(a -a (a,xi)) bbb (a,X:n)) (115)
+ * (X) X) b M (1
(116)
(AA (A 01
(117)
x)
+ c (:, - (/4
(118)
We can derive the efficient estimator for the bounds of the value function through one-step bias correction via
(T: f) + Pm (+++(
(119)
Σ x(a| x) (0
(120)
+ *(A x) (1(77(((.
(121)
+ T(A] (5 ( -at)
(122)
8(A,X)
+ c (A, (A) (YA - -
+ + - (A,)):)))
(vá+ X; - jut (A, x: }]
tA
(123)
=P, Σ Ta.x Qtixx -
êa.x b Lt,x + + xtxxx) )] + +TAX (D Lt, + +bx
(124)
+ TA
(5xx-- -
(Fx - -at) + + (rDr.xx - =xx) + + - p}xx)}}
using our short-hand notation from the main paper.
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F.3 LEARNING GUARANTEE: VALUE FUNCTION
Theorem F.3. Assume [Y] S Cy. Let C. = 20,, (1 (1+1-14 + 1) and let Ra (II) denote the empirical
Rademacher complexity of the policy class II. Then, with probability at least 6,
sup ((())-
S 2C, (TT)
+
5 2 2n
log
(125)
TEll
Equivalently, the same high- probbbbbbbiiiity event, simultaneously for all π E II,
V(T) S (+) + =2,, Rn + 2 2n
5 1
log
(126)
Proof. We start by bounding the sharp upper bound V+* (r) of the value function. By assumption, have that YY| & Cy. Hence, for any IT, we can bound (xt) via
I (T))
(127)
/ Σ (a, x dp(x)
(128)
< sup (x.))XXA
(+.((,,))))))
(129)
sup (x,a)Xix
+ (1-=
X= x, A =d a]
(130)
(1- +1) E Y1 (122)d((tt)) Xx= 1x=r,A=q]
(131)
+ (1-
())a ()) +T))
s sup
(132)
(x,))xxx
sC, 2 + 21 + 2r)
(133)
=Cv-
(134)
Now, for the main result, we seek to find an upper bound for
sup
(++**()) - V(*).
(135)
*ell
By adding zero and sublinearity of the supremum operator, we have that + sup (1)) #ell
sup マサ (TT) (TT) S sup nlll
(136)
well
Further, by validity of our bounds, we know that
sup
V+ (x)- V(=)) 2 0,
(137)
x
such that we only need to focus on
D = sup V+++(T)) - (T).
(138)
sell
Since
1
(x)
Q (Xi, a) X.)
(139)
sample average with |V: (T) C.,, know that D satisfies the bounded difference with Cv/n. Hence, we can apply McDiarmid's inequality (McDiarmid 1989), which yields
2 n
(DD-E(D) 2 S exp
(140)
C
=P1
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Then, solving for gives
C
V
log
(141)
2n
Pl
Hence, we know, with probability at least
P1. that
C2
D E[D] +
log
(142)
2n
pl
Frauen et al. 2024b| Hatt et al., 2022 Kallus & Zhou 2018a), we express the flexibility of our policy class II in terms of the Rademacher complexity. For this, first note that a standard symmetrization argument yields
As in (Athey & Wager 2021
D
S
E
1 2
Σ
sup xell
Σ V.V (T) ==1
(143)
oE(-1,+1)
where oi <vid Unif(-1,++] Then, using the Rademacher-comparisso lemma Ledoux & Talagrand 1989), we have that
B|D] S 2C, Ru (1))
(144)
where
R, (II)
E,
sup xEll
(145)
the empirical Rademacher complexity of policy class IT. Again, R. with 2/n, such that we can apply McDiarmid's inequality (McDiarmid
satisfies bounded difference
1989). This gives
2n
P (II E(R., (II) 2 c) s exp
(146)
=p2
Solving for yields
2
log P2
(147)
such that, with probability at least
P2, we have that
B Rr (T)
(IT)
R. (IT)
2 log
(148)
P2
Combining Equation| 148 with our previous result in Equation 142 P1 P2. that
have, with probability of at
least
sup xell
( - V++((T))
(149)
C2
2 log
22C,RM (II) +
log
+ 2C
(150)
2n
p1
n
P2
2
1
-2,, (1)
log
log
(151)
8n
p2
Now let P1 = P2 = 5/2. Then, we know that with probability least 1 8,
sup rell
(152)
S2C, (IT) + +1 V 8n
1 + +V
1/2
2
log )
log
(153)
5
2 (2))
-2C. (II +
log
(154)
2
2n
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or, equivalently,
(T) S (+*(()) + 20, Rm (n,(II)
+ 2\ 2n
log
(155)
for all æ E II, which concludes the proof.
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F.4 BOUND OF THE REGRET FUNCTION
Corollary FA.. Given Qt.'" (a, (a,x) and our sensitivity constraints P(T) as in Proposition 4.1 an
upper bound for the regret Sunction Rx (TT) is given by
R (T) x //E(((0 ( (a,x) (a (a) x)) dp(x).
(156)
Proof. Our proof follows similar steps as in the proof of Proposition 4.1 For clarity, we repeat the same steps such that the proof is self-contained.
Again, we start by noting that the upper bound on the regret function depends on the choice of our sensitivity constraints and, hence, the set of distributions P(T). Therefore, we can write that
R
R (I, P(T)).
(157)
Following similar steps as in Proposition we can write
R (T)
(158)
= -R (TT, P(T))
(159)
sup
(T,
(160)
PEP(T)
sup (EP(T) (V(T, V(To:p))
(161)
sup (EP(T) /
Σ (Q(a, (,a,,,,)T((L 2(a,,, Mmol (z)) dø(x)
(162)
/
sup PEP(T)
Σ (Q(a, ((la,,,)T((( x) Q(a, x, p)mola (x)) dp(x),
(163)
where Equation 163 again follows from p(D) = (DD for all p E P(T).
Since the optimal bounds (a, are those Qa, p), P(T), for which the supre- mum/infimum are attained, we have that
x, sup Q(a, = + "a, P(T))
(164)
PEP(T)
and
Q(a, 2 inf 2(a,,,,)) = P((T))
(165)
PEP(T)
for all 5 E P(T). Then, since (a, T (T. it follows by dominated convergence that
/ Σ (Q(a,x.
sup
x, ma x)) dp(x)
(166)
PEP(T) x
= =/
sup (EP(T)
Σ (Q(a,
Q(a, x. x. p)T x)) dp(x)
(167)
< Σ sup
Q(a x. ()m(a x d)dp()) / Σ inf
Q(a, x, (a x)) dp(r)
(168)
x PEP(T)
PEP(T)
Σ (Qt (Qt"t(a - -a, x)) dp(x),
(169)
where Equation 168 follows from the sublinearity of the supremun/infimum operator.
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F.5 EFFICIENT ESTIMATOR OF THE REGRET BOUND
Corollary F... Let ( < &C and ply v, a) admit a continuous density bounded away from zero in neighborhood of (at) This holds, for example, the density pyy I a) is strictly positive and continuous. Then, an efficient estimator for the upper bound of the regret function is given by
R (T)
(170)
Σ +Pn Σ Ta,x
[Qaxx ca,x (F (o,x(
+ btxx + năx (DtLix +b
PAX
(171)
+ *A.X
[ - -
Fx (ax + + (YDrax - nāx) + xx
(y - x
ÊA.
(172)
where we use n = π and π To for readability.
Proof.
The upper bound of the regret function is given by
R (+++(m)= V (TT),,
(173)
where
(+) = / (at x)) dp(r).
(174)
By additivity of the efficient influence function, we know that
IF (T) = IF (v++*(=)) - IF -IF(V
(175)
such that we can focus on both terms separately in Supplements F.5.1 and together in Supplement F55.3 in order obtain efficient estimator.
F.5.2 and then plug them
F.5.1 EFFICIENT INFLUENCE FUNCTION OF V+*((T)
We already have the efficient influence function of V++((T) from the proof of Theorem Supplement F2 is given by
4.3 in
IF V (T:))
(176)
= =- - (T) + Σ t(a -((a,
- 2a, ta x:n)
(177)
+ (A X) (A
(178)
*(A | X) (=((A,::)))c(xxxx,)) (Fx (Y, (5::m)--)) (V
(179)
c(A,X)
+ +c((A, (A tot(A, + (Y
=At(A,::))
(180)
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F.5.2 EFFICIENT INFLUENCE FUNCTION OF V=T(())
We can derive the sharp lower bound for the value function analogously V+** Again, make use of the chain rule for deriving efficient influence function (Kennedy 2022). proof of which can be found in (Luedtke 2024) (Lemma S3).
For this, let a - 1/(1 + 1). Then, the sharp lower bound of the CAPO is given by
(a,
(181)
- (11 (1- ())(a))((, + +1) YI YI (rsfalaar) |x= 2, A A= =a
(182)
((1 (1---")e(a,a) x) + r-1 ( X = x, A = g
(183)
Hence, the influence function of Q (a, x) is given by
IF (a,
(184)
(x==) D) (T)((1=))) -e(a,)) YI 1x= x, A a
X
(185)
p(a)
(xx=2AAma) (1-
(186)
p(a,z)
(YSPaddda-))) - E[ (YSF XX= x, A= FF (rsp
(187)
+ p(x) (1- (1- (
x, A=
(188)
(xx=2,A=a)
(189)
x(a,z)
x (1 - E ( (12Fx
X= x, A= a] - ((1 (1 - a
(190)
[(x=a)
b+ (bt (a,x; + b (a,x;1)
(191)
p(x)
+ [
- ( Fa (A (Y,a,x;))a'*
(192)
x(a,x)
+ + +a (VA (YA.:)(A
(a,3;n)) + +o (a,3; (vá (Y=-(Y,,,;;))) i (a, x; 1))
(193)
Following the same steps as for Equation 45 the influence function of V~+((z) is given by
IF V (T: m))
(194)
Vー (m: 1)) + T(a I (00 -((), (f + (a, (o,x:n)
(195)
+ T(A 6t 1 (4x) (F -a-)
(196)
(AA X)
(197)
c(A,X)
+ - (A,X:n) (.xx:n(YYAA (XA....,. - (A.X:))) +o (A.X:)) (V.AA))) p (A, (AA,
(198)
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F.5.3 EFFICIENT ESTIMATOR OF R (TT)
We can derive the efficient estimator for the bounds of the value function through one-step bias correction using our results form Supplements F.5.T and F.222 via
V (T; f) + Pn (T;
(199)
P x(a xx) (a, E(a,, ((c (a,
(200)
+ (A4 X) b (4
(201)
+ x(A] x)
(202)
X)
+ oF (A, ni
( -
(A,
}}
(203)
=P Σ TaiX @2, - êo, b F + b A )
+ +TAX 6F 6 + + xxxxx,)
(204)
+ TA.X Ê
- - -
- -at)) + Cxx (vAVA. Rπx) + xx ( YAX - }}
(205)
using our short-hand notation from the main paper. Hence, the efficient estimator of the upper bound of the regret function is given by
R (1z)
(206)
= R (r; ()) + Pn [F (T:
(207)
- -(vti (T; fj) - (Tu:()) fi)
+ Pa { IF (etivvie (T: - II ( (mo:))))
(208)
Σ tPm Σ - Taxx
@ - ea,x (b + bF m + + ( + +b*
=E(t.-}
(209)
+ x TA,X ÊA.
[ (oãxx - -xxx)
-at) + + (YDiax mtx) + c (y
FAA -
(210)
where we let n. т and π. æ0 for readability.
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F.6 IMPROVEMENT GUARANTEE: REGRET FUNCTION
Corollary E.6. Under the same assumption as in Theorem 4.4 simullaneously for all π E II and fixed baseline policy no II, it holds, with probability 1 ó, that
2
Rxo (T) S R (T) + 4C R2 (IT) + 2
log
(211)
2n
where C. = =02 + (1 --+ + +) and Rn (T) is the empirical Rademacher complexity of policy class II.
Proof. In order to show the main result, we note that
R (T) = y (T) - (TT)
(212)
for arbitrary n, æp II. From Theorem we know that
5
5/1 1
log (2)
V**(m) S (+) 2C, RR, (I)(
+
2
2n
(213)
Since ग, To E II are arbitrary, we can repeat the same arguments for V~** (TTO and obtain
(=o(())) 2 (~7((=))= - 2C, (L)( + 22V 2n
1
log (5)
(214)
Then, we conclude the proof by
Rio (x)
(215)
=V(x)- - V(Ta)
(216)
<V+:*(T) (TO)(
(217)
= (x) + 2C (IT) + 2\ 2n
1
log (2))] (TO) - 2C, R₂ (TI) +
2 2
5
2n
log
8
2
(218)
= -A (x) + 4C, (R. (TT) +
1
)2
(2)
2
log
(219)
2 2n
8
35

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Published as a conference paper at ICLR 2026
G IMPLEMENTATION DETAILS
We summarize the neural instantiations of all estimators Section 5
Runtime: Training the first and second stage models for method took in total approximately 20 seconds using 1000 synthetic data samples and a standard computer with AMD Ryzen 7 Pro CPU and 32GB of RAM.634. All baselines have comparable runtime.
Nutence Sanction
Hyperparander
Contigrraion
Stancbell mettbech
Aallar 20ea
ZUIR 2001
(cam) lificient
extiratoe autrrseee
Layer sine
(64, Gi
HiMes acthvatonn
Rel.3
Proponity
Nanther
Each siopengg
10
Bulck aec
64
(66, bú.
Hidiee actrationn
Kel.l
Creditionll quantile
Leaming
Early
Barch der
64
kyess
Hidles ativationn
RaL.
(Tiocaed) outcome regres sio mode!
Leassing 131
0.001
Butch
byan
liskee actvvaioon
Rcl.2
Ponmertic polcy
Leaming rase
EL001
Narsher spacts
stopenng Butch
Table Neural instantiations of estimated nuisance functions f) and parametric policy Tg. ensure a fair comparison, all methods share the same nuisance function where applicable. For all models, we set the split parameter for training the nuisance and the policy model p 0.5.
36


Reviews:
[{"review_id": "00rDa4Mwg8", "reviewer": "Reviewer_emv2", "summary": "This paper considers the problem of unobserved confounding in offline policy learning. They assume that the unobserved confounding satisfies the marginal sensitivity model (Tan, 2006), which is often used in the sensitivity analysis literature (Aronow and Lee 2013, Miratrix et al. 2018, Zhao et al. 2019, Yadlowsky et al. 2018, Kallus et al. 2018, Kallus and Zhou 2020).", "questions": "N/A", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is clearly written and easy to understand.\n- The main contribution of this paper is a semiparametrically efficient estimator for offline robust policy learning problem, arguing that the approach of Kallus and Zhou 2020 may be unstable due to the dependence on inverse propensity weights. Instability of inverse propensity weights is a known problem that can lead to instability of estimators.\n- They propose a naive plug-in estimator for the optimal robust policy but note that it will suffer from first-order bias. Then, they derive the semiparametrically efficient estimator that does not suffer from the first-order bias from the estimation of nuisance components\n- The theoretical contributions are sound.", "weaknesses": "- The problem of robust offline policy learning under the marginal sensitivity model and Rosenbaum selection model is quite well-studied, e.g. Aronow and Lee 2013, Miratrix et al. 2018, Zhao et al. 2019, Yadlowsky et al. 2018, Kallus et al. 2018, Kallus and Zhou 2020. Furthermore, other works such as Bruns-Smith and Zhou, 2023 consider dynamic policy learning. So, the problem that the authors aim to solve has limited novelty. Nevertheless, this paper does cite and reference many of the relevant works in this area and I believe they do make a technical contribution (in terms of semi-parametric efficiency of their estimator), relative to the Kallus and Zhou 2020.\n- The new estimator appears to only provide modest improvements over the naive plug-in estimator.", "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": "YerWnvzyxi", "reviewer": "Reviewer_d6rY", "summary": "This work offers a method for off-policy learning under unmeasured confounding, whereby unmeasured covariates can jointly affect treatment decisions and the outcome. Concretely, the authors propose a one-step bias-corrected estimator that estimates “sharp” (i.e., tightest possible) bounds around the true policy value under the marginal sensitivity model. These estimated bounds are then used for downstream policy learning. The authors show that their bias-corrected estimation approach is efficient — i.e., obtains the lowest variance among unbiased estimates — and relies upon a simplified minimization objective as compared to the mini-max style objective studied in prior work. The authors validate the approach theoretically and via experiments on synthetic and real-world data.", "questions": "## Questions:\nSee questions raised above. Additionally:\n- Kallus and Zhou also require \"Strong overlap\" to hold w.r.t. the true propensity - i.e., exists $ν > 0$ such that $e_a(x, u) ≥ ν, \\; \\forall a \\in A$. Is such an assumption also needed here given the use of the MSM? \n- Algorithm 2: Can cross-fitting be performed to improve sample efficiency? \n- Figure 5: Why are other approaches not sensitive to the choice of sensitivity parameter? Can you show results for the full range, starting at $\\Gamma=1$?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Policy learning under unmeasured confounding is an important problem with broad applications. The authors identify a key gap in the literature and address it via appropriate methods. Presentation of the work is effective: I especially appreciate Figure 2 illustrating how the concept of sharpness connects to regret. The theoretical results - including identification bounds (4.1), bias-corrected estimator (4.2), and learning guarantees (4.3) are also well suited to this problem setting. The synthetic + real-world empirical validation is also well-suited to the goals of the work.", "weaknesses": "## Connection to prior work & significance\n\nIn general, the authors provide solid coverage of prior work and appropriately situate the contribution in the literature. However, this work can be viewed as a targeted improvement on top of the basic framework established in Kallus & Zhou (2018a; 2021). While I still believe such work is valuable and worthy of publication, this somewhat limits the significance of the results.\n\nMore specifically, it would be helpful if the authors could provide more detailed technical discussion of the differences with (Kallus & Zhou, 2018a; 2021). How does instability in IPW weights propagate up to the estimated policy value/regret, and how is this solved by the proposed approach? Similarly, Kallus & Zhou (2018a; 2021) also show that the regret interval obtained under their proposed approach is sharp. While lines 211-226 offer helpful initial discussion, adding additional technical clarity would strengthen the discussion for the reader. \n\nFurther, Rambachan, Coston, and Kennedy (2022) derive sharp bounds for the policy value under a related Mean Outcome Sensitivity Model (MOSM), which they then estimate via a doubly-robust method. The authors further show that bounds on the MSOM imply MSM bounds. It would be helpful to outline similarities and differences with this approach. To start, I think the method proposed in this work generalizes to non-binary actions, and the bounds in  Rambachan, Coston, and Kennedy (2022) may not remain sharp w.r.t. the MSM after converting from the MOSM framework used in this work. \n\n[1] Robust Design and Evaluation of Predictive Algorithms under Unobserved Confounding, https://arxiv.org/abs/2212.09844, Ashesh Rambachan, Amanda Coston, Edward Kennedy\n\n\n## Empirical validation\n\n\nThe general empirical validation of the work is sound, and the authors demonstrate that the proposed approach yields a benefit over relevant baselines. However, I do have several questions about the empirical validation. \n\nRelated to my point above, could the authors report experiments which illustrate the mechanism by which the proposed approach obtains improved bounds over Kallus & Zhou (2018a; 2021)? For instance, can the authors illustrate how the efficiency of the estimator yields tighter finite-sample bounds, and in turn, improves downstream policy learning? Or, similarly, that error in IPW weights propagates down to learned policies? Evidence along these dimensions would help the reader understand why the proposed method is necessary over Kallus & Zhou (2018a; 2021).\n\nFurther, it appears in several of the empirical results that the proposed method obtains high variance across runs. I find this surprising given that the estimation approach should in principle reduce variance in estimated policy value bounds. For example, confidence intervals are quite wide with Efficient + sharp as compared to baselines. We also see similar behavior in Figure 3. Could the authors explain why this is the case, and also include Kallus & Zhou (2018a; 2021) in Figure 3 for a clear comparison of variance across runs? \n\nAdditionally, while comparing regret against a fully randomized policy is a reasonable starting point, this seems overly simplified for a real-world experiment, especially because non-randomized baseline policies may yield more challenging distribution shift. Can the authors also report comparisons against other baseline policies? \n\nFinally, given that the proposed estimator depends on learned nuisance functions, can the authors report details surrounding the procedure used to fit and select nuisance functions used to construct the doubly-robust estimates? \n\nOverall, I see this work as providing a valuable contribution but do have significant concerns. I am open to re-considering my score if these concerns regarding significance and empirical validation are appropriately addressed.", "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": "VfdRdvPycZ", "reviewer": "Reviewer_P2Zy", "summary": "The paper derives closed-form sharp bounds for policy value under MSM, plus a semi-parametrically efficient estimator. It avoids unstable minimax/IPW and proves optimal confounding-robust policy learning.", "questions": "- In Algorithm 1, Step 6 says Estimate $V^{+,*}$ as in (2), but (2) just defines the propensity.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Closed-form sharp bound for value under MSM\n- One-step bias-corrected estimator hits the efficiency bound\n- Learning guarantees to the optimal confounding-robust policy", "weaknesses": "- The EIF and one-step estimator rely on quantiles $F_{x,a}^{-1}(\\alpha_{\\pm})$. You do not state standard conditions ensuring pathwise differentiability.\n- You claims the estimator “is semi-parametrically efficient” and points to D.2, which provides an influence function expression and cites a chain-rule lemma. But you never identify the canonical gradient in the nonparametric model nor verify your influence function equals it.\n- Theorem 4.4 needs a uniform bound, but the current version is pointwise in $\\pi$.\n- The nuisance $\\eta$ in (14) includes the quantiles, but your EIF contains terms like $(\\Delta-\\alpha)F^{-1}$ that rely on differentiability of these nuisances. Please clarify.\n- Before Theorem 4.4, you write “parametric policy classes (e.g., neural networks) have vanishing $R_n(\\Pi)\\in O(n^{-1/2})$\". For neural networks, this needs norm constraints, otherwise $R_n$ need not decay at root-n rate.", "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)