Learning Representations of Instruments for Partial Identification of Treatment Effects
Authors: Jonas Schweisthal
Dennis Frauen
Maresa Schröder
Konstantin Hess
Niki Kilbertus
Stefan Feuerriegel
Venue: ICML 2025 poster (2025)
Keywords: causal inference
partial identification
instrumental variables
treatment effect
representation learning
Primary area: general_machine_learning->causality
TL;DR: We develop a novel method for partial identification of treatment effects by tailored representation learning of instruments

Abstract: Reliable estimation of treatment effects from observational data is important in many disciplines such as medicine. However, estimation is challenging when unconfoundedness as a standard assumption in the causal inference literature is violated. In this work, we leverage arbitrary (potentially high-dimensional) instruments to estimate bounds on the conditional average treatment effect (CATE). Our contributions are three-fold: (1) We propose a novel approach for partial identification through a mapping of instruments to a discrete representation space so that we yield valid bounds on the CATE. This is crucial for reliable decision-making in real-world applications. (2) We derive a two-step procedure that learns tight bounds using a tailored neural partitioning of the latent instrument space. As a result, we avoid instability issues due to numerical approximations or adversarial training. Furthermore, our procedure aims to reduce the estimation variance in finite-sample settings to yield more reliable estimates. (3) We show theoretically that our procedure obtains valid bounds while reducing estimation variance. We further perform extensive experiments to demonstrate the effectiveness across various settings. Overall, our procedure offers a novel path for practitioners to make use of potentially high-dimensional instruments (e.g., as in Mendelian randomization).

Learning Representations of Instruments for
Partial Identification of Treatment Effects
Jonas Schweisthal
Dennis Frauen 12 Maresa Schröder Stefan Feuerriegel 12 Konstantin Hess Niki Kilbertus 234
Abstract
decision-making in medicine (Feuerriegel et al., 2024). For example, a common question in medicine is to estimate the effect of alcohol consumption on the onset of cardiovascular diseases (Holmes et al., 2014). There are several reasons, including costs and ethical concerns, why CATE estimation is often based on observational data, such as electronic health records and clinical registries.
Reliable estimation of treatment effects from ob- servational data is important in many disciplines such as medicine. However, estimation chal- standard
lenging when unconfoundedness assumption in the causal inference literature is violated. In this work, we leverage arbitrary (po- tentially high-dimensional) instruments to esti- mate bounds on the conditional average treatment effect (CATE). Our contributions are three-fold: (1) We propose a novel approach for partial iden- tification through a mapping of instruments discrete representation space so that we yield valid bounds on the CATE. This is crucial for reli- able peciisionmmaing in real-world applications. (2) We derive a two-step procedure that learns tight bounds using a tailored neural partitioning of the latent instrument space. As result, we avoid instability issues due numerical approxi- mations adversarial training. Furthermore, procedure aims to reduce the estimation variance in finite-sample settings to yield more reliable estimates. (3) We show theoretically that our procedure obtains valid bounds while reducing estimation variance. We further perform exten- sive experiments demonstrate the effectiveness across various settings. Overall, procedure offers a novel path for practitioners to make of potentially high- nigh-dimensional instruments (e.g., as in Mendelian randomization).
However, identify- ing the CATE from observational data is challenging as it typ- ically requires strong assumptions in the form of unconfound- edness (Rubin, 1974). Unconfoundedness assumes there exist no additional unob- served confounders U between treatment A and outcome Y. If the unconfoundedness assumption is violated, a common strategy is to leverage instrumental variables (IVs) Z. IVs affect only the treatment A but exclude unobserved confounding between 2 and Y, which often can be ensured by design such as for randomized studies with non-compliance (Imbens & Angrist, 1994). The causal graph for the IV setting shown in Fig. 1.
Figure Overview of the IV set- ting. We consider complex instru- ments Z (c.g. gene data, text, im- ages), observed confounders X, unobserved confounders U, binary treatment A, and outcome Y.
Motivational example: Mendelian randomization. Mendelian randomization (Pierce et al., 2018) refers to the use of genetic information as instruments Z to estimate the effect of a treatment or exposure A (e.g.. alcohol consumption) on some medical outcome Y (e.g.. cardiovascular diseases). In this setting, there are further patient characteristics that are observed (X) but also unobserved (U), which accounts for through the instrument. Yet, common challenges are that (i) instruments with genetic information are often higl-ddmeesional and (ii) involve complex, non-linear relationships between instruments and treatment intake or exposure.
1. Introduction
Estimating the conditional average treatment effect (CATE) from observational data is an important task for personalized
'LMU Munich 'Munich Center for Machine Learning (MCML) "School of Computation, Information and Technology. TU Mu- nich "Helmholtz Munich. Correspondence Jonas Schweisthal goona..scweesstee Imu.de>.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
However, existing IV methods using machine learning for point estimation of the CATE rely on strong simplifying

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Learning Representations of Instruments for Partial Identification of Treatment Effects
assumptions (++ violating (ii) from above). For example, some methods assume linearity in some feature space in the CATE and make other, strict parametric assumptions on the unobserved confounders such as additivity or homogeneity (Hartford et al., 2017; Singh et al., 2019; Xu et al., 2021). Yet, such simplifying assumptions are often not realistic and can even lead to unreliable and false conclusions by the mis-specification of the CATE.
Complexx NV
Disonoto disstringg Z
Partial kooni of
Nare partaoning U
DX a &
Whde bounce
Instument sOaDE
Reproartaaion leannng
Tight bounde
A potential remedy is to use IVs for partial identification of the CATE where one circumvents any hard parametric assumptions by estimating upper and lower bounds of the CATE (Manski, 1990). This is usually sufficient in med- ical practice when one is merely interested in whether a treatment variable (e.g., exposure as in Mendelian random- ization) has a positive or negative effect. So far, methods for partial identification of the CATE in IV settings are rare. There exist closed-form bounds (i.e., via a fixed target esti- mand that can be learned), yet only for the setting with both discrete instruments and discrete treatments (Balke& & Pearl, 1997).
X
Figure 2. Leveraging complex instruments for partial identifica- tion of the CATE through discrete representations of Z. Naive discretization on the IV input space leads wide, and thus non- informative, bounds. Our method learns a latent representation oi Z) to yield tight bounds.
the performance of our procedure by explicitly reducing the estimation variance in finite-sample settings to yield more reliable estimates. (3) We provide a theoretical analy- sis of our procedure and perform extensive experiments to demonstrate the effectiveness across various settings.
Existing machine learning methods for partial identification are typically designed for simple instruments that are bi- nary or discrete (-> violating (i) from above). Alternatively, methods that extend partial identification for continuous instruments require umstable training paradigms such as ad- versarial learning (Kilbertus et al., 2020; Padh et al., 2023) which becomes even more unstable for more complex in- struments. In contrast, there is a scarcity of methods that can deal robustly with continuous, as well as complex and potentially high-dimensional instruments such as, e.g., gene expressions as in Mendelian randomization but also text, images, or graphs.'
Contributions: (1) To the best of our knowledge, this is the first IV method for partial identification of the CATE based on complex instruments. (2) We derive a two-step procedure to learn tight bounds. (3) We demonstrate the effectiveness of our method both theoretically and numerically.
2. Related Work
Machine learning for CATE estimation with IV: Exist- ing works have different objectives. One literature stream leverages IVs for CATE estimation but focuses on settings where the treatment effect can be point-identified from the data. This includes work that extends the classical two-stage least-squares estimation non-linear settings by learning non- monnlinnar feature spaces (Singh et al., 2019; Xu al., 2021), deep conditional density estimation in the first stage (Hart- ford et al., 2017), or using moment conditions (Bennett et al., 2019). Another literature stream aims at new machine learning methods with favorable propertics such as being doubly robust (Kennedy et al., 2019; Ogburn et al., 2015; Semenova & Chernozhukov, 2021; Syrgkanis al., 2019) or multiply robust (Frauen & Feuerriegel, 2023). Recently, researchers started applying machine learning methods to IVs from Mendelian randomization (Legault et al., 2024; Malina et al., 2022), which is our motivational example from above. However, these works aim at point-identified CATE estimation with IVs. As a result, these rely on hard and generally untestable assumptions on some effects in the causal graph, such as linearity, monotonicity, additivity, or homogeneity (Wang & Tchetgen Tchetgen, 2018). This is
Our paper: In this work, we leverage complex instruments for partial identification the CATE. Specifically, we allow for instruments that can be continuous and potentially high- dimensional (such as gene information) and, on top of that, explicitly allow for complex, non-linear relationships between instruments and treatment intake exposure. In the rest of this paper, we refer to this setting as "complex" instruments.
To this end, we proceed as follows. (1) We propose novel approach for partial identification through a mapping of complex instruments to a discrete representation space so that we yield valid bounds the CATE. We motivate our approach in Fig. 2. (2) We derive two-step procedure that learns tight bounds using neural partitioning of the latent instrument space. As a result, we avoid instability issues due to numerical approximations or adversarial training, which is a key limitation of prior works. We further improve
In Appendix B, we provide an extended discussion about the real-world relevance of our method.
'Code is available https::/ htpps://grt httsss://gttub.ooo JSchweisthall htttweesstaaa//opleeeest expattiaa 1denti f.
2

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Learning Representations of Instruments for Partial Identification of Treatment Effects
unlike our method for partial identification that does not require such hard assumptions and that is non-parametric.
observational dataset D = { Ty dg yo)] of size n. The data is sampled i.i.d. from population (Z, X, Y) ~ P,, with observed confounders X E X S RP, binary treat- ments A E A S {0. 1). and bounded outcomes Y E y S [ C IR. Additionally, we allow for unobserved con- founders U of arbitrary form between A and Y.
Partial identification: Partial identification aims to identify and learn upper and lower bounds of some causal quantity (e.g., the CATE) when the causal quantity itself cannot be point identified from the data and assumptions. In a general setting with binary treatments, Robins (1989) and Manski (1990) derived closed-form bounds on the ATE for bounded outcomes Y. Further work extended these ideas to settings with binary instrumental variables, binary treatments, and bi- nary outcomes (Balke& & Pearl, 1994; 1997) to derive tighter bounds. Newer approaches for discrete variables include the works of Duarte et al. (2023) and Guo et al. (2022). Swan- son et al. (2018) provide an extensive overview of partial identification in this setting, Other works focus on how to leverage additional observed confounders to further tighten bounds on the ATE (see, e.g., Levis et al., 2023). However, these works do not focus on efficiently leveraging continu- ous or even high-dimensional instruments for learning tight bounds, unlike our work that is tailored to such complex instruments.
We further assume a causal structure as shown in Fig. 1. In particular, we assume that Z is an instrumental variable that has an effect the treatment A but no direct effect on the outcome Y except through A. Further, we assume that Z is independent of X. c.g., by randomization. In Appendix B, we provide an extended discussion to show the real-world relevance and validity of our assumptions in di different settings.
Notation: Throughout our work, we denote the response
function by μ (x, E|y|X x, A a, Z = 2] and the propensity score by (Ix = P(4 4 = 1| = 7 =
2).
CATE: We use the potential outcomes framework (Ru- bin, 1974) to formalize our causal inference problem. Let (()) E y denote the potential outcome under treatment A a. We are thus interested in the CATE r(x) (1Y (0) = z/|
Another literature stream focuses on partial identification under general causal graphs (Balazadeh et al., 2022; Frauen et al., 2023; 2024), including IV settings with continuous variables such as continuous treatments (Gunsilius, 2020; Hu et al., 2021; Kilbertus et al., 2020; Padh et al., 2023). However, these methods either make strong assumptions about the treatment response functions or require unstable optimization via adversarial training and/or generative mod- eling such as through using GANs. This can easily result in unreliable estimates of bounds for finite data, especially with high-din -isensiinnii instruments. Further, these methods are not directly tailored for binary treatments, unlike our method.
Identifiability: We make the following standard assump- tions from the literature in partial identification with IVs (Angrist et al., 1996). Assumption 1 (Consistency): Y(A) = Y. Assumption 2 (Exclusion): Z IL Y(A) I A,) Assumption 3 (Indppendence): Z TL (U.
Note that, however, Assumptions 1-3 from the standard IV setting are not sufficient to ensure identifiability of the CATE (Gunsilius, 2020). To ensure identifiability. one would re- quire additional assumptions, such as linearity or, more generally, additive noise assumptions (Hartford et al., 2017: Wang & Tehetgen Tchetgen, 2018). Yet, such assumptions are highly restrictive and are neither testable nor typically ensured in
Research gap: To the best of our knowledge, reliable ma- chine learning methods for partial identification of the CATE with complex instruments are missing. To draw conclusions about CATES (as in, e.g., Mendelian randomization), our method is the first to: (i) make use of the complex instru- ment information (e.g., continuous or high-dimensional), (ii) avoid making strong parametric assumptions by focus- ing on partial identification, and (iii) avoid unstable training procedures such as adversarial learning.
scenarios. Hence, this motivates our objective to perform partial identification instead.
The goal of partial identification can be formal- ized in different ways. First, we can formulate the space we are optimizing over as the distributions (over joint data including unobserved U') that are compatible with the ob- served data distribution by
Objective:
3. Problem Setup
M { pd 2, u. u.) ((2,,,,) = P ((()),,
We focus on the standard IV setting (Angrist et al., 1996; Wooldridge, 2013). Hence, we consider instruments (c.g.. gene data, text, images) given by Z E Z C &d but, un- like previous research, allow the instruments to be complex. As such, we allow the instruments to be continuous and potentially high-dimensional. We further have access to
Setting:
Option 1 (Classical formulation). Recent works (e.g.. (Frauen et al., 2023)) often formulate the goal of partial identification such that by (x) = (bT(x),
(x), (0))((())) (x)) with
(1x) (x) = inf Te- (I), bf (æ) = sup Te (z)
(2)
P.EM
F.EM
3

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Learning Representations of Instruments for Partial Identification of Treatment Effects
gives optimal sharp lower and upper bounds. Here, we explicitly denote the dependency of the CATE r(:) on the unobservable compatible data distributions by the index P
valid (Sec. 4.2). Then, we take finite-sample view and present estimator (Sec. 4.3).
Limitations of existing bounds: There exist different ap- proaches for bounding treatment effects (see Sec. 2) using continuous instruments, yet these either require additional assumptions or can easily become unstable, especially for high- high-dimensional Z. Furthermore, these bounds consider continuous treatments but are not tailored for binary treat- ments (e.g., whether drug is administered). Hence, we derive custom bounds for our setting.
Option 2 (Ours). Alternatively, we can provide the following formulation. First, we define the sets of valid bounds via
V- = {b: X > RR (z) b(x) for all P E EM, E x),
V+ = {b: X + I Tpm (x) S b(x) for P E M,
(3)
Then, we can minimize the average bound width over all valid bounds to get be()) = (b5 (x)), (x)), i.e.,
Why is the derivation non-trivial? For binary treatments, it turns out that there exist closed-form solutions for bounds whenever the instrument bounds for the average treatment effect (ATE) with con- tinuous bounded outcome proposed in (Manski, 1990) can be extended to non non-parametric closed-form bounds for the CATE (Schweisthal et al., 2024). While these bounds are useful in setting with discrete instruments Z, they are not directly applicable to continuous or even high-dimensionall Z due to two main reasons: (1) The bounds need to be evaluated for all combinations 1, m E 2 C Rd x IRd, which is intractable.
b7 by E arg min
Ex b
Ex
(X) b- (X)).
(4)
discrete. That is, the existing
b-Ev D-EVI
Lemma 1. It holds that (x) bx())) =.
Proof. See Appendix A.
More informally, without additional constraints on the func- tional form of the bounds, our Option 2 almost surely gives the sharp bounds from Option 1. In Option 2, we focus on providing valid bounds (b- (x), bt (z)) for the CATE 7 (zz) such that 6 (x) < T(x)= I b+ (x) holds for all possible E X. Furthermore. the bounds should be informative, i.e., we would like to minimize the expected bound width Ex(b+(X-- - b- (X)], while still ensuring validity.
(2) Evaluating the bounds only on a random subset of combinations I, m result in arbitrary high estimation variance for regions with a low joint density of p(X = x, Z = 1) or p(X = x, Z = m). Hence, we must derive a novel method for estimating bounds based on complex instruments (that are, e.g., continuous or high- dimensional), yet this a highly non-trivial task.
For our method, we leverage formulation multiple benefits such as allowing us adopt closed-form bounds from the discrete IV setting and to avoid alternating learning, which we show in the following sections.
this provides
4.2. Population view
In the following theorem, we provide a novel theoretical result of how to obtain valid bounds based on discrete repre- sentations d(Z) of the instrument Z.
4. Partial identification of the CATE with complex instruments
Theorem 1 (Bounds for arbitrary instrument discretizations).
4.1. Overview
Let Z + {0, 1...., k} be an arbitrary mapping from the high-dimensional instrument 7. to discrete representation. We define
We now present our proposed method to solve the partial identification problem from Eq. (4). Solving Eq. (4) directly is infeasible because it involves the unknown CATE (x)) Hence, we propose the following approach:
pg 4) = / 14 21, 87 =)
(5)
P(A a, o(Z) = E)
Outline: ① We learn a discretized representation (also called partitioning) d(Z) of the instrumental variable Z. 2 We then derive closed-form bounds given the discrete representation Φ. 3 We transform the closed-form bounds back to our original bounding problem and, in particular, express all quantities involved as quantities that can be estimated from observational data.
P(A = alZ = z)P(Z = dz and
T E) = ((((2) = AZ = z)
P(Z 2) dz.
P(ő(Z) =
(6)
Then, under Assumptions 1, 2, and 3, the CATE (xx) is bounded by
(x) S 7 (xx) b5(())
Below, we first explain why existing closed-form bounds are not directly applicable and why deriving such bounds is non- trivial. We then proceed by providing the corresponding theory for the above method. Specifically, we first take a population view to show theoretically that our bounds are
with
b (z) min b+ Im
oslm (z) and (x) mmxx b
I.m oi (z)
(8)
4

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Learning Representations of Instruments for Partial Identification of Treatment Effects
where
Once the initial estimators are obtained, we can estimate our second-stage nuisance functions defined in Eq. (34) and
by Dalm (x) (m))===(( (2, 1) (z; 1) (1 (x(xx11))
(9)
(35) via
- -11- (Z m) ()) m.) - (T, m)s1,
n
pg ()
Σ j=1
(x
-ー = L, = =a}
b b (2) =m (1e, Mu (x; 1) + (1 =T( (10)
(12)
71-7 (an m) )r (2) m) (T, m))22
1(((2z) (o((=))=((a(((z)) (1 aa)(1 -((;))))
Proof. See Appendix A.
Ao(a, ΣE lf((2) = = Σ म
f).
Theorem states that, in population, we yield valid closed- form bounds for T(x) for arbitrary representations o. In particular, we can relax the optimization problem from Eq. (4) and obtain valid bounds ba- (X) 2 by (X) and by- (X)= I (X) by solving
(13)
Finally, we can directly 'plug in' these estimators into Eq. (8) to compute estimates of the upper and lower bound 8 (x), 8+ @ (x))
A naive approach would now directly use (b (1), b (1)) to solve the optimization in Eq. (11). However, for finite samples, it turns out this is infeasible without restricting the complexity of the representation function. The reason is outlined in the following theoretical results.
8- E arg min Ex (bg +(X)- by -X)]. ΦEΦ
(11)
Here, we highlight the dependence of variables on the repre- sentation é green show the differences Eq. (4). Note the following differences: In contrast to Eq. (4), we do not impose any validity constraints in Eq. (1I) because Theo- rem 1 automatically ensures the validity of our bounds. Fur- thermore, in contrast to Eq. (4), the objective from Eq. (11) only depends on identifiable quantities that can be estimated from observational data.
trade-off). Let En and Var, denote the expectation and variance with respect to the observational data (of size n). Then, holds
Lemma 2 seightness-bias-vrrinn
E., (b (1))()) - 1 (x))) ) s
(bt(r) - (14)
Pepulatiin rightness
Implications of Theorem 1: naive implementation mini- mizing the bounds following Eq. (11) would require alter- nating learning. The reason is that, after every update step of 6(2). the quantities p (z. I) and (z, 1) are not valid for the updated á anymore and would need to be retrained to ensure valid bounds. This is computationally highly ex- pensive and causes unstable training as well as convergence problems. However, our method circumvents these issues: by using Theorem 1, show that, while training d(=), the quantities ng (a, f) and T (2, E) can be directly calculated. For that, we can simply evaluate the nuisance functions, which only need to be trained once in the first stage. This holds because in our derivation of closed-form bounds for arbitrary discrete representations of complex Z, the bounds only depend on (i) discrete probabilities, (ii) quantities that are independent of é and thus do not change for different di and (iii) the discrete representation mapping to be learned itself. As result, we circumvent the need for adversarial or alternating training, which results in more robust estimation.
+ En (x) - [to)
Var, (bt (x))
+
(10 Estimation bias
(00) Estimation variance
Proof. See Appendix A.
Interpretation of Lemma 2: Lemma 2 shows that the mean squared error (MSE) between the estimated representation- based bound 6z(z) and the ground-truth optimal bound b (z) can be decomposed into the following three com- ponents: (i) population tightness, (ii) estimation bias, and (iii) estimation variance. Term (i) describes the discrep- ancy between the representation based bound in population by & (a) and the ground-truth optimal bound b (x)) It will de- crease if allow for more complex representations क; for example by increasing the number of partitions k. Term (ii) describes the estimation bias due to using inite-sample esti- mators for estimating the bounds. It will generally depend on the type of estimators we employ for ((x, 2), f (z, 2), and ri(z). Finally, term (iii) characterizes the variance due to using finie--sapll estimators. In contrast to term (i), will increase when we allow the representation to be more complex.
4.3. Finite-sample view
In practice, we have to estimate the bounds from Theorem from finite observational data. For this purpose, we start with arbitrary initial estimators: #(x, 2) is the estimator of the propensity score (σz, 2). (z, 2) of the response function yd 2), and i(z) of (1)) = PAA = 1 7 = z).
To make point (iii) more explicit, we derive the asymptotic distributions of the estimators from Eq. (12) and Eq. (13) that are used during training of @ to estimate the final bounds.
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Learning Representations of Instruments for Partial Identification of Treatment Effects
Theorem 2 (Asymptotic distributions of estimators). As- suming orucle estimation of first-stage nuisance functions, such that i = u,, क = n, and fj = 77, it holds that
for some E >> 0 and all E {l,...,k)} neural method to learn tight bounds using the above objec- tive.
k).. We next present
v (p5(1, f)- - 13( 0)) (,(0
(15)
5. Neural method for learning CATE bounds with complex instruments
1
Var(u(Z) 6(Z)= = f)
+ d
Pe,e
c
In this section, we propose neural method for our objective to learn tight and valid bounds. Our method consists of two separate stages (see Algorithm 1): ① we learn initial estimators of the three nuisance functions, and ? we learn an optimal representation s", so that the width of the bounds is minimized. Note that our method is completely model- agnostic. Hence, arbitrary machine learning models can be used in the first and second stages in order to account for the properties of the data. For example, for instruments with gene data, one could use pre- pre-trained encoders to further optimize the downstream performance. We give an overview of the workflow of our method in Fig. 3 (see Algorithm in Appendix H for pseudocode).
- (n - SN((
0,
1
Var(h(Z) ((Z) = f)
pr,
(16)
for = N... = 0f(11-r..60.. o)) such that c, d > 0 and where
pe, = P()(Z) = -). 02.0 = P(A-= = a o(Z) = f), g( =
(=, (a)(() + (1 - (1 A()).. l(Z)= = #(x, Z), and
Oe = El)(Z) (Z) = f.
Proof. See Appendix A.
We observe that the variance of the estimators (and, thus, of the estimated bounds) explodes for small values of Pt.e (Z) = E). Note that we show this behavior assuming oracle first-stage nuisance estimates without any estimation error. Consequently, the variance will be inflated even more when assuming additional estimation error in the first stage. Hence, to reduce the estimation variance, we aim to learn a representation ø that avoids low Pe.s for some E, e.g., by limiting the number of partitions = Altogether, as a consequence of Lemma and Theorem 2, we obtain an inherent trade-off between tightness of the bounds in population and estimation variance in samile-samples?
Initial nuisance estimation: In the first stage, we can use arbitrary machine learning models (e.g., feed-forward neural network) to learn the first-stage nuisance functions o (z, =) P P-A= X = x, Z = z), and fi(z) = IP(A 1 = =).
E | X = A = a, Z = =] *
Recall that we consider Z and X, which are both potentially high-dimensional. Hence, for fa (xr, 2) and R ( 2), we use network architectures that have (i) different encoding layers for X and Z, so that capture structured infor- mation within the variables and (ii) shared layers top of the encoding to learn common structures. Further, for o (st, =). we use two outcome heads for both treatment options A E {0, 13 to ensure that the influence of the treat- ment on the outcome prediction does not 'get lost' in the high dith-siinniinall space of X and Z (Shalit et al., 2017).
Learning objective for the representation o: Due to the inherent trade-off between tightness of the bounds and es- timation variance, the aim for learning the representation is two-fold. On the one hand, we (a) aim to learn tight* bounds, which is given in the objective in Eq. (11). On the other hand, we (b) also have to account for control- ling the variance in sinite-sample settings, especially for high-dimensional Z. Motivated by Theorem 2, we ensure fr. > & for some e > 0, where pr, is an estimator of Pr. = (()(())= = t). Combining both (a) and (b) yields
Representation learning: In the second stage, we train neural network to learn discrete representations of the instruments with the objective of obtaining tight bounds but with constraints on the estimation variance. To learn the function o(z). we use a neural network Φe with trainable parameters 0. Then, on top of the final layer of the encoder, we leverage the Cumbel-softmat trick (Jang et al., 2017), which allows to learn i discrete representations of the latent space of the instruments, where k can be flexibly chosen as a hyperparameter.
the following objective:
d" E arg min Ex (x) - (x0) s.t. Peo >E, E, Φ
(17)
'Importantly, Lemma 2 and Theorem 2 hold for arbitrary & and its bound estimators B (ar), enabling more stable updates by reducing estimation variance during training. Consequently, these results also apply to the finally learned or optimal 6" leading to lower variance in final estimates.
We further transform our objective into a loss function to train the network Φg. For that, we design a compositional loss consisting of three terms:
Custom loss function:
'Here and in the following we refer to "tight" as to minimizing the expected bound width also under potential constraints. This is distinct from the term "sharp" which refers the in theory- tightest achievable bounds.
A bound-width minimization loss that aims at our objec-
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Learning Representations of Instruments for Partial Identification of Treatment Effects
frainable parameters
fixed parameters
Complex instrument Z
Representation network
((z)
A, X
2nd stage nulsance calculation
Bounds on CATE
(a di (+)) M (z, (1)))
dh
T(x)
To. (a, (2) Ms (sr, (2(2))
dr
X
pre-trained 1sf stage nulsence functions
Figure 3. Workflow of the second stage of our method for calculating bounds on the CATE: The representation network do learns discrete latent representations of the complex (e.g., continuous or high-dimensional). By employing the pre-trained μ., â, and fj, we can directly calculate the muisance estimates conditional on the latent representation 6(2) by using Eq. (12) and Eq. (13) to yield the bounds.
tive in Eq. (17), defined via
The key advantage of method is its efficiency and robust- compared to alternatives like alternating learning adversarial training. In the second stage, only the discretiza- tion network de is updated to minimize Lg. while first-stage nuisance estimators remain fixed and are merely evaluated. This enables reusing trained first-stage networks across dif- ferent second stage training settings (e.g., varying k), which makes the training procedure more computationally efficient and robust.
(0) Σ %0 (1x)
6-, (1s) 60
(18)
2 regularization loss enforce the constraints in Eq. (17), i.e., enforcing that Pe,o (Z) f) e, Vf E k,...,k, for some 0. For that, aim to penal- ize the negative log-likelihood Σ log(P(dbg()) j)),
which we can estimate via
6. Experiments
k
Creg(0) (0)= log
Σ (00(())) = =s)).
(19)
Existing methods (see Sec. 2) focus either on (a) point identification with strong assumptions, (b) par- tial identification with continuous treatment variables, or (c) discrete instruments. We instead focus on a setting with complex instruments and binary treatments. Hence, existing methods are not tailored to our setting, because of which a fair comparison is precluded. Instead, we thus demonstrate the validity and tightness of our bounds. Further, for com- parison, we propose an additional NAIVE baseline, which first learns a discretization of the instruments (via k-means clustering) and then learns the nuisance functions wrt. to the discretized instruments to apply the existing bounds for discrete instruments from Lemma 3 on top.'
Baselines:
3 An auxiliary guidance loss Laux (0). which enforces more heterogeneity between IP(Z (2 = 1) and PPZZ| (Z) = m.) for all I, m. To achieve this, we add an additional linear classification head Pς with weights Ç on top of the last hidden layer of de before the discretiza- tion. The auxiliary guidance loss is explicitly defined as the cross-entropy loss via
1 71
Laux (0)
(0o(2)) j} log (pę(2)), (20)
Data: We perform experiments mimicking Mendelian Ran- domization but where we simulate the data to have access to the ground-truth CATE for performance evaluations, so that we can check for coverage and validity of the bounds. We consider three different realistic settings. For Datasets and 2, consider one-dimensional continuous instrument representing polygenic risk score (Pierce et al., 2018).
where Pc (2;) is the predicted probability of assigning 2; to discrete representation j by the additional classification head. While (f) is not strictly necessary for our ob- jective, we empirically observed that it helps to stabilize training by avoiding convergence to informatvie local minima. Hence, we yield our final training loss
(()) = Lz) + Arre (0) + Laux (0),
(21)
Further, in Dataset 1, we model the true (,. 2) as rather simple function to check if our method is already compet- itive in such settings. In Dataset 2, we model ा T =) as
with hyperparameters X and Here, X controls the trade- off between bound tightness and estimation variance, and can thus be tailored depending on the application. The hyperparameter yy can be simply tuned as usual.
We provide comparison to other naively adapted baselines in Appendix E.
7

---

Learning Representations of Instruments for Partial Identification of Treatment Effects
Metric
Dataset
Dataset 2
Naive
Ours
Rel. Improvement
Naive
Ours
Rel. Improvement
Coveragel []
1.00 1 0.00
1.00 = 0.00
0.00%
1.00 1 0.00
1.00 1 0.00
0.00%
Width[_]
1.22 : 0.05
1.05 + 0.01
13.9%
1.31 0.16
1.14 - 0.16
13.0%
MSD[L]
0.28 = 0.0G
0.03 - 0.03
89.3%
0.09 + 0.06
0.06 + 0.06
33.3%
Table 1. Datasets 1 and 2: Comparison of both methods (NAIVE Ours) regarding coverage, width, and MSD. Relative performance improvements in green.
Metrie
Naive
Ours
Rel. Improve
only under the constraint that they yield valid coverage. We thus propose two new metrics, which we call width and MSE*, which denote the corresponding metrics but where we filter for runs with coverage* 2 95%. This allows us to properly compare the ability to learn tight bounds without distortions due to falsely overconfident predictions.
Coverage [1]
0.81 0.37
0.97
0.05
15.3%
Width*
1.88 + 0.06
1.82 + 0.05
3.2%
MSE
0.12 1 0.02
0.10 : 0.02
16.7%
MSD[L]
0.10
0.03 + 0.02
70.3%
Table Dataset 3: Comparison of both methods (NAiVE vs. Ours) regarding the coverage with respect to the oracle bounds, width, and MSD. Relative performance improvements in green.
Implementation details: For our method, we use multi- layer-perceptroos (MLPs) for the first-stage nuisance es- timation and an MLP with Gumbel-softmax (Jang et al., 2017) discretization on the last layer for learning óg. For the NAIVE baseline, we use k-means clustering the first step to learn discretized instruments and then use MLPs with identical architecture for the nuisance estimation to ensure a fair comparison. We provide further details in Appendix C.
19
Figure 4. Datasets 1 and Estimated bounds on the CATE. mean = sd over 5 runs for different k. Left: Dataset I with simple ((, 2). Right: Dataset with a complex (xx, 2).
We present the results of our experiments in Ta- ble (for Datasets and 2) and in Table (for Dataset 3). Therein, we compare our method against the NAIVE base- line averaged over multiple runs and over different choices of clusters k. Here, we report the results averaged over multiple k because, as usual in causal inference, hyperpa- rameter tuning is more challenging without access to the ground truth CATE, and thus there are different strategies for selecting k (see also Appendix F). Thus, taking the aver- age over the reasonably selected & can be seen as reporting the summarized performance over different strategies that would have resulted in selecting the different values of k (e.g.. by expert- sxpert-informed approach).
Results:
a complex function to evaluate the performance in more challenging settings. We use the same CATE for Dataset and Dataset 2 to allow for comparisons between both. In Dataset 3. we model high-dimensional instruments with sin- gle nucleotide polymorphisms (SNPs, i.e., genetic variants (Burgess et al., 2020)) to test our method in an additional realistic and even more complex setting. 6 In all datasets, we model the CATE to be heterogeneously conditioned on X to check whether the bounds adapt to different subpopula- tions. Details are in Appendix D. Additionally, we provide experiments for real-world case study in Appendix E.4.
Overall, we observe the following patterns: (i) Both meth- ods (i.e., ours and the NAIVE baseline) almost always reach a perfect coverage of 100% for the true CATE, which shows the validity of the bounds. For Dataset 3, our method achieves better coverage wrt. to the oracle bounds, which further suggests that our method leads to more reliable estimation. (ii) As expected, average, righter bounds for Datasets 1 and 2 (lower width), and for Dataset our method learns tighter valid bounds that are closer to the oracle bounds (lower width and MSE*). This demonstrates that our method can clearly improve over a discretization that uses solely information of Z in the first step (NAIvE). (iii) Unlike the baseline, our method is robust over different values of k. This is demonstrated by a low MSD in all datasets, with improvements up to 89% over the naive bascline.
Performance metrics: We report the following metrics to assess the validity and robustness of the estimated bounds: (i) The coverage, i.c., how often the true CATE lies within the estimated bounds. (ii) The average width of bounds, where lower values indicate more informative bounds. (iii) The mean squared difference (MSD) of the predicted bounds over different values of k, indicating the robustness wrt. to the selection of the hyperparameter. For Dataset 3, we model (2x, 2) to be dependent on some la- tent discrete representation of the observed Z, such that we can approximate oracle bounds. Thus, we can evaluate the coverage wrt. to the oracle bounds (denoted as cov- erage*) and the MSE to the bounds. Further, for reliable decision-making, we would like to obtain tight bounds but
method learns
"We provide additional experiments show the robustness across instrument dimensions in Appendix
Sensitivity over k: To better understand the robustness
8

---

Learning Representations of Instruments for Partial Identification of Treatment Effects
0.14
Method
Nalee Ours
1.00
0.8
Coveragee
MSER
0.11
Widu
1.85
0.7
180
1.25
Method
0.08
Nalve
Notse
Durs
Durs
12
14
Figure 5. Dataset (high-dimensional):; Sensitivity analysis wrt. to the number of partitions & showing the MSF*[]] (left), width*[[] (middle), and coverage [+]] (right)
Dataset
Method
Coveragel
Widhh|]
increases for more complex datasets and modeling settings. Limitations: Our method for partial identification allows us to relax multiple assumptions that are inherent meth- ods for point identification. Nevertheless, the standard assumptions of IV settings. However, such assumptions often hold by design or can be ensured by ex- pert knowledge such as in Mendelian randomization. We provide an extended discussion in Appendix B.
Natve
1.00 + 0.00
1.62 0.06
0.00
0.83 0.16
Dataset
Ours
1,00 0.00
1.01 0.00
1.00 0.00
1.09 0.04
still rely
Naive
1.00 + 0.00
1.34 0.19
1.00 0.00
1,28
Dataset
Ours
1,00 1 0.00
1,13
1.15
Table Datasets and Sensitivity over
We propose novel method for learning tight bounds on treatment effects by making use of complex in- struments (e.g.. instruments that continuous, potentially high-dimensional, and that have non-trivial relationships with the treatment intake or exposure).
Conclusion:
as well as the source of performance gain of our method, analyze the behavior of the methods for different pa- rameters k. For that, for Datasets and 2, report the performance metrics for varying 1: in Table 3 and the esti- mated bounds in Fig. 4. For high dimensional Dataset 3, we display the MSE*, width* and coverage* over varying in Fig. Overall, we observe robust behavior of our method but unstable behavior of the NAIVE baseline wrt. k. The latter also clearly visible by the large differences in the learned bounds in Fig. 4 the left, and the higher variation in MSE*, width and coverage* in Fig. 5, with rapidly declining coverage* of the naïve method for higher k. In contrast, our method performs robust, with close to optimal coverage* even for higher k. Further, in Fig. 5, we observe lower MSE* and # for our method for all k; demonstrating strong improvements in learning tighter but still reliable bounds of the CATE.
Overall, our method yields bounds that are valid for given k as well as over varying values of k, which is naturally encouraged by our objective of flexibly learning represen- tations while penalizing estimation variance. We provide an extended discussion about the role of & and a practical guideline for selection in Appendix F.
Takeaways: Our method successfully learn bounds that have close to optimal coverage and low width. Further, our method outperforms the NAIVE baseline clearly while ensuring robustness. Here, our results show that the source of the performance gain is the way we learn the represen- tation & and that the performance gain from our method

---

Learning Representations of Instruments for Partial Identification of Treatment Effects
Impact Statement
in discrete settings. Journal of the American Statistical Association, 119, 2023.
While our work relaxes the strict unconfoundedness assump- tion for CATE estimation by leveraging instrumental vari- ables, still requires standard validity conditions for instru- ments, which must be justified through domain knowledge or experiment design. By mapping instruments to discrete representation space, we enable reliable partial identification of treatment effects, which is crucial for pecision-mmkinn when point identification is unrealistic. Our method fur- ther aims to reduce estimation variance, which enhances stability in sinite-sample settings, Additionally, the setting we address estimating partial identification bounds for the CATE with complex instruments- has not been directly tar- geted before, as most existing methods rely on untestable and often unrealistic assumptions. By providing an alterna- tive that avoids these strong assumptions, our approach lays an important foundation for future research and offers a start- ing point for further exploration in this underexplored area. This contributes more robust causal inference, particularly in medicine, where unobserved confounding is common and reliable treatment effect estimation is essential.
Feuerriegel, S., Frauen, D., Melnychuk, V., Schweisthal, J., Hess, K., Curth, A., Bauer, S., Kilbertus, N., Kohane, I. S., and van der Schaar, M. Causal machine learning for predicting treatment outcomes. Nature Medicine, 30(4): 958-968, 2024.
Frauen, D. and Feuerriegel, S. Estimating individual treat- ment effects under unobserved confounding using binary instruments. In ICLR, 2023.
Frauen, D., Melnychuk, V., and Feuerriegel, S. Sharp bounds for generalized causal sensitivity analysis. In NeurIPS, 2023.
Frauen, D., Imrie, F., Curth, A., Melnychuk, V., Feuerriegel, S., and van der Schaar, M. A neural framework for generalized causal sensitivity analysis. In ICLR, 2024.
Glymour, M. M., Tchetgen Tchetgen, E. J., and Robins, J. M. Credible mendelian randomization studies: ap- proaches for evaluating the instrumental variable assump- tions. American Journal of Epidemiology, 175(4):332- 339, 2012.
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Learning Representations of Instruments for Partial Identification of Treatment Effects
A. Proofs
A.I. Proof of Lemma 1
Proof of Lemma 1. Proof by contradiction. By Equations (2), (3), and (4), holds that
(z) 2 by (z) Vz E X and
(22)
(x) bj (z) Az E X.
(23)
Assume that Lemma would not hold by
(x) = bx())) < 1. = C P(S) > 0 and by (x) > by (x) Va E E
(24)
(25)
EBT (X) - b7 (X)
(26)
1 bū (I) - bj (x)dP(x) + / bx()) bj (x)
(27)
1 S0 09 (a) - by (x) + / b+(()) by (T)
(28)
1b5 (X) - (x)),
(29)
because S5 bi -
bj (x)dP(a) < (x) by (1) (z) due to the definition of S and Sge (x) - bj (x))P(<) s
Ige b5 (x) by (x) due to the definition of by b This contradicts the definition of 62 by as minimizers of the
average bound width via Eq. (4). The argument for the upper bounds bj b" follows analogously.
A.2. Proof of Theorem 1
We begin by stating result from the literature that obtains valid bounds for discrete instruments. Lemma ((Swanson et al., 2018; Schweisthal et al., 2024)). Under Assumptions I and 2, the CATE is bounded via
b (x) S ((x) S (x),,
(30)
with
b+(x) min bi (z)
and b bz(x)= max bim (z)
(31)
1,m
1.m
where
bi (x) = (x, (z. 1) + (z, - - π(x m)) ( m) - + (2, b (x) = π(, 1) (z, 1) + (1
(32)
- (1 π(x m) )4 (z,
(z, m) x (2, m)
(33)
Proof of Theorem 1. First, note that, for given representation Φ, the representation (Z) is still a valid (discrete) instrument that satisfies Assumptions and 2. Hence, we can apply Lemma 3 using (() as an instrument and immediately obtain the bounds from Theorem 1, but with representation indiieedddddedddddd nuisance functions 1g E) = E X x,A A a, o(Z) = E
and T (z( E)- P(A 1|X x. <(Z) E) for E E {O....,k} .}..
We can write the representatiooninnuuce response function as
z /
= = 2 B| = x, - o
- =,
(() = E) de
x, A= = P(o(Z)= = - =2) dz
P(A= a a((()) ()
(34)
/ 10VX==,AA000 a = - = al7 = =)P(Z = 2) dz
A=
P(A= a dl((Z) = () =
E| x, = =2= =)= = al = =)P(( 2)
12

---

Learning Representations of Instruments for Partial Identification of Treatment Effects
and the representaaioon-induedd propensity score as
P(A = 1 = =z = = / P(A 1|x=1|X= = x, = 2) = 2|0(Z) = f)
P(A - 1|X - =x, x2 2 ==)p((((() - Z/2 - =z) PíZ = 1 22
PP()(?) P) dz
(35)
P()(Z) = E) / P(A 1|x x, Z= =))(((() 1 2) 2) d2,
1
which completes the proof.
A.3. Proof of Lemma 2
Proof. The result follows from
En
(b (x: (x))
27 En (b (z) bz. (x) + bz. (x) (x)
- 83 -b0) 9
(36)
(bt(s)) -t((0) 2 + En (bt. (1r) b (1))
2
(37)
(=)2 (1st()) 6g + Bn bt. bb. (x) 6t())] 2
(
+ Var (8. (o)))
(38)
where we used the vari-nariance decomposition for the MSE for (x).
A.4. Proof of Theorem 2
Proof. We derive the asymptotic distributions of the estimators (2, E) from Eq. (12) and (x, E) from Eq. (13) when assuming oracle estimation of first-stage nuisance functions, such that ft μr, Ω = n, and ij We proceed by analyzing the numerator and denominator of each estimator. First, we show that both are asymptotically normal and then we apply the delta method to obtain the asymptotic distribution the ratios.
Distribution (2x E): Recall from Equation (12) that we can write (z, E) as
S.
Ag(2, E) Nn.
(39)
where
8, 1
W with w pz 2, ((((2)) ( Han((z) + (1 a) (22,))1
(40)
j=1
N₂
ΣΣ Dj,
with D=-1(0(2) a, = a
(41)
We define the moments
wW [[W]= = peO
(42)
W = Var(W) = pr(7e - -P:0?)
(43)
HD = E[D] = pege
(44)
o Var D) = pege(1 pege)
(45)
CWD = Cov D) = 11 - pt),
(46)
where p. =
( 0). - E[) x( = E). and 7/ = Eig(Z)² ((2) = f. with
g(Z) = (z) + (11 - a)(1 (()).. Note that, for better readability, in this proof we avoid the double indexing
showing the dependency on ái which we used in the theorem in the main paper.
13

---

Learning Representations of Instruments for Partial Identification of Treatment Effects
By the central limit theorem, know that
vn
Sn jaw No pD
4 N2
(()) --- (
oo CWD ob
cwD)
)
(47)
Let f() 2. We are interested in the asymptotic distribution of the ratio (z !) = N.)) The delta method states that
(f(((S,,NN)) N.) f(ww, ((ww,DD)) 4 No (o, V PD)
(48)
Using that the gradient is V (AW,KD)= = -12-
we can obtain the asymptotic variance via
HD
2
v WWW
=
AWWWD
(49)
FD
1DD
HDD
1
(7e 07)
07 prge)
(50)
Pt
qi
q
1
Var(g(Z)
Φ(lZ) E)
0 pEge)
(51)
pr
di
Distribution (x, E): Recall from Equation (13) that we can write ( E)
Sn
(x,
(52)
N₂
where
S, W with
W, # 21)111)1(0(2)) =(),
(53)
N₂
Σ
Djr with - 1}.
(54)
n j=1
We define the moments
μW [[ prde
(55)
OW Var(W) = pelce - p207)
(56)
PD E[D] = Pt
(57)
0 = Var D) = ppe(1 - pP)
(58)
CWD D Cov(W, D) -pp0p(1 (1 - pc):
(59)
where p. P(ó(Z) E). E[h(Z) 6(Z) f). and ne E(h(Z)2 ((Z) E]. with h(Z) 5 Z).
By the central limit theorem, know that
Vn N, PD
Sm k1
4 N₂
(()) Σ
ow
CWD)
(60)
CWD D
o
14

---

Learning Representations of Instruments for Partial Identification of Treatment Effects
We then calculate the asymptotic variance using the delta method as above and obtain
Vf (WW HD)
T
2
HwOWD
o D
(61)
u
1
(7e 02²)
(62)
Pt
Var(h(2) | (( E).
(63)
Pt
15

---

Learning Representations of Instruments for Partial Identification of Treatment Effects
B. Real-world relevance and validity of assumptions
In this section, we elaborate on the real-world relevance of our considered setting and show that our assumptions often hold and are even weaker than the ones of existing approaches. For that, we draw upon two real-world settings.
B.1. Mendelian randomization
Mendelian randomization (MR; the main motivational example from our paper) is a widely used method from biostatistics to estimate the causal effect of some treatment or exposure (such as alcohol consumption) on some outcome (such as cardiovascular diseases). We refer to (Pierce et al., 2018) for an introduction to MR, which also shows that MR is widely used in medicine. For that, genetic variants (such different single nucleotide polymorphisms, SNPs) are used as instruments where it is known that they only influence the exposure but not directly the outcome. Our method for partial identification with complex instruments is perfectly suited for this common real-works application. Depending on the use case, either a predefined genetic risk score (Burgess et al., 2020) as a continuous variable, or up to hundreds of SNPs are used simultaneously as IVs to strengthen the power of the analysis, resulting in high-dimensional instruments (Pierce et al., 2018).
Validity of assumptions: The IV assumptions used in our paper such as the exclusion and independence assumptions can be ensured by expert knowledge (e.g., given some observed confounder age (X), genetic variations (Z) do not affect age) or, in some cases, they can be even directly tested for (Glymour et al., 2012). In contrast, as explained in Sec. 2, existing methods for MR rely on additional hard assumptions on top such as the knowledge about the parametric form of the underlying data-generating process. Especially with such high-dimensional IVs, misspecification of these models may result in significantly biased effect estimates. In contrast, our method does not rely on any parametric assumption and also no additional assumptions compared to previous methods, thus enabling more reliable causal inferences in the real-world application of MR by using strictly weaker assumptions than existing work.
B.2. Indirect experiments
With indirect experiments (IEs), we show that, in principle, our method not constrained to medical applications but also highly useful in various other domains. IEs are widely applied in various areas such social sciences or public health to estimate causal effects in settings with non-adherence, i.e., where people cannot be forced to take treatments but rather be encouraged by some nudge (Pearl, 1995). For instance, researchers might be interested in estimating the effect of some treatment such as participating in healthcare program (T) on some health outcome Y by randomly assigning nudges Z (IVs) in the form of different text messages on social media promoting participation. Here, common nudges (IVs) are in the form of, for instance, text or even image data and thus high- dimensional, showing the necessity of a method capable of handling complex IVs such as ours.
In principle, our method can be applied every setting with continuous or multi-dimensional IVs where one wants avoid making the hard untestable assumptions necessary for point identification such as linearity or additivity (e.g., Hartford et al. (2017)). Specific examples for applications with high-dimensional IVs are text-based nudges for encouraging vaccinations (Milkman et al., 2021), or various kinds of experiments where text nudges are generated by different strategies such as for political microtargeting (Hackenburg & Margetts, 2024) or for personalized persuasion in general (Matz et al., 2024).
Another important application area is online marketing. Concrete use cases involve extended A/B testing for evaluating the benefits of new features, e.g., when one interested in the effect of a new version of an app on user engagement. Here, users with features such age, gender, and content preferences (X) can be nudged by emails or push notifications (Z) to test new feature such as using new version of an app (A) to estimate its effect engagement metrics such as screen time (Y). Further, our method could also be extended to improve current methods for optimizing instrument designs for indirect experiments that for now assume identifiability is possible (e.g., Chandak et al. (2023)).
Validity of assumptions: As a major benefit of IEs, the IV assumptions are ensured per design as the IVs are randomly assigned, and, thus they always hold. Hence, our method provides a promising tool for evaluating the effects of IEs.
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Learning Representations of Instruments for Partial Identification of Treatment Effects
C. Implementation and training details
Model architecture: For all our models, we use MLPs with ReLU activation function. For ugg, we use layers to encode X and layers to encode Z. Then, we concatenate the outputs and add 2 additional shared layers. Finally, we calculate the outputs by separate treatment head for A 0 and = ensure the expressiveness of A for predicting Y. For f, we use the same architecture. For ii use 3 layers. For Φ9, we also use 3 layers and apply discretization top of the K outputs (Jang et al., 2017). For the nuisance parameters of the k-means baseline, we use the same models as for ing and ã for fair comparison. We use a neuron size of 10 for all hidden layers.
Training details:
For training our nuisance functions, we use an MSE loss for the functions learning the continuous outcome Y and a cross-entropy loss for functions learning the binary treatment For all models, we use Adam optimizer with a learning rate of 0.03. We train our models for a maximum of 100 epochs and apply early stopping. For our method, we fixed À 1 and performed random search tune for [0, 1] for 7. We use PyTorch Lightning for implementation. Each training run of the experiments could be performed on CPU with 8 cores in under 15 minutes.
17

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Learning Representations of Instruments for Partial Identification of Treatment Effects
D. Data description
Dataset I: We simulate an observed confounder
Uniform[- |-1, 1] and an unobserved confounder U Uniform[-1 1].
The instrument Z is defined
(1 Uniform[-1,11 Beta(2,2) - =4" ( BBBeaa(2 2))
Mixture
(64)
We define p as
P = 1 + exp (- ([2)Z] - max 2)) + + U))
(65)
Then, the propensity score is given by
π = (p 0.5) 0.9 + 0.5.
(66)
We then sample our treatment assignments from the propensity scores as
A ~ Bernouli( (T).
(67)
The conditional average treatment effect (CATE) is defined as
(2.5X)) + 12sin(6XX + 0.5 + 0.5.
r(X)
(68)
80
The outcome Y is then generated by
Y= (X 0.50 + 0.1 Laplace(0, 1)) 0.25 + T(X) A.
(69)
Dataset 2: We keep the other properties but change the propensity score to be more complex, which results in harder-to-leam optimal representations of Z for tightening the bounds. The propensity score is given by
0.04
sin + U 0.48 + 0.48 1 exp(-3
(70)
Dataset 3: We simulate and as above. Then, we sample a d-dimensional Z E {0, 1]dd with 20 as
Z ~ Binomial (d, 0.5).
(71)
Thus, our modeling is here inspired by using multiple SNPs (appearances of genetic variations) as instruments (Burgess et al., 2020), where we simulate potential variations for 20 genes.
Then, we define
(72)
and the propensity score, inspired by the more complex setting Dataset 2,
0.04
= 0.48 + UU) 0.48 + + exp(-335ppp
(73)
Then, we define the CATE as
T(X) (1.66X 0.5)* + 12 1.5) cos(( (X)
+ 0.5.
(74)
80
and the outcome dependent on т, and analogously as for Datasets and
18

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Learning Representations of Instruments for Partial Identification of Treatment Effects
Dataset 4: To test method in higher-dimennionall settings, we consider 4th dataset with 100-dimensional IVs. For that, adapt the DGP from dataset 3 but set 100. Then adjust the latent discrete IV score
L S 25}
(75)
By Eq. (72) and Eq. (75), we ensure that some the modeled SNPs are irrelevant for π and thus do not affect the treatment or exposure A. Thereby, we focus on realistic settings in practice, where the relevance instruments cannot always be ensured which imposes challenges especially for existing methods for point identification, but not for approach. Further, ensure that the latent score P can only take discrete levels for dataset and 25 discrete levels for dataset 4. This allows us to approximate oracle bounds using the discrete bounds top of p by leveraging Lemma 3 such that we can evaluate our method and the baseline in comparison to oracle bounds.
To create the simulated data used in Sec. 6, we sample n = 2000 from the data- tatta-enerating process above. We then split the data into train (40%), (20%), and test (40%) sets such that the bounds and deviation can be calculated on the same amount of data for training and testing.
19

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Learning Representations of Instruments for Partial Identification of Treatment Effects
E. Additional Results
E.1. Additional baselines
As mentioned in the main paper, existing methods are not designed for our considered setting of continuous or high- dimensional IVs with binary treatments. However, to further show the advantages and necessity of our tailored method, we compare with two additional baselines that were not developed for our task but which adapted for our task, namely, one from uncertainty quantification for point estimates and one from the discrete instruments setting:
(i) Deep/V with bootstrapped confidence intervals. DeepIV (Hartford et al., 2017) is neural method tailored for high- dimensional instruments when point identification can be ensured. This requires the additional assumption of additivity of the unobserved confounding, which usually cannot be ensured and is not necessary for our method. For DeepIV, we can approximate confidence intervals using bootstrapping. Here, we approximate confidence intervals with a confidence level of 95%, indicating an expected coverage of 95% if assumptions were not violated. However, note that these intervals can only adjust for statistical uncertainty, but not for identifiability uncertainty due to the violation of causal assumptions. Thus, this baseline acts as additional motivation for why bound estimators such as method are important.
(ii) Discretized IVs: As further additional baseline, we proceed by directly discretizing the high-dimensional IVs and then estimating the existing bounds for discrete IVs. Hence, one loses information from the IV due to the discretization. Our implementation here is the same as for the naïve baseline, however, the l partitions are not learned by k-means clustering but instead defined by a simple grouping rule. To ensure a fair comparison, we average the results of experiments conducted with the same number of partitions k for all methods.
Metric
DeeplV (CI)
Discretized
Naive
Ours
Rel. Improvement
0.52 = 0.29
1.00 0.00
1.00 0.00
1.00 0.00
0.0%
Coverage*1
0.99 0.00
0.84 4: 0.37
0.97 1 0.05
0.0%
Width*[[]
1.91 1. 0.07
1.88 1 0.06
1.82 1 0.04
3.2%
MSK*LLI
0.13 + 0.02
0.12
+ 0.02
0.10
+ 0.02
16.6%
MSD[J]
0.08 1 0.03
0.10
0.10
0.03
+ 0.02
70.3%
Table Dataset 3: Comparison methods (Naive vs Ours) on coverage and width metrics with relative performance improvement. Note: means that there are no reliable runs for which the corresponding performance metrics could be calculated.
Results: We report our results for Dataset 3 in Table 4. We observe that the DeepIV method, as expected, falsely overconfident bounds with only about 53% coverage of the true CATE and no coverage of the oracle bounds. Thus, there no reliable runs for which the other metrics could be calculated (denoted by " "_" in the tables). This emphasizes the necessity for using bound estimators. Further, observe that the discretized baseline gives more conservative and wider bounds under similar coverage (higher Width and MSE*) and performs less robustly with regard to k (higher MSD). In sum, the results confirm the strong performance of our method.
E.2. High-dimension dataset
Metric
DeeptV (CI)
Discretized
Naive
Ours
Rel. Improvement
Coveragel
0.01 0.00
1.00 + 0.00
1.00 : 0.00
1.00 : 0.00
0.0%
Coverage*11
0.00 0.00
1.00 1 0.00
1.00 + 0.00
1.00 1 0.00
0.0%
Width*[U]
1.90 1 0.08
1.82 0.13
1.75 + 0.08
3.7%
MSR*[L]
0.26 + 0.03
0.23 0.05
0.21 1: 0.03
10.9%
MSD(J]
0.05 1. 0.03
0.10 1 0.04
0.05 1 0.01
48.2%
Table Dataset 4 dineesiinnii IVs): Comparison of methods (Naive Ours) coverage and width metries with relative performance improvement. Note: means that there are no reliable runs for which the corresponding performance metrics could be calculated.
To show the validity of our method in even more nigh-dimensiooal settings, we added additional experiments with 100- dimensional IVs. For that, we introduced our Dataset 4 (see Appendix D). We report the results for our method and the same baselines as in the previous section. Further, for the higher-dimensional setting, we varied the hyperparameter k over [2, 7, 10, 20) for all bound estimation methods. We observe similar patterns as for our other dataset. In particular, the DeepIV baseline fails entirely to provide reliable bounds. In summary, our method shows robust performance by providing tighter and more reliable bounds than the bascline, even in high nigh-diimnsional settings. This emphasizes the applicability of our bounds in even more complex settings.
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Learning Representations of Instruments for Partial Identification of Treatment Effects
E.3. Ablation studys
To further examine the robustness of our method in non-standard settings, we perform two additional ablation studies, one for varying the DGP and one for varying the selected nuisance models.
Linear DGP: To analyze if our flexible method also performs robustly in simple settings, we evaluate our method which uses neural networks at every stage on a simple linear DGP. For that we adapt our Dataset 3 and use linear functions for the dependencies between the variables. We report the results in Table 6. As expected, our method performs also robustly in the simpler linear setting and outperforms the baseline by a clear margin again. Summarized, our method shows strong performance which emphasizes its applicability to datasets of various complexity levels.
Metric
Naive
Ours
Rel. Improve
Coverage[{]
1.00 = 0.00
1.00 = 0.00
0.0
Coverage"[11
0.92 0.18
1.00
8.6%
Width+IJ1
2.07 2 0.04
1.99
3.9% + 0.05
MSE
0.01
0.08
20.0%
MSD[L]
0.08 + 0.08
0.04
+ 0.03
50.0%
Table Linear DGP: Comparison of methods across key metrics. Relative performance improvements in green.
Non-linear DGP with linear models: In our method, we leverage neural networks at all stages to allow for consistent and flexible estimation of all properties. However, since our method is model-agnostic in principle, we analyze the behavior of our method when using non-flexible (mis--pecified) models. For that, we implement our method and the baseline by using linear models for the nuisance estimates and evaluate the performance on our non-linear Dataset 3 (i.e., the nuisances and the bounds misspecified). We report the results in Table 7. As expected, because of the misspecification of the nuisance models, full coverage of the bounds cannot be guaranteed. However, our method still outperforms the naïve baseline evidently with respect to coverage and MSD while yielding similar bound tightness. Further, with coverage to the oracle bounds over 90% and low MSD, our method still predicts close to valid bounds robustly over different runs which is unlike the naive baseline. This shows that our method is also robust against misspecification of the nuisance models as when using linear models for non-linear datasets.
Metric
Naive
Ours
Rel. Improve
Coverage[f]
0.96 = 0.06
1.00 - 0.00
4.1%
Coverage"[11
0.28
0.91
51.2%
Width+
1.91 1 0.02
1.91
- 0.03
0.0%
MSE
0.04
0.14
0.02
0.0%
MSD[L]
0.20 # 0.11
0.02 - 0.01
90.0%
Table Non-linear DGP with linear nuisance models: Comparison of methods across key metries. Relative performance improvements in green.
E.4. Real world experiments: ADJUVANT study
We provide results using real-world data from an ADJUVANT chemotherapy study (Liu et al., 2021) as provided in https://ggt thub. com/ cancer oncogenomi real-world data to study the effect of exposure (smoking) on progression-free survival in cancer. This yields a typical MR setting; i.e., we use 22 genetic variations IVs, smoking status (binarized as yes/no) the exposure, and disease-free progression (DFP) as the outcome, while controlling for possible confounders such as age. We report the results in Table 8, While yielding slightly tighter bounds, our method shows clearly lower variation with respect to k, indicating robust behavior.
cs mwi///inaaaaaa/eeemmmemmeeeaaaem wot.rrro-oooogoooooiio
0.00 C. We use the
Further, in Fig. 6, we display the estimated bounds on the CATE of disease-free progression (DFP) in months conditional on age averaged over multiple k. We observe clearly lower variation in bound estimates for our method compared to the naive baseline. Further, our method shows more stable estimates over different ages and centering around an effect of 0. This is as expected, the genetic variations in the study are not selected to be strongly correlated with the exposure (smoking) and we do not expect strong heterogeneity in age on DFP.
In Fig. 7, we show the estimated width for different k. As expected, both methods show lower width for higher k, but also increased variance. Thus, for real-world application a sensitive field like medicine and without access to ground truth CATES for evaluation, a possible strategy would be to select a lower i such as & = 2 or 3. This results in a bit wider bounds but also clearly reduced variance and more reliable estimates than for k 7.
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Learning Representations of Instruments for Partial Identification of Treatment Effects
Metric
Naive
Ours
Rel. Improve
Width[U]
48.10 0.76
47.72 + 0.90
0.79%
MSD[J]
9.43 3
2.18 + 0.80
76.88%
Table 8. Real-world data: Comparison of both methods (NAivE vs. Ours) regarding width, and MSD. The other metrics such as coverage cannot evaluated due to lack knowledge the ground truth CATE. Relative performance improvements in green. Results are reported for lo (2, 3, 5, 7)
Method
Dura
CAVE (DPS In Months)
48,0
20
47.5
width
-20
46,0
Method
Ours
-30
45.5
Figure 6. Estimated bounds on the CATE of fisease-free pro- gression (DFP) in months conditional on age. Averaged over multiple k.
Figure 7. Estimated width for different k.
F. Role of number of partitions k
F.1 Why our method is robust to different choice of k
One major advantage of our method is that it is clearly less sensitive to the hyperparameter & than, for example, the naïve baseline. Empirically, we demonstrate this in our experiments by lower variance and stable behavior over varying especially visible in the low values of MSD. This is due to the combination of learning flexible representations tailored to minimize bound width (allowing us to estimate tight bounds already for low k) while ensuring reliable estimates of the nuisance functions in the second stage by using our regularization loss in Eq. (19) (ensuring robust behavior also for higher k).
Note that the robustness of our method is especially beneficial when applying our method to real-world settings in causal inference. In real-world settings from causal inference, hyperparameter tuning and model evaluation are not directly possible because oracle CATE or oracle bounds are not known. Thus, the robustness against suboptimal selection of hyperparameters such as k is crucial. In the following, we provide further high-level theoretical insights into the role of and propose practical recommendations for selecting & in real-world applications.
Estimation error for different h: The hyperparameter À controls the regularization loss in Eq. (19), i.e., it tries to maximize P., = (Z) = E) > for all E 1,..., A. Thus, if we choose À high enough, then we enforce that P.. 1/k for all E l.....kk k. Plugged into Theorem 12, the asymptotic variances for the nuisance estimators are k: \aarg(()) + d) for (z, E), and is (Var(h(Z) <(Z) = 2)) for (2, E), respectively. Thus, for large enough X, the variance of the nuisance estimators (and, thus, also likely of the final bounds) will increase for increasing However, as an interesting side note, for fixed (not too large) X, the penalization term Eq. (19) will also grow with growing & due the same reason, which yields automated stabilizztiin for higher lt. This is also shown in our experiments where higher values of & do not necessarily result in higher variance.
Bound tightness for different k: On a population level, the bounds get tighter with growing k. This follows straight- forwardly from Theorem 1, since using more increases the flexibility of Φ. While the exact bound width is highly non non-trivial, we can use results from Schweisthal et al. (2024) about bounds for the CATE with discrete instruments to give some intuition. Specifically, in our setting, for some v, the bound width bounded by by (x) b5 (x) S
22

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Learning Representations of Instruments for Partial Identification of Treatment Effects
ming, ((0g (2 (2(2, (1 (2 m)))} with m A)). This has two major implications. First, if for some n, Φ is learned such that (x, E) is close 1 for some I and <0( m) is close for some m, the bound width is close to zero ("point identification"). Second, if the optimal partitioning function e is the same for all (implying b(x) b). then setting & = 3 can be sufficient to yield the tightest bounds. This is because, by using a flexible network for o, the partitions can be learned such that partition yields propensity scores as close as possible to zero (as the data allows), partition 2 yields propensity scores as close as possible to I, and partition 3 contains all & resulting in propensity scores between those values. Note, however, that this only valid population but can result highly unreliable estimation in finite sample data.
F.2. Practical guidelines for selecting k
Although we showed that our method designed to be robust against different selections of k, we provide two potential guidelines for how choose & in real-world settings where ground-truth CATE or bounds are not available for model selection.
Approach 1: Expert-informed approach. In some medical applications, physicians might already know or make an educated guess about a number of underlying clusters of patient characteristics such as genetic variants. For instance, this is common assumption in subgroup identification or latent class analysis in medicine where patient groups are characterized by having similar responses to treatments or showing similar associations with diseases (Kongsted & Nielsen, 2017). Thus, no data-driven approach necessary here but one can integrate existing domain knowledge.
Approach 2: Data-driven for hypothesis confirmation. Often, physicians are interested in whether some treatment exposure has positive negative effect (i.e., lower bound 0 or upper bound < 0) for at least some observations Thus, k can be selected by increasing k until such effect can be observed while holding the variance minimal. Then, the variance can be approximated (e.g., by bootstrapping to test for the reliability of the corresponding bound model and its effect). Thus, this approach can be used when our method is used as a support tool for hypothesis confirmation.
Last, straightforwardlyy, from an exploratory perspective, all hyperparameters (k;, X, 7) can be altered together to examine the behavior of bound width and estimation variance to post-hoc find suitable hyperparameter configuration for dataset that fulfills the subjective preferences of the practitioner.
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Learning Representations of Instruments for Partial Identification of Treatment Effects
G. Sensitivity analysis
We perform a sensitivity analysis over the hyperparameters in our custom loss function. We report the results Fig. and Fig. 9 for dataset 3 and for l We observe that y does not affect the bound size but can be optimized reduce estimation variance, as mentioned in the motivation of our auxiliary guidance loss. Thus, À demonstrates the trade-off' between tightness and variance and shows the importance of our regularization loss. Here, À can be increased to reduce the variance. In our experiments, the optimal trade-off between reduced variance and bound tightness also results in optimal oracle coverage, showing the practicability of our regularization.
1.9
1.35
1.00
1.25
145
04
12"
10"
12"
00"
10"
19"
10'
12'
kerbdda (log Sraln)
aabiiaa j.ng Scats)
Figure 8. Sensitivity over X. Left: Average bound width. Right: Oracle coverage. Averaged over 5 runs + sd.
1.85
Orack
1.80
1.80
0.0
1.29
1.20
1.85
1.90
13"
12"
12"
16
12'
ganima Soals)
gamma Stals)
Figure 9. Sensitivity over y. I.eft: Average bound width. Right: Oracle coverage. Averaged over 5 runs + sd.
24

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Learning Representations of Instruments for Partial Identification of Treatment Effects
H. Training procedure
Algorithm 1: Two-stage learner for estimating bounds with complex instruments
Inpul sobservational data sampled from (2, A, Y), spochs e, buich size 7ay, neural network dy with parameterss learning rale Output: hounds 0gg (2), 67 (a)
// sirst stage (muisance estimation)
(") B/y X a, =
((=)) - 1 ==
-
Second-stage itar=ition learning and bound calculation!
for (I,...,e) botches do
for (I.....k) do
P (z, (L,
E, (1 = (a((())) +11 a)(1 ((zy)))
Rgg (x, End ((0 Σn ((dos)
end
bgg (a) mini. 8 a (x), 800 (w) maxi bo...m for (I....,) K}
((6) - (x(0) + (0) (0) + CCam (0) as per Sec.5
end
/7 Final bounds
return 87 (w)) 6 (z)
25


Reviews:
[{"review_id": "KLLtTL1Soh", "reviewer": "Reviewer_d3FE", "summary": "To partially identify and estimate the bounds of the conditional average treatment effect (CATE) with potentially high-dimensional instruments, the authors propose a two-step approach. This method first learns discrete representations of the complex instrumental variables $Z$, then derives closed-form bounds based on these representations, and finally maps the bounds back to the original problem. The authors present two theorems for computing these bounds: one from a population perspective and the other from a finite-sample perspective. Additionally, they introduce a neural approach for learning CATE bounds when dealing with complex instruments. Experimental results demonstrate that the proposed method achieves superior validity and robustness.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed feedback and the overall positive evaluation of our manuscript! We will take all your comments to heart and improve our manuscript accordingly. Below, we provide answers to all your questions\n\n# Response to Methods and Evaluation Criteria\n\n- **Benchmark datasets**: We agree that providing insights on real-world data is an interesting extension for our paper. However, it is standard in causal inference literature to rely on _synthetic data-generating_ processes for benchmarking, since the CATE (and for our setup, also optimal bounds on the CATE) can **never be observed in real-world data**. Thus, we provide benchmarking using synthetic DGPs that are **closely tailored to the real-world settings in Mendelian randomization (MR)**, such as polygenic risk scores and SNPs, and provide results for different levels of complexities. Further, by using our DGP for datasets 3 (and 4 in the Appendix), we can approximate _oracle bounds_, allowing us to check for validity also with respect to optimal bounds and not only compared to the CATE, **which is an extension over previous work** in similar settings. \n\n**Action**: To further strengthen our paper, we provide **additional insights on real world data** in MR (https://anonymous.4open.science/r/IVRep4PartId-714C/rebuttal/rebuttal_experiments.pdf). **Overall, our findings are consistent with our synthetic setup and indicate robust performance of our method**\n\n# Response to Experimental Designs or Analyses\n\n- **Additional baselines and experimental setups**: Thank you! Importantly, we focus on the highly relevant setting with _complex instruments_ and _binary treatments_ as it has been neglected in prior work. On the other hand, this implies, that our method is _not tailored or applicable_ to setups considered by existing papers (i.e., our method cannot be naively adopted to continuous treatments; and, for simple discrete IVs, no tailored representation learning is necessary since the closed form bounds can be estimated directly). Thus, we focus on our considered datasets. However, we agree that additional baselines besides the naive one are interesting. Thus, in **Appendix E**, we provide results for additional adapted baselines to show that our tailored method is clearly superior and necessary.\n\n# Response to Strengths and Weaknesses\n\n- **Contribution**: We agree that the reformulation of $\\mu$ and $\\pi$ is one of our main contributions as this allows us to avoid alternating learning. However, we would kindly highlight our major other contributions: (i) **Novel setting**: We are the **first** to directly focus on **partial identification with complex instruments such as in MR**, (ii) **Our learning algorithm**: We do _not_ only avoid alternating learning and leverage discrete representation learning but also encourage reduced estimation variance by our _tailored loss function_ as shown in Theorem 2.\n\n- **Dataset 1, Table 3 performance**: As the DGP for Dataset 1 is simple, a naive clustering with $k=3$ could already lead to close-to-optimal bounds. However, importantly, for Datasets 1 and 2, _lower width does not necessarily indicate better performance_. By using only finite data for estimation, the bounds could also be **falsely overconfident**, even if they lead to full coverage of the true CATE, as the naive baseline does not try to reduce estimation variance. Since, in these DGPs, we use _continuous IVs_, we cannot approximate the oracle bounds (as for Datasets 3 and 4). Instead, the main finding here is that **our method leads to robust estimates for different $k$, while the naive baseline highly depends on the selection of $k$**.\n\n- **How to set $k$ in practice**: As mentioned, a major benefit of our method compared to the baseline is the _robustness regarding $k$_. However, since hyperparameter selection is hard in applied causal inference due to a _lack of access to the ground truth CATE_, we provide practical guidelines in **Appendix F2**. Therein, we propose two approaches: (1) an expert-informed approach and (2) a data-driven approach, which can be used seamlessly in practice.\n\n# Response to other Comments\n\nThank you for your careful reading! All of your points are very helpful and we will include them in the camera-ready version of our paper.\n\n# Response to Questions\n- Thank you for your questions! For the first two questions, we kindly refer to our responses above.\n- **Invalid IVs**:  As in typical MR and IV settings, we need to ensure that the exclusion and independence assumptions hold. However, unlike usual methods, we do **not** require the relevance assumption: we can – in principle – also use SNPs that are not associated with the exposure /treatment. Since these irrelevant IVs do not lead to closer bounds, our method will just not use their information. Our datasets 3 and 4 indeed contain 75% of irrelevant IVs, which demonstrates the **strong performance of our method in this relevant scenario**.", "claims_and_evidence": "Yes. I did not find any problematic claims.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria are well-aligned with the problem at hand. The method effectively captures the key characteristics of CATE by complex $Z$. The evaluation metrics such as convergence, width, and MSD effectively illustrate the performance of the proposed methods. However, the benchmark dataset used in the study is a simulated dataset, while it is better to perform experiments on a real-world dataset.", "theoretical_claims": "I checked Theorem 2 and found no discussed issues.", "experimental_designs_or_analyses": "The authors simulated three datasets to conduct MR simulations. The evaluation metrics include coverage, width, and MSD for Datasets 1 and 2, while for Dataset 3, they introduced three analogous metrics. Additionally, the authors provided implementation details and conducted a sensitivity analysis on different $k$. However, I believe the comparison is limited to only a naive baseline, lacking comparisons with more alternative methods. Furthermore, the experimental settings could be adjusted to settings where existing methods are applicable, facilitating a fairer comparison.", "relation_to_broader_scientific_literature": "This paper introduces a two-stage approach for partial identification of treatment effects using high-dimensional instruments, leveraging discrete representations to obtain closed-form bounds. \nPrevious works on Machine Learning for CATE estimation (e.g., Singh et al., 2019) focused on settings where the treatment effect can be identified and used ML methods with favorable properties (e.g., Kennedy et al., 2019). \nIn partial identification with binary treatment, Robin and Manski proposed closed-form ATE bounds for bounded $Y$, and in the binary case, Balke and Pearl derived tighter bounds. \nFor discrete variables, some researchers have provided broad overviews, but the issue is that these methods fail to effectively leverage continuous and high-dimensional instrumental variables to learn tighter bounds. In contrast, the method proposed by the authors expands the applicable scope and achieves tighter bounds.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "- The paper is well-written and clearly structured, making it easy to follow the key arguments and contributions. The figures and illustrations are well-designed and effectively convey the key insights of the paper.\n- The contribution in this paper appears to be somewhat limited. The core contribution primarily involves redefining the forms of $\\mu$ and $\\pi$ from Lemma 2 under a discrete representation $\\phi$, directly leading to Theorem 1. \n- In Table 3, the naïve method for dataset 1 performs significantly better in terms of width. Can the authors provide an explanation for this observation?\n- It seems that $k$ is important for the performances, from the sensitivity analysis. Performances are different with different $k$. How to set $k$ in practice?", "other_comments_or_suggestions": "1.\tThe sentence 'There are many reasons, including costs …' reads awkwardly and contains a grammatical issue. \n\n2.\tSome notations in Theorem 2, such as $q$ and $\\theta$, only appear in the proof and are not found in the main text, which may \nconfuse readers.\n\n3.\tThe abbreviation \"wrt.\" should be explained the first time it appears.\n\n4.\tIf possible, it would be helpful to place Figure 4 near Figure 5.", "questions_for_authors": "- The experimental section lacks real-world datasets and relies solely on simulated experiments, which do not demonstrate the performance of the method in real-world settings. \n- Additionally, the comparison with other methods is insufficient. Since the authors claim that other methods are not suitable for their setting, and their method can handle more complex instruments $Z$, why not apply their method in a simpler setting and compare it with other methods? This would make the argument more convincing.\n- There are many invalid IVs in MR. In this case, does the proposed method still work well? Or how to deal with these invalid IVs?", "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": "vuv4PsUFWd", "reviewer": "Reviewer_QCKo", "summary": "The paper proposes a method for partial identification, that is, bounding, of treatment effects in the instrumental variable (IV) setting. Specifically, the paper studies the scenario where the instruments are continuous and potentially high-dimensional. It proposes an approach for partial identification through a mapping of instruments to a discrete representation space, and learning the discrete representation by minimizing the width of the bounds. Theoretical results are provided claiming the validity of the bounds. Experiments are performed to show the effectiveness of the proposed method against a naive baseline.\n\n## update after rebuttal\nThanks for the rebuttal to clarify a few things. Overall, the novelty of the paper is acceptable although somewhat incremental. The experimental results show that the proposed method is less sensitive to $k$ but are not convincing in showing that the proposed method is actually better than the naive baseline when a best $k$ is selected. I've updated my score to Weak Reject.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review and the opportunity to clarify multiple points of our paper! We took your comments at heart and improved our paper as follows. \n\n# Response to concerns around claims and evidence, theoretical claims, and relation to broader scientific literature\n\n- **Novelty of our paper and implications of Theorem 1**:  We would like to emphasize that, while the extension of existing closed-form solutions to complex instruments is novel (i.e., even the naive baseline has not been considered yet), this is only _a small part_ of our contribution. Importantly, instead of **only deriving** valid bounds, we focus on a method for **improved estimation**. Based on this, we introduce the new objective of directly minimizing bound width during representation learning (Eq. 8). \n\nImportantly, in Theorem 1, we show that this objective can be optimized _without alternating learning_: we can directly evaluate the necessary quantities from pre-trained nuisance functions by leveraging Eq. 3 and Eq. 4. (otherwise, we would need to retrain $\\mu_\\phi$ and $\\pi_\\phi$ after every update step of $\\phi$ in an alternating way). Thus, with our formulation, **we avoid instability and high computational cost**. (see also “Implications of Theorem 1” in our paper). Lastly, to ensure robustness in finite samples, we regularize for balanced partition probabilities (Eq. 14), _reducing estimation variance as shown in our Theorem 2._\n\n- **Clarifications on our objective in Eq. (1), and claims about “tightness” and “validity”**: The validity of our bounds in population follows straightforwardly from Theorem 1. In finite samples, as for every method, full validity (i.e., guaranteed 100% coverage) can never be proven. However, by using our regularization loss, we mitigate the risk of overconfident estimation, which is unlike existing methods that do not account for estimation uncertainty. \n\nIn addition, we refer to tightness by minimizing the expected bound width as much as possible given some representation $\\phi$. Importantly, this usually does not lead to **sharp bounds** (i.e., the “tightest possible ones” for all $x$ which would only minimize the population tightness in Eq. 12), since this would not allow us to reduce the estimation variance. This is also our motivation for our objective in Eq. (1): _We aim to reduce the average bound width to get robust estimates of “informative” bounds for the largest part of the population_. Here, “informative” refers to “being useful for the underlying decision-making problem”. E.g., for classical CATE estimation, informative means that the lower bound is greater or the upper bound is below some decision threshold (often 0). \n\n**Action:** While we are consistent in our terminology with previous work on partial identification, we will add a discussion about the different terms to improve the clarity of our work. \n\n# Response to experimental designs and analyses\n\n- **Adaptation of existing methods**: As these methods often include additional assumptions over the continuous interventional distribution in the form of additional constraints, the adaptation is not straightforward. However, in **Appendix E**, we provide results for additional adapted baselines, such as extensions of methods that were designed for point identification.\n- **Significance of results**: As standard in machine learning, we avoid making inferences about statistical significance since this would require careful adjustment for dependencies and multiple testing, but instead report mean and standard deviation over multiple seeds.\n- **Role of $k$ in Table 3**: Dataset 1 is defined in a simple way such that a naive partitioning with $k=3$ can already closely approximate optimal bounds. Interestingly, with k=4 the partitioning gets worse, yielding only a coverage of 0.49 for the naive baseline, while still yielding 1.00 for our method. _This shows the robustness of our method_, and even may indicate that the narrow bound width of 0.83 for the baseline for k=3 may be due to _falsely overconfident_ bound estimates.\n- **Evaluation and selection of $k$**: We average over the k-values considered in our sensitivity analysis as reported in Table 3 and Figure 5. The intuition here is that, in real-world applications, one does not have access to the ground-truth CATE or oracle bounds, and thus coverage cannot be checked to select the best k. When only considering width as the criterion, this would lead to always choosing the tightest bounds, which are likely to not yield full coverage, especially for the baseline. In **Appendix F**, we provide an extended discussion over the role of $k$ and give guidelines on how to select k in practice.\n\n# Response to other comments or suggestions\n- Thanks for your careful remarks! We will adjust our paper accordingly. Regarding the “Implications of Theorem 1”, we kindly refer to our answers above.\n\n# Response to Questions\n- Thank you for your questions! We addressed all of them in our answers above.", "claims_and_evidence": "Overall, it feels that the paper exaggerates the novelty of the approach. It looks to me the proposed method is a somewhat direct extension of the existing closed-form bounds with discrete instruments. Rather than simply discretizing continuous instruments and using the existing bounds (the naive baseline), the paper proposes to learn a discrete representation of the continuous instruments that optimize bound width. The existing bounding results (Lemma 2) should be presented in the main paper.\n\nIn addition, I'm not sure about the claims regarding the \"tightness\" and \"validity\" of the bounds obtained by the proposed method throughout the paper. I think these claims are rather misleading.  \n\n-I don't understand the objective given in (1). To my understanding, we want valid and \"tight\" bound $b^-(x)$, $b^+(x)$ for each given $x$. Why would one want to minimize the \"expected\" bound width? In what sense are you claiming the \"tightness\" of the bounds given by (1)? Why do you call this \"informative\"? To my understanding, \"informative\" has a different meaning.\n\n-Overall, I'm not sure about claiming tightness and validity under discretization and finite samples.", "methods_and_evaluation_criteria": "Overall, the proposed method and evaluation criteria, including test datasets, are reasonable and make sense for the problem at hand.", "theoretical_claims": "I did not check the details of the proofs for theoretical claims. They look reasonable. Theorem 1 is a somewhat direct extension of existing results. Lemma 1 is standard.", "experimental_designs_or_analyses": "The overall experimental designs and analyses are reasonable. However, I have the following concerns:\n\n-Can existing methods in IV settings with continuous treatments be tailored for binary treatment and therefore used as baselines (even if they are not directly tailored for binary treatment)?\n\n-In Tables 1 and 2, some of the improvements in Width and MSE are small. Are the improvements actually statistically significant?\n\n-In Table 3, it looks like the Naive baseline improves significantly with increasing $k$ values. What are the results for $k\\geq 3$?\n\n-I'm not sure it makes sense to show the results that are averaged over different $k$ values (what $k$ values are the results averaged over?). It makes more sense to me to select the best $k$ value. Have you tried choosing the best $k$ instead?", "relation_to_broader_scientific_literature": "To my understanding, the work directly extends the existing closed-form bounds with discrete instruments given in Lemma 2.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "none", "other_comments_or_suggestions": "-I believe assuming the causal structure in Fig. 1 implies the identifiability Assumptions 2 and 3.\n\n-$s_1$, $s_2$ in (7) and (8) are not defined.\n\n-I couldn't follow what the paragraph \"Implications of Theorem 1\" on page 4 is talking about.\n\n-Eq. (24), (25) in Line 237 should be (3) and (4).", "questions_for_authors": "1. I don't understand the objective given in (1). To my understanding, we want valid and \"tight\" bound $b^-(x)$, $b^+(x)$ for each given $x$. Why would one want to minimize the \"expected\" bound width? In what sense are you claiming the \"tightness\" of the bounds given by (1)? Why do you call this \"informative\"? To my understanding, \"informative\" has a different meaning.\n\n2. Can existing methods in IV settings with continuous treatments be tailored for binary treatment and therefore used as baselines (even if they are not directly tailored for binary treatment)?\n\n3. I'm not sure it makes sense to show the results that are averaged over different $k$ values. It makes more sense to me to select the best $k$ value. Have you tried choosing the best $k$ instead?", "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": "OkMEziSvAL", "reviewer": "Reviewer_XTJ8", "summary": "The authors tackle the problem of partial identification of treatment effect with high dimensional instrument variables that have a complex relationship with the treatment. They do this by learning a discrete representation of the instrument variable and deriving a learning objective that does not require retraining the nuisance functions at each iteration step. A loss then introduced that includes extra regulariser terms that ensure good performance in the finite sample setting. They compare their method against a \"naive\" baseline that discretises the instrument (via kmeans) and applys existing bounds for discrete instruments. The performance of their estimator improves over this baseline.\n\n## update after rebuttal\n\nI will keep my positive score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you a lot for your positive review and your helpful feedback! We took your comments at heart and improved our paper as follows.\n\n# Response to Claims and Evidence\n\n**Wording of contribution**:  Thank you for this remark! We use the term “effective” to reference the good performance of our method when optimizing for our different objectives: (1) learning tight bounds with low average width, (2) providing valid bounds with full coverage, and (3) demonstrating robust estimation without requiring alternating learning or retraining of nuisance functions while providing stable performance with low MSD between different values of $k$. However, we agree that this terminology is a bit loose. \n\n**Action:** We will spell out more clearly which property of our method we refer to, and we will thereby spell out our contributions more clearly to avoid confusion.\n\n\n\n# Response to Methods and Evaluation Criteria\n\n\n**MSE*** **and Width*** : We agree that the MSE* and Width* would be also interesting metrics for Datasets 1 and 2. However, to calculate these metrics, we need to approximate the oracle bounds. Since we use continuous IVs for datasets 1 and 2 to show diverse settings, ground truth bounds cannot be estimated by closed-form solutions. For the high-dimensional Dataset 3 (and Dataset 4 in the additional experiments in **Appendix E**), we model the IVs to be dependent on a lower-dimensional discrete space such that we can approximate the oracle bounds. However, we agree that our paper would benefit from explaining this more thoroughly. \n\n**Action:** We will include an additional paragraph where we discuss our considered metrics in more detail.\n\n# Response to Weaknesses and Other Comments Or Suggestions\n- **Improved Presentation:** Thank you for your careful checks! **Action:** We will use the additional page of the camera-ready version to improve our presentation significantly including FIgures 3 and 4.\n- **Figure 3**: As you correctly pointed out and as presented in Appendix H, our current Figure 3 represents one training step. **Action:** To improve clarity, we will add our loss function to the final Figure and distinguish between (i) the training loop to update the parameters, and (ii) the final forward pass to estimate the final bounds.\n- **Motivation behind metrics**: As mentioned above, we will provide an additional discussion behind our choice of metrics. Here, we shortly want to summarize the most important aspects: The **Coverage**(# CATE within estimated bounds / n), and **Coverage*** (# ground truth bounds within estimated bounds / n)  should always be 1 and as high as possible to allow for reliable decision-making. The **Width** (sum of upper minus lower bound / n) and **Width*** (“Width for runs with Coverage* of at least 95%) express certainty about the CATE, and should be, in principle low. However, note that through estimation variance, low values of Width can become _falsely overconfident_ (tighter than the oracle bounds), which is why we introduce the Width* for Datasets 3 and 4. The **MSD** shows the variation for different values of $k$. Low values indicate robust behavior. This metric is especially important in real-world applications where we do not have access to the oracle CATE or oracle bounds, and thus cannot calculate the Coverage, Coverage*, or Width*. This makes it hard to select a $k$ which guarantees validity but also a certain tightness of the bounds. In **Appendix F**, we provide an extended discussion about the role of $k$ and practical guidelines for selection.\n\n# Response to Questions\n\n- **Results Dataset 3**: On average, **we show improvements of about 9 % for** **MSE*** **and 2% for Width***. For the latter, even though we filter for runs with coverage above 95%, there still can be up to 5% observations with overconfident bound estimations with low width for the naive baseline, biasing the gap to appear smaller. Further, the standard deviations are inflated by summarizing the runs over different $k$. Within $k$, these effects appear stronger, as shown in Figure 5. Here, we can also see that for low values of $k$, coverage is optimal for both methods while only for the rarer evaluated high values of $k$ does the coverage start to decline rapidly for the naive baseline. \n- **MSD score and role of $k$**: For the MSD score and the reporting of different $k$, we kindly refer to our answer from above. In **Appendix F**, we provide an extended discussion about $k$ and provide practical guidelines for selection that are suitable depending on the specific problem setting: (1) an expert-informed approach and (2) a data-driven approach. In our experiments, we provide the results for the respective values of $k$ in Table 3 and Figure 5, such that we can check the performance under different selection strategies.\n\n**Action: To address all points, we will add an extended discussion of the metrics and results to our paper.**", "claims_and_evidence": "The claims of the paper are:\n- Allow for partial identification with continuous, high dimensional instrument variables that may have complex relationship with the treatment.\n- Introduce an algorithm that avoids adversarial learning.\n- Demonstrate effectiveness both theoretically and empirically.\n\nThe claims are validated through the design of their algorithm as well as the good performance on simple and more complex tasks. It's a bit unclear how the theory shows that the algorithm is \"effective\", this contributions could be worded better.", "methods_and_evaluation_criteria": "The paper compares on benchmarks with simple propensity and a more complex propensity to show that their method can gain performance in the first setting without losing performance in the first setting.\n\nThey introduce multiple metrics that tackle different properties of the bounds. However the MSE* and width* metric are only shown for Dataset 3.", "theoretical_claims": "The theoretical claims relate to the closed form computation of the nuisance functions, and the finite sample properties of the proposed algorithm. They seem correct.", "experimental_designs_or_analyses": "The experimental design makes sense for the problem at hand, increasing the complexity to show that the proposed changes are necessary.", "relation_to_broader_scientific_literature": "There is ample discussion of the difference in the settings compared to previous work. The fact that adversarial learning is not required is also a contibution.", "essential_references_not_discussed": "Not to my knowledge.", "other_strengths_and_weaknesses": "Strengths\n- The paper tackles a well formed problem that is realistic and show that their solutions works.\n\nWeaknesses:\n- Some of the presentation could be improved.", "other_comments_or_suggestions": "- Figure 4: The writing in this figure is too small. The figure needs to be a lot larger. I would suggest even moving it to the appendix. \n- Figure 3: This is not clear, is the training of the representation and updating of nuisance parameters not repeated multiple times? As is shown in Appendix H. The figure makes it seem like the representation is trained, then the nuisance updated which then results in the bounds.\n- Motivation behind the metrics could be made clearer. What is a higher or lower value of a metric saying about the algorithm/performance under a decision?", "questions_for_authors": "There does not seem to be much of an improvement on MSE* in dataset 3, which is the purported use case. Although you claim the the MSE* performance is better over the naive baseline, it seems to me that they are the same (within error). Am I missing something here? Why would the performance be so similar? Agains, the width is the same as well within error, and the coverage of the naive baseline is not much lower. How should I interpret these results? \n\nWhy do we care about the MSD score? In practice I will chose a single k and I want the performance for that k. If there is a procedure for choosing the best k, I would want to see the performance for that best k. For example, one method could be ok for all k, whereas another could be great, but only for a single k. \n\nA clearer discussion of the metrics and the results would really help the narrative.", "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": "uxbIfPUqRh", "reviewer": "Reviewer_m2Ds", "summary": "This paper proposes a new method for partial identification of the conditional treatment effect (CATE) when working with continuous or high-dimensional instruments. By mapping complex instruments into a learned discrete representation, the authors apply Manski-style bounds while mitigating the instability that arises in adversarial or iterative training. Their two-stage procedure first estimates nuisance functions, then partitions the instrument space in a way that reduces variance in finite-sample settings. The key theoretical result is that these learned partitions yield valid, reasonably tight bounds for the CATE under the usual IV assumptions, but with fewer structural constraints than point-identification approaches. The proposed method is empirically evaluated on synthetic datasets.\n\n## update after rebuttal\n\nThe authors' rebuttal addressed most of my concerns. However, I still view the work as somewhat incremental and, as such, I will maintain my borderline positive score (**3: Weak accept**).", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive and very actionable review! We took your comments at heart and improved our paper as follows.\n\n# Response to Claims and Evidence\n\n- **Stability of our method compared to adversarial approaches**: This is a very interesting point! As _widely investigated_ in the literature, adversarial (typically min/max) optimization suffers from _instability due to joint optimization of competing objectives_, which often leads to **slow convergence, gradient issues, and sensitivity to hyperparameters**. In contrast, our two-step approach decouples nuisance function estimation and partitioning to provide a more controlled, stable learning process. Even in simple **empirical** cases, when we estimated the bounds in an adversarial manner by retraining $\\mu_\\phi$ and $\\pi_\\phi$ after updating $\\phi$ (instead of using our Eq. 3 and 4), we could _not even achieve convergence_. This highlights the superior stability of our method. \n\n**Action**: For the final version, we will report results with more exhaustive tuning of the adversarial baseline to provide a fair comparison.\n\n# Response to Methods and Evaluation Criteria\n- **Real-world data**: For benchmarking, we use synthetic DGPs that are **closely tailored to the real-world settings in Mendelian randomization (MR)**, such as polygenic risk scores and SNPs, and provide results for different levels of complexities. **We added new experiments with real-world data from a chemotherapy study containing genetic variants**. This allows us to the effect of exposure (smoking) on cancer progression (outcome).  **We provide the data description, results, and short interpretation here:** https://anonymous.4open.science/r/IVRep4PartId-714C/rebuttal/rebuttal_experiments.pdf. In sum, we observe that our method provides expected results while showing similar Width and **clearly more robust estimation** compared to the naive baseline, which confirms the effectiveness of our method.\n\n**Action:** We will include the real-world experiments in our paper.\n\n\n# Response to Weaknesses\n- Regarding the points “improved stability against adversarial methods” and “real-world experiments”, we kindly refer to our response from above.\n- **Limited benefits in simpler scenarios**: We agree that the benefits in terms of improved tightness are limited in the simpler setting. However, even though our method is designed for more complex settings, we still achieve similar performance in tightness (Width), while also providing more robust estimation (MSD) here. This shows that our method is suitable in both simpler and more complex settings.\n\n# Comments suggestions\n- **Loss formulation for $L_b$ in Eq. (16)**: While it might seem counterintuitive, our formulation in Eq. (16) is correct.  By our **Theorem 1**, we ensure that in population, the upper bound is always _greater than the lower bound_, and thus, the loss **cannot become negative**. In finite samples, by having large estimation errors in our bound estimates, this could change in theory. However, by using our **regularization loss** in Eq. (17), we directly avoid such unreliable estimations. Also, when running our experiments, we _never_ observed a negative loss, **undermining its applicability**.\n- **Intuition for $L_{aux}$**: Here, as a motivation for cross-entropy, we followed the heuristic that if the activations before the discretizations can be separated more easily, they should be more diverse. We also did some runs with the empirical Wasserstein distance loss to directly enforce more distributional distance in Z given the partitions. However, the cross-entropy loss performed stable. Further, the auxiliary loss only reduces the _estimation variance_, but does not affect the average _width or coverage_ (**see also our experiments in Appendix K**), thus turning out to be **useful but not crucial** for our method.\n\n# Response to questions\n- For loss mechanics and stability, we kindly refer to our answers above\n- **Interpretation of Partitions**: In principle, the clustering is only an internal mechanism to optimize our learning objective. However, under certain assumptions (see also **Appendix F.1**), maximizing for similar propensity scores within clusters and high heterogeneity between the clusters will lead to the closest bounds. Thus, one could implicitly identify genetic variations with similar influence on the exposure within one cluster.\n- **Auto-Tuning**:  Indeed a major benefit of our method compared to the baseline is the _robustness regarding $k$, since hyperparameter selection is hard in applied causal inference due to a _lack of access to the ground truth CATE_. However, we nevertheless provide practical guidelines in **Appendix F.2** for the optimal selection of $k$. Therein, we propose two approaches: (1) an expert-informed approach and (2) a data-driven approach, which can be used seamlessly in practice.", "claims_and_evidence": "The paper makes three primary contributions: \n\n(1) a discrete representation approach that yields valid bounds on the CATE for complex Ivs\n\n(2) a two-step estimation procedure claimed to be more stable than adversarial approaches\n\n(3) theoretical guarantees that their learned bounds are valid under standard IV assumptions. \n\nContributions (1) and (3) are well-supported by evidence: detailed proofs and empirical results confirm that the proposed method produces narrower bounds compared to naive discretizations, and simulations demonstrate good alignment with the true CATE or oracle bounds. However, contribution (2) is not substantiated, as the authors neither provide direct comparative experiments nor theoretical justification for this claim. Thus, while the paper effectively motivates the stability advantage conceptually, it lacks sufficient evidence to show that this holds in theory/applications.", "methods_and_evaluation_criteria": "The authors propose a two-stage approach: first, they train neural networks for nuisance functions (propensity and outcome regressions), then they learn discrete partitions of the instrument space to minimize bound width while balancing finite-sample variance. They evaluate performance primarily by coverage and average bound width, with additional metrics like MSD to gauge robustness under different partition sizes. These benchmarks are sensible for partial-identification methods in synthetic and semi-synthetic scenarios, though it would be valuable to see how the approach performs on a real-world dataset—even if we would only rely on qualitative assessments for validation.", "theoretical_claims": "The theoretical claims are supported by detailed proofs, for which I commend the authors. I carefully reviewed Theorem 1 and Lemma 1 and they appeared correct to me. I skimmed the proof of Theorem 2, and it appeared correct as well, although I did not verify every detail thoroughly.", "experimental_designs_or_analyses": "The authors evaluate their method on simulated data, including scenarios with high-dimensional instruments (e.g., SNP-like variables) and known ground-truth oracles. I checked the design’s overall soundness—splits, metrics, and comparisons to baseline methods—and found nothing amiss. The coverage metrics and bound widths are computed in a consistent way for partial identification, and the analyses appear robust with no evident methodological flaws.", "relation_to_broader_scientific_literature": "The approach addresses an existing gap in the literature where prior methods either required strong structural assumptions for point identification with IVs or relied on adversarial techniques to build valid bounds. The core idea—partitioning the instrument space and applying Manski-type bounds to the CATE function—is straightforward and heavily builds upon existing literature. Although not highly novel, the theoretical results and empirical validations are convincing, making this work a strong candidate for inclusion in the conference.", "essential_references_not_discussed": "It could be beneficial to reference additional recent works on partial identification beyond Padh et al. (2023) and Levis et al. (2023) if they exist—for instance, studies that attempt to bound treatment effects with minimal assumptions in observational data, or new theoretical advances that might provide alternative bounding approaches. But otherwise, I think the related literature section is pretty comprehensive and includes the relevant references.", "other_strengths_and_weaknesses": "[Summary of the thoughts from the above sections]\n\nStrengths\n\n* The paper presents a straightforward method for applying Manski bounds to complex IV settings, supported by clear theoretical justifications.\n* Empirical evaluations demonstrate effective coverage and tighter bounds in challenging, high-dimensional scenarios.\n* The proposed method is modular, allowing the integration of different models for the first-stage nuisance estimation.\n\nWeaknesses\n\n* Although the authors claim improved stability over adversarial methods, no direct comparative experiments or formal proofs substantiate this claim.\n* The benefits over naive clustering methods can be modest in simpler scenarios, with improvements becoming more pronounced in complex cases.\n* While benchmarks presented are appropriate for partial-identification methods in synthetic and semi-synthetic contexts, it would be valuable to evaluate the method's performance on real-world datasets, even if such assessments were qualitative.", "other_comments_or_suggestions": "*  I believe there is a typo in the definition of $\\mathcal{L}_b$ in Eq (16) since he loss seems to decrease when the upper bound is lower than the lower bound which is not valid given that the objective is to minimize this loss. \n\n* Additional intuition for the auxiliary loss $\\mathcal{L}_{aux}$​ would be helpful, especially why cross-entropy is the best penalty to induce diverse partitioning.", "questions_for_authors": "[Summary of the thoughts from the above sections]\n\n* Loss Mechanics: How does your loss function handle scenarios where the upper bound might dip below the lower bound, given that $\\mathcal{L}_b$ is minimized directly?\n\n* Stability Claim: You mention more stability than adversarial methods—could you expand on whether you attempted a direct comparison or if you can point to any theoretical rationale beyond avoiding min-max loops?\n\n* Interpretation of Partitions: In a domain like genetics, do you see real-world interpretability in these learned clusters (e.g., discrete genotypes), or is it purely an internal mechanism?\n\n* Auto-Tuning: Choosing the number of partitions $k$ seems crucial for balancing bound tightness and sample size per partition. Have you considered automatic selection methods (e.g., a validation-based approach)?\nOverall, I consider this a strong partial-identification paper, and I support acceptance—pending clarifications on stability and the auxiliary loss.\n\nOverall, I consider this a strong partial-identification paper, and I support acceptance—pending clarifications on stability and the auxiliary loss.", "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": "ar6RKvfzBX", "reviewer": "Reviewer_xPaP", "summary": "The paper presents a method for learning bounds on CATE (conditional average treatment effect) in the event of observed covariates X, unobserved confounding U, binary treatment A, scalar outcome Y,  and an instrument Z which may be high dimnensional.\n\nThe proposed method extends:\n\nSchweisthal, J., Frauen, D., van der Schaar, M., and Feuer- riegel, S. Meta-learners for partially-identified treatment effects across multiple environments. In ICML, 2024.\n\nwhich deals with the case of discrete instruments, and provides the core result of Lemma 2. The contribution of the paper is a neural net approach to mapping the high dimensional continuous variable Z to a set of k partitions. Consequent bounds on the CATE are then found straightforwardly (Theorem 1).", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you a lot for your positive feedback and your comments! We are happy to see that our claims, methods, theory, and experiments are well received and do not raise any additional concerns. \n\nHere, we would kindly like to elaborate on why our paper is **not** an incremental contribution of the work of Schweisthal et al. but instead a standalone work with multiple novelties in a different setting.\n\n(i) **Applicability to arbitrary IVs and target bound width minimization**: We show that the existing bounds for discrete instruments from Lemma 1 (which Schweisthal et al. used and which are also just an adaptation of the Manski bounds) can be applied to other instrument types (continuous, high-dimensional) by using _arbitrary partitioning functions_, enabling to transfer and generalize the bounds to new unconsidered settings such as Mendelian randomization (MR) or indirect experiments with complex nudges. **This is a novel theoretical finding**. While this may seem straightforward at first sight, to the best of our knowledge, we are not aware of any prior work considering that connection, i.e., _even our naive baseline leveraging $k$-means clustering has not been considered before_. \n\nThis is orthogonal to the work of Schweisthal et al. who derived model-agnostic learners in the **discrete** IV setting, while we focus on **complex and continuous** IVs. Further, this finding allows us to develop the **new objective of directly targeting bound width minimization** during representation learning to learn optimal partitions (Eq. (8)). Based on this, **we  make two major theoretical contributions regarding optimized training**:\n\n(ii) **Stability by avoiding alternating learning**: A straightforward implementation minimizing the bounds following Eq. (8) would require alternating learning. The reason is that, after every update step of $\\phi(z)$, the quantities $\\mu_\\phi^a(x, \\ell)$ and $\\pi_\\phi(x, \\ell)$ are not valid for the updated $\\phi$ anymore and would need to be retrained to ensure valid bounds. This is computationally highly expensive and results in unstable training and convergence problems. However, our method circumvents these issues: by using our novel Theorem 1, we show that, while training $\\phi(z)$, the quantities $\\mu_\\phi^a(x, \\ell)$ and $\\pi_\\phi(x, \\ell)$ can be directly calculated. (see also our subsection “Implications of Theorem 1” on page 4).\n\nFor that, we can simply evaluate the nuisance functions, which only need to be trained once in the first stage. Therefore, we avoid any need for alternating learning, resulting in more efficient and stable training. Here, also note that Theorem 1 and its proof in Appendix A1 target effective _estimation_ of our target quantities, and **thus is orthogonal to the works about discrete instruments** which aim for the _derivation_ of bounds.\n\n(iii) **Improved finite sample robustness**: Even using our stable training procedure from above, optimizing for Eq. (4) only yields valid bounds _in expectation on the population level_. However, if the discrete representation learning results in highly imbalanced marginal probabilities during training (i.e., $\\mathbb{P}(\\phi(Z)=\\ell)$ is small for some $\\ell$), this can result in high estimation variance of the nuisance estimates and thus unreliable bound estimates. **We show this more formally in our Theorem 2 where we provide theoretical guarantees for the asymptotic behavior**.\n\n In contrast, we avoid these problems: by using our theoretically motivated custom loss from Eq. 19 with the respective regularization from Eq. 17, _we enforce lower estimation variance during training and thus more reliable bound estimates_.\n\nIn sum, we only leverage the formulation of the closed-form bounds of Schweisthal et al. from the discrete IV setting ( - which is also **not** their main contribution but only extending Manski bounds to the CATE - ) as a simple starting point for our method. Thus _our major contributions are independent of the contributions of Schweisthal et al_., which are model-agnostic meta-learners in a different setting (discrete vs. complex continuous IVs).\n\n**Action**: We further improved the comparison to previous work in our paper to clarify the novelty of our method.", "claims_and_evidence": "This paper presents a reasonable approach to a setting with some real-world application, and has some methodological improvements over the prior work by Schweisthal et al. that it extends. However, the contribution might be considered incremental.", "methods_and_evaluation_criteria": "A number of strategies are employed to train the instrument discretization network. These combine: 1) minimizing the bound width, 2) ensuring no bin has too small a mass, 3) enforcing that the distribution of Z given its bin is heterogeneous across bins.  These approaches are all reasonable so I have no questions or objections.\n\nThe evaluation is conducted on three synthetic benchmarks, to ensure a known ground truth. Since no method yet addresses the setting, the main point of comparis on is the naive k-means discretization on the one hand, as well as the known ground-truth CATE.  Again, this choice is reasonable, and I have no questions on it.", "theoretical_claims": "Asumptotic normality of the conditional mean and propensity score estimates is shown by combining CLT and delta method (Theorem 2). \n\nI did not read the proofs in detail but standard techniques are used.", "experimental_designs_or_analyses": "see above", "relation_to_broader_scientific_literature": "see above", "essential_references_not_discussed": "n/a", "other_strengths_and_weaknesses": "n/a", "other_comments_or_suggestions": "n/a", "questions_for_authors": "n/a", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper provides a methodology and analysis for estimating (potentially) high-dimensional Conditional Average Treatment Effects (CATE) under partial identification. As a means of identification, the authors derive bounds with the discretized instrumental variables and analyze the associated estimators and their asymptotic properties. A neural approach for estimating these bounds is also presented.

The paper is clearly written and accessible, and it has been positively received overall. The comprehensiveness of the experimental setup is another strength. However, as noted by several reviewers—including expert reviewers—the technical contributions of this paper are rather limited. As a result, the review scores are divided.

The area chair has studied this issue in depth and concurs that the main results are somewhat similar to existing work. Although the paper does offer several additional contributions, the limited novelty of the core results diminishes the overall impact of these additional elements. Nonetheless, the quality of the paper itself is recognized.
Decision: Accept (poster)