Tightening Causal Bounds via Covariate-Aware Optimal Transport
Authors: Sirui Lin
Zijun Gao
Jose Blanchet
Peter Glynn
Venue: ICML 2025 poster (2025)
Keywords: Partial identification
causal inference
conditional optimal transport
Primary area: general_machine_learning->causality

Abstract: Causal estimands can vary significantly depending on the relationship between outcomes in treatment and control groups, leading to wide partial identification (PI) intervals that impede decision making. Incorporating covariates can substantially tighten these bounds, but requires determining the range of PI over probability models consistent with the joint distributions of observed covariates and outcomes in treatment and control groups. This problem is known to be equivalent to a conditional optimal transport (COT) optimization task, which is more challenging than standard optimal transport (OT) due to the additional conditioning constraints. In this work, we study a tight relaxation of COT that effectively reduces it to standard OT, leveraging its well-established computational and theoretical foundations. Our relaxation incorporates covariate information and ensures narrower PI intervals for any value of the penalty parameter, while becoming asymptotically exact as a penalty increases to infinity. This approach preserves the benefits of covariate adjustment in PI and results in a data-driven estimator for the PI set that is easy to implement using existing OT packages. We analyze the convergence rate of our estimator and demonstrate the effectiveness of our approach through extensive simulations, highlighting its practical use and superior performance compared to existing methods.

Tightening Causal Bounds via Covariate-Aware Optimal Transport
Sirui Lin Zijun Gao
José Blanchet Peter Glynn'
Abstract
outcomes: one under the control condition and one under the treatment condition. However, for each unit, only the potential outcome corresponding to the received treatment is observed, and the other potential outcome is missing. As a result, the joint distribution of the potential outcomes is never observable, while the marginal distributions of the two potential outcomes can often identified'. This poses fundamental challenge: causal estimands that depend on the joint distribution are only partially identifiable, meaning that their exact values cannot be determined.
Causal estimands can vary significantly depend- ing on the relationship between outcomes treat- ment and control groups. potentially leading to wide partial identification (PI) intervals that im- pede decision making. Incorporating covariates can substantially tighten these bounds, but re- quires determining the range of Pl over probabil- ity models consistent with the joint distributions of observed covariates and outcomes in treatment and control groups. This problem is known to be equivalent conditional optimal transport (COT) optimization task, which is more challeng- ing than standard optimal transport (OT) due the additional conditioning constraints. In this work, we study a tight relaxation of COT that effectively reduces it to standard OT, leveraging well-established computational and theoretical foundations. Our relaxation incorporates covari- ate information and ensures narrower PI inter- vals for any value of the penalty parameter, while becoming asymptotically exact as a penalty in- creases to infinity. This approach preserves the benefits of covariate adjustment in PI and results in a data-driven estimator for the PI set that is easy to implement using existing OT packages. We analyze the convergence rate of our estimator and demonstrate the effectiveness of our approach through extensive simulations, highlighting its practical use and superior performance compared to existing methods.
Previous works (Chernozhukov et al., 2017: Torous et al., 2024; Gao et al., 2025) have used OT, a rapidly growing field with various applications machine learning (Agueh & Carlier, 2011; Choi et al., 2018; Hui al., 2018), to obtain the PI set the range of plausible values that the causal estimand may take. This thread of research is built upon the key observation: for causal estimands defined as the expectation of a function of potential outcomes, the PI set an interval whose boundaries are effectively the optimal objective values of two OT problems. Explicitly, in these two OT problems, the function the causal estimand (or its negative counterpart) serves as the cost function, and the marginal distributions of the two potential outcomes serve as the source and target distributions, respectively.
In many applications, each unit is associated with a set of co- variates. Leveraging the covariate information typically can reduce the uncertainty in the outcome's 's joint distribution thus the width of the PI interval. From an OT perspec- tive, this reduction in uncertainty involves optimizing an expected cost function over all couplings (i.e. joint dis- tributions) that respect the conditional marginal outcome distributions framework termed as Conditional Optimal Transport (COT). A comparison of optimal transport cou- pling for OT and COT is illustrated in Figure 1. Compared to standard OT, which disregards covariates, COT enforces stricter coupling constraints, leading to tighter PI bounds. This refinement is particularly beneficial when covariates have substantial influence on the distribution of potential outcomes.
1. Introduction
In the potential outcome model (Rubin, 1974) with a bi- nary treatment, each unit is associated with two potential
'Equal contribution 'Department of Management Science and Engineering, Stanford University. USA AMarshall School of Business. University of Southern California, USA. Correspon- dence Sirui Lin <siruilin @ edd Zijun Gao <zijun- gao wss.shall.uu.ccu....
'The marginal distributions the two potential outcomes can be identified in randomized experiments or in observational studies where the experiment is unconfounded- the potential outcomes independent of the treatment assignment conditional the observed covariates.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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Covariatc-Aware Transport for Causal Bounds
Unconditional (OT)
Conditional (COT)
infinity. This implies that the PI interval of our framework with finite penalty parameter is always valid and narrower than (or equal to) the interval obtained without using the covariates. Provided with a finite sample, we construct a plug-in estimator for the optimal objective value of our framework and establish its convergence rate. Computation- ally, the proposed approach compatible with off off-the-shelf OT solvers (Flamary al., 2021). Empirically, we show that our method significantly improves the PI set from OT without covariates and demonstrate our method's efficiency (i.e., narrower PI intervals) and robustness (to the model misspecification of the conditional marginal distributions) compared to alternative COT-base methods.
((1)
Y())
Y(0)
Y(o)
2=1
2=
Figure 1. Comparison of optimal transport coupling for OT and COT when the covariate Z impacts the distribution of the potential outcomes (0), (1).
Contributions.
However, COT is considerably more challenging than the already complex OT for the following reasons.
(i)
We propose a model-lean covariate-aware OT method for solving Pl sets in causal inference with an efficient computation algorithm (Algorithm 1) and a rigorous statistical guarantee (Theorem 4.3).
(i). The conditional marginal distributions given the covari- ates, which come in the constraint of the COT problem, are unknown and often challenging to estimate from a fi- nite sample. This difficulty is especially pronounced with continuous covariates, where each covariate value typically corresponds to only one observation.
iii) We establish an interpolation between OT and COT (Proposition 3.6), where COT is more challenging and less explored, enabling the study of both the statisti- cal and computational properties of COT using well- developed tools from OT.
(ii). Unlike OT, straightforward plug-in estimator for COT using finite sample may not converge to meaningful limit as the sample size goes to infinity (see Example 9 in (Chemseddine et al., 2024)). In this case, imposing well- specified modeling assumptions can enable the construction of a consistent estimator for the COT from finite samples, which, however, entails a significant risk of model misspeci- fication.
Organization. Section 2 provides preliminaries on stan- dard OT and COT with their connection to PI. Section introduces our proposed method with population level anal- ysis. Section presents a data-driven estimator and the finite-sample analysis. Section presents the experiments synthetic and real data. Section 6 collects some final remarks and future research directions. Discussion on re- lated work is deferred to the Appendix A; Notations are collected in the Appendix B; All the proofs are collected in the Appendix C.
(iii). Even if the true conditional distribution is available, solving COT involves addressing the computationally in- tensive OT problem for each possible covariate value. For continuous covariates, enumerating all possible values is infeasible. Approximation through fine covariate discretiza- tion is required, which is computationally demanding and introduces additional approximation errors.
2. Preliminaries
2.1. Unconditional Optimal Transport
In this work, we study model-lean ewvariate-adjuster opti- mal transport method to obtain PI sets, which avoids solving COT problems. The key idea is to introduce "mirror" co- variates and reformulate the problem as standard (uncondi- tional) OT, using the original cost function plus an additional penalty term that accounts for the gap between the mirror covariates. Our framework rooted in the method by (Car- lier et al., 2010) for seeking a block triangular map using OT, and is further extended by (Hosseini et al., 2025; Bap- tista et al., 2024b; Chemseddine et al., 2024) for different tasks. At the population level, the optimal objective value of our framework is shown to improve upon the standard OT optimal value that ignores covariates and increases to the COT optimal value as the penalty parameter approaches
Consider unit associated with two potential outcomes (0) and (1).. The causal estimand of interest is V* := E.- () ( for some function h y x y + R. where x denotes the joint distribution of (O) (1)), Y S Rd Let W denote the treatment indicator. If the unit receives the treatment W = 1,, the observed outcome is Y (1); otherwise, Y Y Y()). A fundamental limita- tion in causal inference is that ( and Y()) are never observed simultaneously for the same unit, making the joint distribution xr" of (O), Y()) unidentifiable. Conse- quently, V* is typically not identifiable, and we instead aim to obtain a partial identification set for V*.
The set of joint distributions consistent with the observable
2

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Covariatc-Aware Transport for Causal Bounds
marginal distributions is
distribution Y(-) Z varies with the value of z, the op- timal joint distribution of the the unconditional OT problem and that of the COT differ significantly, and the partial iden- tification interval under COT is notably shorter as result. Despite these benefits, the estimation and computation of COT are challenging, as discussed in Section 1.
I1, = { E P(D3) : (Y) Py(o)= TY(1) Proj).
where (o) denotes the marginal distributions of Y (0) un- der n,, and similarly for Ty()); Py(o). (r(1) are the marginal distribution of (()), Y (1). Thus, the partial identification set of V takes the form (A<(( Y(1)) which is convex and thus an interval. The lower boundary of the partial identification interval is the optimal objective value of the following optimization problem
E TT.),
3. Mirror Relaxation of COT
In this section, we introduce relaxation of COT using mirror covariates, namely, Vip(r) as an alternative PI lower bound. We additionally present the interpolation of (i) between the PI lower bounds V. and Ve. The subscript "ip" in Vip(n) stands for "interpolation".
(Vo)= min E (O), (I)). FEIL,
(1)
This formulation establishes a natural connection between the causal partial identification and the OT problem (Villani et al., 2009), where we want to couple the measures of (o and (y()) with the minimal cost measured by the function h. More motivating examples can be found in (Ji et al., 2023; Gao et al., 2025), which propose obtain the partial identification sets by solving the related (multi-marginal) OT problems.
3.1. Assumptions
Assumption 3.1 (Completely randomized design). For each
unit i E I, (Y:(0), (1)),, Z.) i...d. follows ( Suppose m units are assigned to the treatment group (W 1), and units are assigned to the control group (W = 0).
Assumption 3.2 (Compact support and density). Assume that y, Z are convex and compact, and Pr(o).z. Pr(1).z both admit densities with respect to the Lebesgue measure +dy+
2.2. Conditional Optimal Transport
Suppose each unit is also associated with set of covari- ates, denoted by Z E Z S Rd4, such as height or age. The covariate information can be leveraged to shorten the partial identification interval. In a nutshell, the conditional marginal distribution of Y (0) | Z = z is identifiable under the unconfoundedness assumption, which imposes a con- dition the joint distribution ı more stringent than only preserving the unconditional distribution (oy similarly for (1). Consequently, the class of consistent joint distri- butions shrinks, leading to shorter partial identification interval. Mathematically, we define the set of joint distribu- tions consistent with the observable conditional distributions of (0) | Z, Y(1) Z, and as:
Assumption 3.3 (Smooth cost). Assume that the cost func- tion h CR((2dy), ) and V -y. injective for Vy E y.
Assumption 3.2 is introduced to facilitate theoretical anal- ysis while covering a wide range of practical scenarios. Assumption 3.3 ensures the uniqueness of the optimal trans- port maps, which serves the foundation for defining our estimand, and is satisfied by, e.g., quadratic costs.
3.2. Definition of (T)
Definition 3.4 (Mirror relaxation Vnp(n)). For [0, let TT (1)
Te = ( E P()2 2): TY(0)2 = (Y) TY = }.
argmin E, [A(Y(O) + + 1l z(1)1 #Ellp
where Pr(o denotes the marginal distribution of (Y()), Z): similarly for (1( Z). The partial identifi- cation interval {E, [h(Y (0), (1)) and thus again an interval, and the lower boundary of this interval is the optimal objective value of the COT problem:
where the set 1lp of joint distributions consistent with marginals of outcomes and associated mirror covariates is defined as
E I1. is convex
1T = = x x²)) (T(0)) = Py(o).z,
TY(I)2(1) Pr(nzz]
(V.) min E, (0), FEIT.
(2)
Then the interpolating OT lower bound Vip(t) is defined as
The subscript "c" in Vo stands for "conditional", and the subseript "u" in V, from Equation (1) stands for "uncondi- tional". In Figure 1, we illustrate that when the conditional
Vip(T) = (Y))
"The upper boundary can be obtained using- - (0), (())..
The following proposition shows that V well-defined.
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Covariatc-Aware Transport for Causal Bounds
be found in the Appendix C.I1):
Definition
Optimal coupling coupling under constraints
Vu ( (v 38 + t V 81 + o
2
Vu (m h)
m argmin E,, (N(Y Tell,
Vo (mo h))
"e E argmin F2 (1)( mEIl.
Ve (go + +1) + (oo o1)²
Vip(n) (8) + 39 o of)
Vip(n) =
Tip (r) = argmin E, (0),
(g6 + (B) of) - n/80,31 + 10o0n
xelli
((8) + oZ) + of) - 2n10081 + 2n0001 + 172)
(m (1)
+ -
straightforward analysis shows that the properties in Proposition 3.6 hold in this case. Further, in this case we have Ve - Vip(n) = 2(0oa 2(00ntth100) + + as nl + cc. as illustrated in Fig. 2.
Table Definitions of partial identification OT lower bounds.
Proposition 3.5 (Existence and uniqueness of x²). Under Assp. 3.2, 3.3, oh (n) exists and is uniquely determined.
We remark that -u: गh. exist as well, and mH is also uniquely determined. In Table 1, we summarize the three lower bounds for comparison, where we denote (T,, hi) := Ex, [h].
Vipin) Ve
Vu
3.3. Interpolation Between PI Lower Bounds
The following result shows that (n) interpolates the lower bounds Vú and as 17 ranges from 0 to oc.
Proposition 3.6 (Interpolation between lower bounds). Un- der Assp. 3.2, 3.3, for 1) 2 0, we have
Figure 2. Comparison Vi (7), V. and V. for h(yo, 91) = (30 + y1)². The model is (0) 0.8Z Eo, Y() 1.62 + E1. Z, E1 are i.d. ~ N'0 1).
(i). (Monotonicity) V, S (v) S Vo, and Vip(n) non- decreasing and continuous with respect 17.
(ii). (Interpolation) (0) Vu, 2 (ip(n) Ve.
4. Finite-Sample Analysis
Remark 3.7 (Implications in Pl). For partial identification in causal inference, the interpolation (T) between Vú and V. demonstrates that (1) strictly improves upon the lower bound V,, which disregards the covariate information. As 77 increases, the improvement becomes more pronounced, with Vi (17) converging to Vo-the sharpest lower bound achievable leveraging the covariates.
In this section, we introduce our empirical estimator of Vn)(n) based on a finite sample and discuss the statistical convergence rate of this estimator.
4.1. Data-Driven plug-in Estimator
Given a finite sample ((Y, Z,, W E Z) where there are n units with W; = and m units with W, = 1, we denote the empirical marginal distributions as
When the compactness assumption (Assp. 3.2) is not sat- isfied, the interpolation can still hold as shown in the next example, where the lower bounds possess closed-form ex- pressions.
Pay(O)) = 1 - dy. z. 1W, = 0),
(3a)
iET
Example 3.8 (Gaussian linear model). Consider the data generating mechanism Y()) 3oZ+op8n Y(1) = B 7+ 01E1, where Z, E,50,11 are i.idd. (0, 1). Let h(yo, 91) = (yo + ?. Applying the formula of 2- Wasserstein distance between Gaussian random variables (see e.g. (Dowson &
P,,Y(1,2 = m
8y 1(W, = 1). (3b)
IEI
Our empirical estimator Vpp.n.m (1) is then based on the plug-in of (a,y (resp. Pm,y(1)) (1) to replace Pr(o).z (resp. Py(), in the definition of Vyp (n). We demonstrate
Landau, 1982)), we get the following results (the proof can

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Now, we present the result on the convergence rate of our plug-in estimator for Vip(T). Our analysis relies on quadratic cost functions so that the optimal transport map Covariatc-Aware Transport for Causal Bounds
Algorithm 1 Derivation of Empirical Estimator Vip.n.m (1) sample (OY, Ze, W.), i E I), cost function h,
[An A12]; [A]2: And) A12 E Rdyxxy is non-singular. Under Assp. 3.1, 3.2 (excluding the compactness assump- tion), further assume that the Brenier potential between P PAT,(0)):22 2 and Pr(i),z. denoted as Φ, satisfies & e Ala for X Then,
Input:
parameter 77
Construct (0).2. Pm, (1)) by (3a), (3b). Compute the cost matrix H E [nxxm, with
C² 2) and I 3 (2G(x) 3
we have
H(j. k) h(y6
) ()) vi + 17 || (060))
21 (8)
E VVp.. (n) Vip (1)] S CAn . NN,d.
between the n points (denoted as (yu 20 )0)) E (rr) in the support of (0), and the m points (denoted as
N-
d 5 3 = d 2 5
where
YN.d" = N v Iog(NN)
(v1%), ø1 2(M)),, k E [m]) in the support of P. (1).2.
N
Solve the optimal transport problem: for 1 (1,
Here, = min(n, m), d = dy + dz, and C is constant depending on d, |I ol |l llop | oloop as well as the fourth moments of (0), ( Z.
min
Σ (ū,,) J.K
(4)
Remark 4.4 (Examples of quadratic costs). Many causal quantities are the expectation of quadratic functions of po- tential outcomes, whose associated Vip(1) falls into the um- brella of Theorem 4.3. Examples include the Neymanian confidence interval "Splawa-Neyman 1923), testing a null effect using ([(Y((1) - []] O, the correlation index of potential outcomes (Fan et al., 2023), and the covariance of treatment effects of different potential outcomes. We include the details of the examples in Appendix E.
and denote the optimal solution as mip.n.m (1)) Compute
Vep. (1) = =
mm. (n) 1)) h(v6") (3) .9 )(
J.k
Output: Vip.n.m( (1)
the estimator in Algorithm 1. Note that (4) may not have a unique optimal solution, while our analysis in this section is valid for any optimal solution "ip...m The next proposition states that the plug-in estimator is consistent.
The compactness assumption in Assp. 3.2, together with certein smoothness assumptions on the supports and densi- ties of the marginal distributions (introduced in (Caffarelli, 1992a;b; 1996)), can be used to guarantee the existence of a positive curvature (Corollary 3.2 of (Gigli, 2011)).
Proposition 4.1 (Consistency of the plug-in estimator). Un- der Assp. 3.1-3.3, Vip.n.m((1) (1) is a consistent estimator of (i)( as n, m + co.
Lemma 4.5 (Caffarelli's regularity theory for curvature). Assume that the supports of both P PAI,Y (0),n( (Z),z and Py are C2 and uniformly convex, and Assp. 3.2 holds. Fur- ther, assume that the densities of both are C for some B E (0, 1) and bounded away from zero and infinity. Then the associated Brenier potential : C and L 3 <²p(()) 1 Ald, for some X > 0.
4.2. Convergence rate
enjoys a convex characterization (Knott& & Smith, 1984; Brenier, 1991).
Here, the definitions of C², uniformly convex sets and C functions and will be explained in Appendix B.
Proposition 4.2 (Brenier's theorem). Under Assp. 3.2, for the cost function lly - y' v'112 + ||2 2/12 there exists a Pr(o),z-a.. a.s. unique OT map T between (r and Py(1). z such that T V for convex function s.
Unfortunately, we currently lack general method bound the curvature À with respect to 17. Ideally, as 7) co, X would grow at most linearly with respect to 1) (see Ap- pendix C.7 for a discussion), as we will prove for jointly Gaussian r.v.'s
Here, op is the so-called Brenier's potential between (Y) and Py()).2 Also, the Brenier's theorem can be extended to general quadratic functions. Then, inspired by the proof strategy in (Manole et al., 2024) that uses the smoothness and strong convexity of s,, we demonstrate the convergence rate of Vin,n,m.(1) (1) in the case of general quadratic functions.
Lemma 4.6 (Curvature of OT maps for Gaussian marginals). Suppose that (Y (0), 2). (Y(1), Z) are both Gaussian vec- tors. For r > O, there is constant C, è that only depends the covariance matrices of (Y (0), Z) and (Y Z), such that ě(n A 1)Ia < (² = O(n V 1)Is.
Theorem 4.3 samile complexity). Suppose that hyn, 12) = TTAy. Ay, where y = (21, and A =
In certain scenarios, potential outcomes can be dependent. One example is sequential data collection, where earlier
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Covariatc-Aware Transport for Causal Bounds
units influence which units are selected in later stages. An- other example is where units are connected through a net- work, and their potential outcomes are dependent based on the network structure. In view of these situations, we extend Theorem 4.3 to a special case of stationary but non- i..d. samples.
(i) For the location model (k- -1,
Ve =B [1150(2) + [((2)11 + +S(,, E1).
(i) For the scale model (k = 2), Ve = Ez (S(DU)(2)) ())) ((((())))
Theorem 4.7 sample complexity, non-i.i.d)). Un- der the same assumptions as Theorem 4.3, suppose that the units satisfy the a-mixing condition for some func- tion a N- Rt, satisfying a(k) < so and corr(f(Y,, 7.), g(, 7n)) I alli j() for Vi.j and all bounded functions f. g. Then, we have
where S(E,, Σ,) = Tr Eg (r(E
- and
D(v) is the matrix with the vector u placed at the diagonal.
Now, we provide the experiment results for synthetic data, comparing our proposal with (i) an existing method for estimating V. in our causal setting, that is, (Ji et al., 2023) using their python package DualBounds 3, and (ii) the unconditional OT method (Gao al., 2025) estimating Va- In Figure 3 and Appendix D, plot the results for the following three models:
E VI (n) (i)(1 S CAn dN,
where C' is a constant depending on o, d, ||Azzlopp || llop Azallop well as the fourth moments of (0), Y (1),
5. Experiments
(a) Linear location model, fo(z) = 0.62, f1(2) = 1.62.
(b) Quadratic location model, fo(z) 0.2*² f1(z) 0.6z².
The code can be found in the link: https://github: com/
/aasaa/ git.
(c) Scale model, fo(2) = 0.52 - 0.35, f1(z) 1.12 + 0.35.
5.1. Computation
From Figure 3, we can see that the Vip.n.m (1) curve de- creases quickly for small and then stabilizes as 1) increases, implying that incorporating covariate information signifi- cantly improves over the estimator for Vú (denoted with label "OT"". For the linear location model (Figure 3a))d)), the Dual with -baded nuisance estimator actually utilizes apriori outcome model knowledge; in con- trast, our non-parametric method, not leveraging this knowl- edge, can effectively achieve comparable L1 error with rea- sonably large v). For the quadratic location model (Fig- ure 3(b)())) and the scale model (Figure ((c)())) com- pared to DualBounds with Fridge"-basse nuisance es- timator that relies on an invalid model assumption, our method demonstrates its robustness with significantly higher accuracy; compared to DualBounde with "knn""bbsed nuisance estimator that is also -on-aarmetriic. our non- parametric method provides closer and more stable ap- proximation to V.. Particularly, for the scale model (Fig- ure (f)) Dual l Bounds with "knn"-based nuisance esti- mator may even yicld negative-valued estimator for V. >> O (see Figure 4 (c)(f) in Appendix D), indicating significant bias.
In Algorithm (4) can be solved by standard linear pro- gramming (LP) solver. Alternatively, popular approxi- mation algorithm for solving this problem is apply the Sinkhorn algorithm (Cuturi, 2013), which induces small gap from the desired optimal transport solution. However, the LP method is sufficiently efficient to solve the above problem for typical datasets in causal inference (with data size [Z] = m + n S 104). Thus, in this work, we directly compute Vip.n.m (1) without using the Sinkhorn approxima- tion.
5.2. Synthetic Data
Our synthetic data generating mechanism based on the following generic location (k: 1) and scale (k 2) model: (0) Eo)) (1) = (Z(f(()),E)) where
(0), (1), E dR Z E dg2 for fi Rd + gdy, and (1 u) = u + v. G2(u, v) u @ v (uzey.} E [d Specifically, 7,80, é are independent with each other.
When the noise variables Ep. #1 follow Gaussian distribu- tions and the cost function h is quadratic, V. is computable in closed form. Thus, the true value of V. will serve as oracle benchmark for evaluating the experimental results.
5.3. Real Data
We utilized data from the Student Achievement and Reten- tion (STAR) Demonstration Project (Angrist et al., 2009), an
Lemma 5.1 (Ve of models with Gaussian noise). As- sume that hyy, 9) = |ly + T the noise variables Eo ~ (0, Eo), E1 ~ ((0, Σx) and Z, E E1 are independent with each other. Then,
httpp:://duaabbbudd. readthedocs io/en/ attst/ indexxiddxx...ee html
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Covariatc-Aware Transport for Causal Bounds
66
Vde Liroge
Valee
Vaice
20
106
23
100
(a) Linear location model
(h) Quadratic location model
(c) Scale model
0.5
24
2.4
02
(d) Linear location model
(c) Quadratic location model
Scale model
Figure 3. Li error for estimation of Va (i.e. E [lestimator Vel]. is computed by averaging over 800 repetitions) for linear location model, a quadratic location model, and a scale model. The first row: m = n 500, the second row: m = n = 1500. Each column corresponding to model. "Vip(n)" stands for the v, (1)) estimator, "LL "LL_ridge" stands for the method Dua Bounds (Ji al., 2023) with outcome model chosen as "ridge", "LL_knn" stands for the method DualBounds with outcome model chosen as "knn", "OT" stands for the unconditional OT method (Gao al., 2025).
Table 2. Neymanian confidence upper bound.
Lidddee
LLxmn
1 = 0
10
20
30
= 40
= 50
upper bound (x10*
3.334
3,596
2.762
2.683
2.657
2.645
2.638
2.628
relative sample size
1.207
1.302
1.0
0.971
0.962
0.957
0.955
0.951
initiative aimed at understanding the effects of scholarship incentive programs on academic performance. The treat- ment (i.e. access a scholarship incentive) was randomly assigned, and approximately 30% of the students received the treatment. Academic performance in the STAR data is measured using the GPA for the first academic year. In addi- tion to the treatment assignment and the outcome variable, the dataset also includes the baseline GPA (GPA collected before the treatment assignment), which is considerably influential on the outcome variable.
5.3. NEYMANIAN CONFIDENCE INTERVAL
We revisit the classic Neymanian confidence interval (Splawa-Neyman, 1923). Consider the finite-sample av- erage treatment effect estimand == Y (1) (0), where Y = - Yikk)/N represents the finite maitee mean of potential outcomes at treatment level For completely randomized experiment with m treated units, n control units, the Neymanian confidence interval for the difference in means estimator is
S
S
S?
We explore two applications of the PI set with distinct pur- poses. The first application improves the variance estima- tor used by the Neymanian confidence interval (Splawa- Neyman, 1923), thus boosting statistical power and reducing the required sample size. The second application estimates the PI set for the correlation of two potential outcomes. If the treatment effect is constant, the correlation will be equal to one. Therefore, upper bound that is strietly less than one indicates heterogeneity in the treatment effect.
m
N
(Y₁(k) Y()))²
S
KE (0,1),
(5)
N -
N 1= (Y2(1) - -Y((0) - (1 - YO)))
8
N-1
Here, St represents the sample variance of outcomes for the treatment or the control group, which can be estimated using the empirical variance units with W, = k. How- ever, S? remains unidentifiblee because (1)1,, Y:(0) are not

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Covariatc-Aware Transport for Causal Bounds
Table 3. Pl sets for correlation.
LI ridge
L
7 =
" = 10
7 = 20
? = 30
7 = 40
1) == 50
upper bound
1.005
1.187
0.997
0.932
0.911
0.900
0.895
0.887
lower bound
-0.707
-1.036
-0.997
-0.646
-0.623
-0.607
-0.597
-0.592
Pl interval length
1.712
2.223
1.994
1.578
1.534
1.507
1.492
1.479
simultaneously observable for the same individual.
those of Vy and DualBounds (our PI set is only 74% of that achieved by Va), indicating better understanding of the correlation between potential outcomes.
The conventional variance estimator for V replaces Sg. Sf with its empirical estimate and substitutes Sž with trivial lower bound of zero. This approach leads to an upward- bias variance estimator and, consequently, to overly conservative confidence intervals. In (Aronow et al., 2014), the sharp lower bound for S without using covariates is provided, which corresponds to our Vu Vip (0). Our Vip(n). 77 > incorporates the covariate information and can offer sharp lower bound for SV. leading to even narrower confidence interval and a more powerful test. In practice, narrower confidence intervals are highly desirable as they often imply that smaller sample sizes are sufficient to achieve statistical significance. For example, clinical trials, this can reduce the number of patients needed, thereby lowering costs and expediting discoveries.
6. Conclusion And Discussion
In this work, we propose a model-lean covariate- awarre OT method for solving PI sets in causal inference by introducing the mirror covariates. In traditional causal inference, covari- ates are typically considered unaffected by the treatment and are not assigned counterfactual values. Our mirror covari- ates can be effectively interpreted as the counterfactual coun- terparts of observed covariates. To ensure the invariance of pre-treatment covariates, we introduce a penalty on the deviation between the observed and mirror covariates. Our approach unifies covariates and potential outcomes within common framework and also allows for incorporating post-treatment covariates.
In Table 2, we present our estimated Neymanian variance estimator in comparison with alternative methods discussed in Section 5.2. All variance estimators are normalized by Vu, allowing the resulting ratio to be interpreted as the ratio of the sample size required to achieve a specific level of statistical significance. A smaller ratio indicates that fewer units are needed. Our procedure, with 1/ = 50, reduces the required sample size by approximately 5% compared to Vu ignoring covariates. In contrast, the Dua lualBs BBunds methods result in relative sample size ratios exceeding one, suggesting their efficiency is less favorable than both Vú and our proposal.
We point out several promising directions for future re- search.
Variable selection for tighter Pls. As additional covariates are included, the population-level interval narrows. How- ever, the finite-sample estimator of the PI deteriorates in accuracy, particularly in terms of bias. To address this, it is beneficial to only adjust for a selected subset of covariates that substantially influences the distribution of potential out- comes. Ideally, the population PI adjusted for this subset should be close to that adjusted for all covariates, and by limiting the analysis this subset, the estimation error of the PI is significantly reduced compared using the full set of covariates.
5.3.2. PI SET OF CORRELATION INDEX
We consider the correlation between two potential outcomes defined
Potential approaches to achieve this include (1) forward stage-wise approach, where unselected covariates with the greatest impact are added sequentially; (2) a single-step selection method minimizing the PI width with an appropri- ate penalty for the number of covariates used, conceptually analogous to LASSO.
p E| -
Var
Since it depends on the joint distribution of the two potential outcomes, the quantity only partially identified.
In Table 3. we present our estimated PI sets for the correla- tion index in comparison with alternative methods discussed in Section 5.2. First, notice that the upper bound of p being strictly smaller than 1 indicates treatment effect heterogene- ity. The upper bounds of for Vn and DualBounde are close to 1, while our method achieves an upper bound of 0.887 (1 = 50) sufficiently smaller than 1. Second, the Pl interval from our method is significantly narrower than
Model-assisted PI. When knowledge of the model regarding the relationship between the potential outcome and covari- ates is available, this information could further refine the PI compared to our model-lean approach. In particular, is promising direction to develop model-assisted estima- tors that are both robust and efficient. Here the robustness means that the model-assisted PI should remain valid, least asymptotically, even when the underlying model as-
8

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Covariatc-Aware Transport for Causal Bounds
sumptions are incorrect; the efficiency means that when the model assumptions are correct, the model-assisted Pl should be narrower. A potential approach is using the one-
erative adversarial networks. In International Conference Machine Learning, pp. 214-223. PMLR, 2017.
step correction estimator from the semi aami-parmmttrc literature (Van der Vaart, 2000).
Aronow, P. M., Green, D. P., and Lee, D. K. Sharp bounds the variance in randomized experiments. Annals of Statistics, 42(3):850-8777 2014.
Relaxation using Causal Optimal Transport. Causal OT (or udapted OT) extends the standard OT framework by incorporating additional temporal causality constraints (Ya- mada & Watanabe, 1971; Xu et al., 2020). In our context, this temporal constraint corresponds to the property that (0) (0), (1) has the same distribution as Y (0) ((),, or in other words, (0) independent of (1) given (())- This assumption is reasonable because (0) and (1) exist in different "worlds", and any dependence between them should arise through (0). In Appendix F, outline relaxation that takes advantage of Causal OT (Veusal(r)) to effectively address these constraints. We also present result analogous to Proposition 3.6. In particular, Vemssat is guaranteed to improve upon (T), and admits an ap- proximation eπror to V. of order 1/n. However, despite this desirable property of Causal OT, the computation of its consistent estimator is much more challenging compared our current proposal due additional causal constraints (Eckstein & Pammer, 2024).
Balakrishnan, S., Kennedy, E., and Wasserman, L. Con- servative inference for counterfactuals. arXiv preprint urXiv:2310.127555 2023.
Baptista, R., Marzouk, Y., and Zahm, O. On the repre- sentation and learning of monotone triangular transport maps. Foundations of Computational Mathematies, 24 (6):2063-2108, 2024a.
Baptista, R., Pooladian, A.-A., Brennan, M., Marzouk, Y., and Meess J. Conditional simulation via en- tropic optimal transport: Toward non-parametric esti- mation of conditional Brenier maps. arXiv preprint arXiv:2411.071552 2024b.
Bernton, E., Jacob, P. E., Gerber, M., and Robert, C. P. On parameter estimation with the Wasserstein distance. Information and Inference: Journal of the IMA, 8(4): 657-676, 2019.
Acknowledgements
Bickel, P. J., Klaassen, C. A., Bickel, J., Ritov, Y., Klaassen, J., Wellner, J. A., and Ritov, Y. Efficient and Adaptive Estimation for Semiparametric Models, vol- ume 4. Springer, 1993.
The material this paper is partly supported by the Air Force Office of Scientific Research under award number FA9550-20-14 20-9779---77 and ONR N000142412672. Support from NSF 2229012, 2312204, 2403007 is also gratefully acknowledged.
Billingsley, Convergence of Probability Measures. John Wiley & Sons, 2013.
Impact Statement
Black, E., Yeom, S., and Fredrikson, M. Fliptest: fair- ness testing via optimal transport. In Proceedings of the 2020 Conference Fairness, Accountability, and Transparency, pp. 111-121,
This paper presents work whose goal is advance the field of Causal Inference and Optimal Transport as well as Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Blanchet, J., Murthy, K., and Zhang, F. Optimal transport- based distributionally robust optimization: Structural properties and iterative schemes. Mathematics of Op- erations Research, 47(2):1500-1522 2022.
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Covariatc-Aware Transport for Causal Bounds
A. Related Works
A.1. PartiaalEdentiffcaiion
In the literature of causal inference, identifiability is typically the first issue addressed when a new causal estimand is proposed. Under the potential outcome model, causal estimands depending on cross-world potential outcomes are generally not identifiable due to the inherent missingness. In this case, the range of parameter values that are compatible with the data is of interest. A multitude of works have been dedicated finding causal estimands that can be identified, where set of assumptions is usually necessary. For instance, the identifiability of the average treatment effect requires the assumption of no unobserved confounders, and using instrumental variables to identify the local treatment effect requires the exogeneity (Imbens& & Rubin, 2015). However, verifying these assumptions can be challenging or even impossible based solely on the data, and the violation of the assumptions will render the parameter unidentified.
In the literature of econometrics, the partial identification problem has been a topic of heated discussion for decades (see (Kline & Tamer, 2023) for a comprehensive review and references therein). The works generally begin with model depending on the parameter of interest and possibly other nuisance parameters functions. The partially identified sets are specified to be the values that comply with set of moment inequalities, parameters that maximize a target function, or intervals with specifically defined endpoints. In causal inference, there is a tendency to avoid heavy model assumptions imposed on the potential outcomes to ensure the robustness and generalizability of causal conclusions. Methods therein often involve leveraging non-pann paraameeettic or semi-parametric methods that make few structural assumptions about the data-generating process (Bickel et al., 1993; Van der Vaart, 2000; Laan & Robins, 2003).
We detail three particularly relevant threads of works among the studies on partial identification problems. Copula models (Nelsen, 2006) are used to describe the dependency between multiple random variables with known marginal distributions, which also aligns with the potential outcome model with marginals accessible and the coupling unknown. The Fréchet-Hoefdding copula bounds can be used to characterize the joint distribution of the potential outcomes (Heckman et al., 1997; Manski, 1997; Fan & Park, 2010). However, when dealing with more than two margins, one side of the Freffit-Hoeffdingg theorem's 's bound is only point-wise sharp: moreover, copula models are generally constrained to unidimensional random variables; in addition, the bounds on the joint distribution do not necessarily translate to those of causal estimands, such as the variance of the difference of two potential outcomes. The second thread of related works explicitly employs the optimal transport to address the partial identification problem in econometries and causal inference. (Galichon, 2018) considers model that includes observed variables and latent variables, and the parameter of interest regulates the distribution of the latent variables and the relationship between the observed and unobserved variables, which differ from our problem formulation. (Balakrishnan et al., 2023) introduce the quadratic and differential effects for continuous treatments (infinite levels of treatment) that can be considered as examples of the lower limits of our formulation with a quadratic objective function. (Ji et al., 2023) tackle the PI problem using the COT framework. Their approach involves first estimating the conditional marginal distribution and then solving for the optimal dual function at each observed covariate value. While this method consistently provides valid Pls, accurately estimating the conditional marginal distribution (nuisance function) is challenging, and inaccuracies in the nuisance function estimation may result in conservative PI sets. Additionally, the method is computationally demanding.
A.2. Optimal Transport
Optimal transport (OT) provides an effective way to compare two probability distributions (namely, source and target) by leveraging the geometry of the sample space. The OT cost between two distributions is the minimal expected cost of transporting mass from one distribution to the other, where a ground cost function dictates the cost of moving a mass from one point to another. OT is a powerful tool in machine learning and statistics, and it is used in model fitting (Kolouri et al., 2017; Peyré & Cuturi, 2019), distributionally robust optimization (Mohajerin Esfahani & Kuhn, 2018; Blanchet et al., 2022; Gao, 2023), fair machine learning (Taskesen et al., 2021: Si al., 2021), generative modeling (Arjovsky et al., 2017: Gulrajani et al., 2017; Liu et al., 2018), goodness-of-fit tests and two-sample tests (Ramdas et al., 2017; Sommerfeld & Munk, 2018; Bernton et al., 2019; Tameling et al., 2019), among others. In particular, in the causal inference literature, optimal transport has been widely recognized as an effective tool for counterfactual modeling (Black et al., 2020; Charpentier et al., 2023; De Lara et al.. 2024; Torous et al., 2024).
When matching two probability distributions, conditional optimal transport (COT) arises naturally in the presence of covariates, as it transports measure between the corresponding conditional distributions for each covariate value. COT provides triangular map between the source and target, which is useful for conditional sampling (Wang et al., 2023;
12

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Covariatc-Aware Transport for Causal Bounds
Hosseini et al., 2025; Baptista al., 2024a). Further, COT has been applied in training conditional generative models, including domain adaptation (Rakotomamonijy et al., 2022), and flow-based generative modeling (Kerrigan et al., 2024). To address the computational challenges of COT, several methodological approaches have been developed: GAN-based optimization (e.g. (Baptista et al., 2024a)), Kullback- Leibller (KL) divergence-basse relaxation (Tabak et al., 2021; Manupriya et al., 2024), entropic relaxation (Baptista et al., 2024b), OT relaxation (Chemseddine et al., 2024). While several simulation-bedl based computational methods have been proposed, the statistical guarantees for estimating the COT value remain largely unexplored.
B. Notation Table
For any domain Ω C IRd, we denote P(2) as the space of probability measures on Ωi we alse denote Pac to be the set of measures absolutely continuous with respect to the Lebesgue measure. For measure /' and map T, we denote the push- puww-forward measure of P using T as T P, where T. P(A) = (A)- for any measurable set A; If the map is linear, i.e. T(x)= = B x for matrix B, then we use B P to denote the associated push-forward measure. For a function f. xp" denotes its convex conjugate, i.e. (²(v)) = sup,, fx - -(x)}. For matrix A, we denote (A) to be its largest eigenvalue and (A) to be its smallest eigenvalue; we denote ||A oop to be the operator norm defined as || supayt0 ||Ax]||/ ||=|||
A closed set is said to be C if its boundary OS is a twice continuously differentiable manifold.
A convex set S C. IR" is said to be uniformly convex if for every > 0, there is a >> 0 such that for any two points
y,y E S || - ull2 2 6, the point 2 = [x + y) satisfies the condition:
inf | wl12 2
wsds
The space (h,)(()), where & e N and (0, 1). is defined as:
D - D* Ir y
Ch. = u E (12)
sup zEQ
D*)(()) < oa for all |a
S k, sup sup jal=kry
< {00).
where n denotes the partial derivative of u of multi-index order and |al| S l indicates that a is multi-index of order up to k.
The Wasserstein distances between P,Q E P(X) are defined as
=EP(X3) W (P,
inf
(En [|| - X 1)
p PEN.
Pang
The optimal transport distance between P,Q P(X) with cost function f xx² + R is defined as
inf (P.P.Q)= -EP(
Ex ((X
C. Proof of Main Results
In this section, we provide our proof and the theoretical results we will refer in our proof.
C.1. Brenier's Theorem
In this section, we collect the results of Brenier's theorem (Brenier, 1991)) for optimal transport using quadratic costs. Theorem C.1 (Brenier's theorem). Let y and be two probability measures on Ω C I&d, where μ E (a))) Then, there exists convex function p : R² - I. such that the optimal transport map T R= + R" pushing P to v is given by T(x) Vi)(x) where Vip is uniquely defined p-almost everywhere. Moreover, T is the unique optimal transport map solving the the Monge problem for transporting onto
inf T&TTH=R R"
|| -(xxx)((dp(()))
13

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Covariatc-Aware Transport for Causal Bounds
we further assume that (Q), then Vi²" is the gradient of a convex function, uniquely determined v-almost everywhere, satisfying V v 1e, and providing solution to the Monge problem for transporting onto μ. Moreover, for Lebesgue-almost every I. y E S2, the following holds:
Vp" Vp(z)=z, VpoVp'(y)= =y
C.2. Founier and Guillin's Result
Suppose that the sample (X,, E [N]) ii.d. follows μ and denote Σ ōx.- Theorem 1 in (Fournier & Guillin, 2015) characterizes a nearly sharp convergence rate for empirical Wasserstein distances.
Theorem C.2 (Empirical Wasserstein convergence rate, Theorem (Fournier & Guillin, 2015)). Let μ E P(K)) and let p O. Assume that Mz(x)= = f ||>|| du < do for some p. There exists constant C depending only p, d. such that, for all N 2 1,
N-1/2 1/2 + N (9-1n///
if > d/2 and + 2p.
B[WB(uN,)) S CMP/1()) x
N-1 log(1 N) N-
if d/2 and + 2p.
N- + N (1--1//
if E (0, d/2) and # d/(d-) p). pp).
Theorem C.3 (Empirical Wasserstein convergence rate, non-iid., Theorem 14 (Fournier & Guillin, 2015)), Assume that the sample is c--mixing, i.e. there is function a N - R2, satisfying Σσ (k) < co and corr( (())) S a(li jl) for j and all bounded functions f,g, then the results in Theorem C.2 still holds, with the constant C may additionally depend on a.
C.3. Proof of Proposition 3.5
Proof of Proposition 3.5. By Theorem 4.1 (and its proof) in (Villani et al., 2009), 1lp is tight and closed in the weak topology. Further, since My,yy) v 1| 2²112 is bounded and continuous the compact domain of y 2², the set of minimizers 11 (1) is non-empty. Similar reasoning
be applied argmin [, (0), Y (1)) and argmin,ee, E, (o) (1))). Therefore, xu me (n) exist.
We then proceed to show that the uniqueneness of m Tip (1)) We only prove the uniqueness of TiP (17) and similar reasoning can be applied to -.-
We apply Theorem 10.28 (Villani et al., 2009), and denote the three conditions in Theorem 10.28 as cond(i), (ii). (ii)). By Assumption 3.3, hy, + - ||= 2|l is differentiable (thus superdifferentiable) everywhere, thus cond(i) satisfied. By Assumption 3.3, the gradient
V 3,8 (MV. + ||= -1 (V Ny 2) =))
injective with respect to J. à for vy, Z. thus cond(ii) is satisfied. Finally, by Assumption 3.2.3.3, locally Lipschitz, Py(o),z, z, (y(1),z z compactly supported, and Py z is absolutely continuous with respect to the Lebesgue measure on Bd thus by Remark 10.33 in (Villani et al., 2009), cond(iii) satistified.
As result of Theorem 10.28, m (n) is uniquely defined.
C.4. Proof of Proposition 3.6
Proof of Proposition 3.6. We prove the three properties as follows.
(i). Since V0 can be written as
min F. (Y *EII;
where
(TE x 2): : Proo, = Pro)
14

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Covariatc-Aware Transport for Causal Bounds
it follows that hp S IT Therefore, we get Vu Vip(n))
Since V. can equivalently written as
min Er Tell?
(1)((1)))++11))) |1100) 2(1)
(6)
where
I
T P(V 2 x 2): ( TY(1).7) Pr
Z ((1)
r-almost surely).
it follows that II S Hp. Hence, we have
Vip(n)
min E, [(((0))) +000))))+200)))) 2(1) s Ve. rElle
To prove that (n) non-decreasing with respect to 17. we note that, for 1)1 S 12.
E 0 (m))
((Y(O),Y(1)) + 7 [| Z
(7)
E (m2)
[(Y(()), Y()) + [| 2(1)
(8)
< 5 (ma)
((Y (()) + llZ()) Z
[(Y(()), Y(1) + || ((1)
(10)
where the first and last inequalities are due to the optimality of (m). Tip (m2),, and the second inequality is due to M S 122
Therefore, we have (9) - (8) < (10) (7), that is,
(112 1/1 mmm))))))
||z(0) Z(1)ll² S (n2 71) [1Z(0) 2(1)
and thus, (m)) [| - 2(1) S Bng(n [12(0) -1)1)) Then, combining this with (7) S (8), we get
E Eng(on [b(Y (0), s Bx (0), (())
i.e., (T)) S (n2)
Next, we will prove the continuity by showing that Vip(1) is left and right-continuous with respect to
For left-continuity, consider any 77 E (0, oo). By Theorem 4.1 (and its proof) in (Villani et al., 2009), IIp is tight and closed in the weak topology. Then, consider sequence (Ns, & 2 1) such that 7 -> foo- Applying the Prokhorov's theorem (Theorem 5.1, (Billingsley, 2013)), we deduce that there is subsequence of (T (7%)) k 2 1) that weakly converges measure = P(()' 2) Without loss of generality, we assume the subsequence is (# (m))) 1) itself.
Again, since V/ 2 compact, then c, |1-112 are bounded functions confined on the corresponding domains. Therefore, we have
lim Eag(on) k0000
(M(Y())) + 7()) 2(1) = Erz [NYY()) no I|((0) Z(1 211)
On the other hand, have
[|Z(0] 2(1)12] = =0
lim (n - -0) x Eng(on))) 8000
(mx)
Combine the above two limits together, we get
lim Ex (m)) (mx) k-000
(O(Y())) + Th [((0) (1) = Exz [N)Y()), Y()) + +% [Z 2(1))13]
15

---

Covariatc-Aware Transport for Causal Bounds
Since we have
Bag((n)) [(Y(()) m m [Z z(1)
(n0))
(L)) + no IZ(0) z(1)1I²
<
((Y + no (2((0) =
thus, we get
Exg((0)) ((Y(O), + no [1 - 2(1) = EEng (0), (()) + +oo [ 2(1)
implying The = (70) due to the uniqueness of the minimizer (Proposition 3.5). Hence, we get limg-+00 Vip (Tx) (mp)..)
As a result, for any sequence (m,, 1 2 1) such that limy-+ac = 7o. we can find a subsequence of (n. 2 1), denoted as (M, k 2 1), such that limg-poo Vip ( = (1))) implying that lim Vip(n) = Vyp(1
For the right-continutty, similarly, we can similarly take sequence (ns, k 2 1) such that 77s - no+, such that Tip (1k)) converges to T. E PP(O' x z) in the weak topology.
For any x' E P(()' x )) by definition of mi (mu), we have
Eng((n) (O) Y(1) + The IZ(0) 2(1)1 s E. [(((()) + Ta Il((0) 2(1)1
Send &s oo, similarly as above, we have
lim Exg(mn)) (N(Y(0), + 7 [| z(1)1]] = =Er; [N(Y()) no IZ(() 2(1)1
thus
E [(Y()), + no || - _(1) < E [(Y + nu [| - Z(1)
which implies that Th = क (1o). . due to the uniqueness of the minimizer (Proposition 3.5).
As a result, we get lim (nx) = Vip (1o). Then, using similar reasoning as in the proof for lim, Vip(n) Vip (mo)) we conclude that lim (n) = Vip()
(ii. For 1 = 0, since Va S (O), it remains to show that Vi)(0) < Vu.
By the gluing lemma in Chapter 1 of (Villani et al., 2009), we can "glue" the three distributions nØ Pr(o), Pr()), constructing coupling ı E PP)' x 2)², such that
TT (1)) n. (U),2( = 10 = Pr
Thus, π E Thp which results in (0) < [= = (M Y(1) = Vu-
For 17 oo, by (i), we have
E Bng((n) [((Y()) + + - 2 - Vo < x
Vn 2000
Therefore, we get
lim Engon 170000
[Z(00) 2 = 0.
Since ()² x -)² is compact, then take a sequence (M,, k: 2 1) such that limg- = co, and apply the Prokhorov's theorem, we get: there is a subsequence of (Ti (7x), k2 2 1) that weakly converges to a measure oh E P(Y x 2)²).
Without loss of generality, we assume the subsequence is (ग (mx), & 2 1) itself.
Since [|Z 211)l22 is a bounded function on 2², then we get E,, [[] -(1)) = 0. i... Z = Z(1)
T -almost surely. Therefore, n5 E IT' where IT' defined in (6), and thus V. S F. ()),)((()))))))))
16

---

Covariatc-Aware Transport for Causal Bounds
As result, we get
(e (O), & (ag(ne)) ((Y(0), + m/ [2(0) 2(0) SV.. S B (MY()))
Since lim Brg( [ (0), ( = E (1(Y())) (1))). then as approaches infinity, the above inequalities become equations, implying that
lim Vip(m)) Ve. k-+00
As result, for any sequence (m, 2 1) such that limz 7N = co, we can find subsequence of (m, 2 1), denoted (Ma & 2 1), such that limg-too Vip (m))) Ve. implying that lim Vn)(n) Ve.
C.5. Proof of Proposition 4.1
Proof of Proposition J. Under Assumption 3.1, as m - oo, we have P,,Y(0),z 4 Py(o).2 the weak topology almost surely, and also ( 4 Py()).2 a.s. Then, by Assumption 3.3 that h is continuous, and Theorem 5.20 in (Villani et al., 2009), we get: up to extraction of a subsequence, Tip. (1) almost surely weakly converges to an optimal coupling of the OT problem
min Er (L)) + + - 2(1) TElly
By Proposition 3.5 that the optimal coupling (1) of the above problem is unique, we get the subsequence of T (n) 4 Th (1) a.s., and thus Ti n.m (n) 3 (n) a.s. (for every subsequence, we can extract the converging subsubsequence to converge weakly to mip(n)) As a result, Vip.n.m (1) converges (1)(1) almost surely.
C.6. Proof of Theorem 4.3
In this section, we provide the proof of Theorem 4.3, which extends Proposition 14 in (Manole et al., 2024) to general quadratic functions. To proceed, we first establish the following result as foundation. Note that in the following discussion, we use (and æ) denote source random variable and X (and T)) to denote the target random variable, such that X, X X' Rdd where X' is convex.
Proposition C.4 (Stability bound of quadratic optimal transport). Let (X,, Ki,siis n) be i.ddd samples samplesfollowing distribution P and (X,, < < m) be i..dd samples following distribution Q. Let P. Ei 6x,, Qm m E5=1 ōx,, and suppose that ăे (fig, 1 <is n S m) is coupling of Pas Qm- Additionally, suppose that T is coupling map such that T P Q.
Let the cost function defined by hg(2, F) = x Bx. where x = (2, and
B11
B
B
B1, B2g, B12 Id
B
B22
Then, we have
E
,)))((X,
Ep [he(X,
-E
[2X] Bia(X, - T(X)))]
i=1 1=1
17

---

Covariatc-Aware Transport for Causal Bounds
Proof of Proposition C.4. We first expand the term (x Xx) follows:
hB(X;, X3)
=X, B11X; - 2X; B12Xy + x B
BuX, 2x, (x) 2X; -
(1)
(2)
+ +X())) (x)) +(X;+
+ T(X:)) B T(X:))
= hB(X,,T((X)) +(X, T(X))) B22 - T(X.)) + 2 B12 + T(X)) Bx2) -((X))).
(1)+(2)+(3)
Note that E/=1 mij Vi since is a coupling between Pns Qm. thus
E
(X,,
Ep Eb(hh((X,T(x))
Therefore, we get
E
tng (X1,
-Bp [XX,
1= f=1
Aig [8, - -((x)) Be2(X, - - B1 + T(X,) ( (11)
Also, note that
2T(X.) (x, T(X;)) = -XX, T(X.)) B2a(X; T(X:)) + XyB2XX T(Xi)* (x2)(X;).
Thus, continuing from Equation (11), we get
E
(X, X,)
Er (x(,x(X,T(XX)
E
my (-2x, B12(X, T(X))) + -2,
- T(X)) (BeT(()))
(12)
Again, since Ej R,j 1Vi and T
Q, we have
Y2
m
E
8 T(X,) [xx)
j=1
n B22 - Σ T(xx)
E
1=1
-EQ
[g2
Ep
[T()] B [X)]
-0.
18

---

Covariatc-Aware Transport for Causal Bounds
Then, from Equation (12), we get
R
m
1
E
Mey
Fegh
Ep [(,((x -B
Tay
[2X] - -xx(x;TT(X))]
(13)
j=1
When the the cost function is defined by (n, σ) = x
Bx, where x - (I, and
B11
-12
B
B11, B22, B1 12xd
B
B22
the optimal transport problem is actually equivalent to the problem that uses the cost function -2x" B120 since the remaining
parts in hg are determined by the marginals. Then, if we consider a linear transformation x - B 1|22 & E, then the cost
function becomes - 2 J,, which possesses a Brenier map as optimal solution (see Section C.I). To utilize the convexity of the corresponding Brenier potential, we present the following result.
Proposition C,5 (Coordinate-shift optimal transport). the same setting as Proposition C.4, additionally,
(i). Assume that P, have densities on X.
(ii). For the Monge's problem
E B pX (x, (())),
min (((BB22 P=Q
where the cost function is defined by (1, 3)
x Bxx, with (x,
and
B B11 Bi2 T
Ia
B
-Ia
Bza
(Here, T is map defined by x - (B),) and BI2 is a linear map defined by - BIgz), assume that the Brenier map of T*, and the corresponding Brenier potential Φ, satisfies that & E (2) Further, there exists constant > 0, such that
1 - L Vz(x)) AI
Va B supp(P).
(ii). Assume that B12 is non-singular.
Then, we have
(a). There is a P-a.s. unique solution T I sup
Further, T* = (B(()()) ( for P-a..
(b). Let (fij, E [m],j E [m]) be the solution of
n
m
Wha (Pas Qm) = min
tiyho(X,,
#Ell....
i=1 j=1
where
11 { P(X X) Ax Pas Qm)
Then, have
E
Tt [X,
T (x))1
< A S (Pn Qm)
Wha (P.(((Q) ()]
i=1
==
19

---

Covariatc-Aware Transport for Causal Bounds
Proof of Proposition C.5. (a). Note that ng (B (t) Then, have
min T:(fB/24
E B p (X,
min
Ep(h g X,,
(change of measure: B P P)
T.( P=Q
min (xplt((X,T(BBxx
(due to: hg (B12,,,)) (g(2,
f((fn[)+ P-Q
= min Ep (X, T=Q
(since B12 is invertible).
The above deduction implies that there is a one-to-one correspondence between T': :TBBB # P = Q and T T4 P Q. such that the corresponding maps share the same value of the objective functions equipped with h and h,, respectively. Therefore, we get T* (a)) = T*( (B [[]2) is the P-a.s. unique solution of argming: P=Q Ep (hahh(X T(X)) since T is the unique Brenier map of the corresponding Monge's problem.
(b). By (a), can write
x, BIz(X, - T(X;))
X, Bia(X, - -T(B)x)))
=V (T(B(xx)) (X]-T(B[2X))
(14)
where 6 is the convex conjugate of c and Vø (T(x)) = x Vz B supp(P) (Theorem C.I). By (c) and Theorem 4.2.2 in (HiriartUUrrtty & Lemaréchal, 2004), we have
1 (og)
Va E supp(Q).
X
Then apply the second order Taylor expansion, we get
-(([(x,)).
2X
Thus, by (14), we get
2X |x -
- -X_ - T(Xi))
Note that by Tw P Q, we have
- ((())))((((X)))) (x) - Er(p 0.
Therefore, by Proposition C.4, we have
E
|X, T(X:)11
j=1
<
E
[2X. BI0(8,- T(X,]]]
t h RayhB(X,, (x,, X,)
- -Ep Eh ((X))
W (Pa Qm) - Wha (P,Q)).
20

---

Covariatc-Aware Transport for Causal Bounds
The following result a direct corollary of Proposition 13 in (Manole al., 2024).
Proposition C.6 (Bounding W2 distances by curvature). In the same setting Proposition C.5, for any random measure P.Q such that
J sap = saP, E a
saa) - laq v..
E
we have
< Whe (P, 1.--Wo Wha (P.Q)
(
<xxB W2 (2 P, Blew P) + Wa((, 0))²
Proof of Proposition C.6. Let hi (x,5) -2x F. we have
E
Who (P,4) -wn (P.0))
E
Whn (Bi2 P,
Wh, (B P. Q1
E
min (-,, h&)
min (TT, h&)
Tell
xEIl
E
min (T,, h')
min hz) EEn
Lell
(2 P, w P.
where
IT' { E P(22) xx B P. π ②}
and
T P(X(3 Tx = Bi24 P. T = 2
Proposition 13 (Manole et al., 2024) results in:
0 W (2 P. Q) W3 P.Q) Bnd P- BB P) P) 1--odll vodl(- Q)
<Xx (Wa P, B (Q)))))
Taking expectation over the terms of this inequality finishes the proof.
Now we are ready to prove our main result Theorem 4.3 stated again as follows.
Proof of Theorem 4.3. Equivalently, we can write
(,, Sy.z. Pm,Y(1),2 1 ,
where ((Y,, & (n]) are i.i.d. samples following Py and ((V, Zz). E [m]) are i.i.d. samples following Py())
As previously discussed, we write the source variable = (Y (0), Z(0)) and the target variable X = (Y(1), Z(1)). Then we introduce the cost function is (2 Ø) x Bx. where x (x. .) and
A
A12
m/
B
A
A22
I
rp
21

---

Covariatc-Aware Transport for Causal Bounds
Recall that
Vip.n.m = E w.n.m (1) (A) Y())).
(15)
where
mmp. (1) E argmin En
7ellp.n.m
I pm { xxppxxxxxxx> TY 2 Pm,y())2)
and
Vip(1) - E. (0),
(16)
where
T (1) = argmin E, (O), (Y((),
=
1 ( P(Y 2 x y x 2): (Y(0)2(0) 2y(0),2. Pr())
By Proposition 3.5, there is an OT map T coupling Py Pr such that mp(()) = (la. T) Pr(0) Therefore, we have, o almmost surely,
(1)), (0), Z(0)).
(17)
Note that h(y. x B,x =: hB, (x, where x = (x,z)²
((y. 2), (Ø. B)) and
A11
12
0
B₂
A
A22
0
0]
Then, let X; é (Ya, Z., X, - (Y,, 2,) & (Y()), Z(0)), & 2(1)), we have Vi (n) (1)
Riyho, (X, - Elhn, ((x)(
(due to (17))
:=1
2
Y A (YY0))))227(()))
1=1
(Term
Y AnY ( An (0)
(due to
Σ Rij
n
1
Y
AzsY, T A Y
(due to Σ Rzz
For (Term A), we further expand be
(Term A)
<
m Y, A T(2.
==
(Tem B)
(due to Σ = )
+
Y A Z,) -1Y0)))))2 2(0) 2(0)]
n
1=1
22

---

Covariatc-Aware Transport for Causal Bounds
For (Term B), we have
E[ B)
S
Rag Y, T, A12(Y,
T(Y;, T(Y1
im
b Tij |1, T(G. 20)12
< $| E
ts |i:ll
E
(Cauchy-Schwarz ineq.)
S A E [1)0))
E
Kij
|x,
T(X;)
(due to Σ = n
)
(Term
For the expectations of the terms above except for (Term A), (Term B) and (Term C), we apply Markov's inequality. For (Term C), by Proposition C.5, we have
E
Σ Σ # ||, T(X.)
SAxE Wnn Pm, Pm,Y(1),2)
(P: (18)
Therefore, we get
E || V Vp (T) Vip(n)]]
< 2x Tlllo Epy [1)) 21
An,m
Var(Y (0) AAnY
Var(Y Y (1) An (
(0) (Y), Z(0))
(19)
Vn
Vm
Vn
(Term
where
A E Who Pm.10))) WNO (Prio, Pr(),z))
p.m
(Term D) can be further bounded by
(Term D) <
(IAmllop (l EIIY + ||A22llop (1) + + + (20)
where = min (n, m).
As for An,m applying Proposition C.6, we have
An,m IA x E
W W2((P P + W2(Pm.
Pr(),2)) (r(,,,,)),PP()2)))))
22 W (P ( PALy(O))))) + W2((m.y(a.e) Pr().2))
(21)
Finally, apply Theorem C.2 to directly bound the terms E Wi(Pp..5.y00)azz PaniiPPa PApr(O)))) (and similarly for W3((m. Pv(),2)), we get
E PZ
SC A12l (I [|) + [1212]
(22)
where C' is a constant only depending on dy + dz, Ind is defined in the statement of Theorem 4.3. Combine Eq. (20) (21) (22), and bring back to Eq. (19), we get the desired result.
23

---

Covariatc-Aware Transport for Causal Bounds
C.7. Curvature Condition in Theorem 4.3
In this section, We provide an informal discussion on the general relationship between the curvature and 7). Without loss of generality, we suppose A I.
1) > oc, (n) converges to V. stated in Proposition 3.6, does the corresponding OT maps. That is, the OT map between (1))) and (0) converges to the conditional optimal transport map for the COT problem Ve. Consequently, the Brenier map Tn : y x 2 + y x 2 between (r()) and (r(0).12 satifies that
I'
/ 1-11
[sy, z) + o())] [= 2 o
T (v, 2)
as 17 - co.
where f)) A fa(1) the Brenier map (Theorem C.I) of the conditional optimal transport problem
min xel(())
Ex [(Y (0) (1))
where 11x)) = {TT P()
(0) Py Ty() = Projj1=3)
As result, we may get the Hessian matrix of the associated Brenier map be
0yf(y. 2) + o(1) 0.f(y, = +o(1)
VTn
0. 2) + +0(1) o(1)
0yf(v,))+(()
1 (0,f + o(1))
I
Va
vTj
vn (0.[(y,)) (1))
1 1+0(1)
vnI
Then, we have Amax VT,,) < (0,))) V 1). where , converges to matrix independent of 17. As for Amin. similar argument can be applied with reversed inequalities, thus Amin (V7,) 2 Amin (0,) A 1). In Section C.8, we provide a rigorous proof following the above discussion the case that (Y 2) and (Y(,,, Z) are Gaussian vectors.
C.8. Proof of Lemma 4.6
Proof of Lemma 4.6. Denote the covariance matrix of (Y))), 2) to be [Eo, Eo,2]: ( Σg]]. and that of (Y Z) to be [[>,, 21.2]: 1T M2]]. Suppose the two covariance matrices non-singular. We shall analyze the curvature of the Brenier potential between Pr()), and PAT, (0) Without loss of generality, we let A12 =I. I.
By Eq. (25), the Hessian matrix of the Brenier potential is:
V イ
where
Σg
Σ
E1
Σg =
Σ
11200,2
1)
17 Dز
Σg
First, we have
V2pp
(5)
(Eq
E 53 }
(8 xx)
||
2
||0 1/
thus Amax (V2))= < Amax (5
S 52 Amax ( (5i
Apply the similar reasoning, we have Amax
(Ei) Einist < Amaa (mio)susx (En).
Note that
I
Σn
Eo,2
I
$0
mI
2
Σn
nI
An
5%
24

---

Covariatc-Aware Transport for Causal Bounds
Then, Amax (Eo) S Amax (E)) (A2)) Amax Amax(()) (E)) (17 V 1)a. Therefore, Amax (V) S C(n 1) for constant C.
For Amin. the same argument above can be applied, except that the inequalities are reversed. As result, get Amin (V+)) 2 c(1 A 1).
C.9. Proof of Theorem 4.7
Proof of Theorem 4.7. Follow the same reasoning in the proof of Theorem 4.3, derive the following inequality similar to (19).
EB [|V m.m (1) Vip(n)]]
< 2
A op Epyco [1\(0)1|]] VAn.
+ E
Y An 1 - ( An (()
+ E
Yj - (1)
+ E
Y T22, Z 20000)
Similarly, by Proposition C.6 and Theorem C.3, we have VAn.m S CXtmnNa for a constant C that depends on a, d, ||A12llap as well as the second moments of (0) (1), Z.
For the remaining terms, we have
Y,T Any. - B[ An 2
a Y
n n
E
A - - An (()
S
(Jensen's inequality)
12
Cov(Y, AnY, YJ AnYY)
Var Var(Y(()) An (O))
n k)a(k))
(Definition of a-mixing)
Var(Y(0) An (O))
Σ akk)
Vn
Var(Y(0) An (0))
a(k)
(due to a(k) < oo).
k=u
The other terms can be bounded similarly, which finishes the proof.
C.10. Proof of Lemma 5.1 Proof of Lemma 5.1. We have
V. = min E, ((Y)
*ENl.
min
Epz
E (YY)
*ell,
EEz
min
E. (X(00)
x (Ell(()
25

---

Covariatc-Aware Transport for Causal Bounds
where IT(=) {nr P()2) (0) 7Y(0))==::7Y11) Pr(s)1==s-
Recall that
| - xx112] = 1140 - Mill2 1:11-1115 Tr Eo +1 -
min
Ex
(23)
8:5 Ng Ea),
Xx (1)
Therefore, when
= =
E1),
E1), then we get
min E. -X())) (160))) + S(Z. 21),
x x'El((a)
where S(Eo, E1) = Tr (2. + 1 2
When & =, 2, Y(0) = ~ ((fo)))(())))))))))))))))))))))) --- (1DK:/))))VVY0000-------0-111111 then we get E. (V Y()) -1D.,//()))5(P(((()))),,,ii()))0
min x'El(z))
where D(v) is the matrix with the vector placed at the diagonal. The desired results are proved.
C.11. Computation Details in Example 3.8
Recall that
Ex [ - -x113] = 11100 m1l12 Tr Eo + E1 - 2( M, )
min X Na), xx (
(24)
The optimal coupling nt is defined by the distribution of (Xo, A(Xo - no) + 41), where
A= Eの 1
(25)
In Example 3.8, we have hyyo y1) = (yo + y1)2². Then, we have
(1). For Vn, since we have (0)) ~ (0, % + + -Y + direct application of (24) leads to the result.
(2). For Ve, we can directly apply Lemma 5.1.
(3). For Vip, we have (() A (Y()), Z(0)) ((0))
(O, X(1) A (-Y(1),((1) ~ (O, E1), where
[30 + of V "
[89 +
-
No
Σ =
-v78
"
Consequently, the optimal coupling of (Y))), -Y))), i.c (n), is the distribution of (Y T AX(()), where - A is defined in (25). Then, we have
Vip(1) =[ 2E (O0-))))) (B) + + - 22 (())))(
Specifically, we compute
E Bnj(o) [()) = Exj(nle A(())X()TAAT))
Erj((n ((()xx()) (ATee)))
Ex3,0 (O) TA
A e)
Tr((B=go) (X()) A ee e)
-TrE) ) dee
26

---

Covariatc-Aware Transport for Causal Bounds
A quick algebra shows that for 2-by-2 matrix B,
B
(B + v det (2)
(2)
Tr(B) + 2/ det(B)
where I2 is the 2-by-2 identity matrix.
Then, plug in B = 2 E, 2 with
Tr T = Tr(2o)1) = (6) ag) + - 2n +
det - det(Eo) det(En) 17 o'oj
we get
((00)(())) Eree det(B))e)
TT((B) + del(BB
(g) - oz) + +o,) 1730111 + 10001
(0 + (B) tff) 21130 + 112 + 270001)
We obtain the desired result.
Remark C.7 (Multi-dimensional covariates). The above computation can be directly extended to the case when Z is multi-dimensional. Specifically, without simple closed-form, the quantity
Tr
イ =ce
can be computed by using the eigenvalue decomposition compute the square root of matrices, for (Jay: Ody xdz) ) and No, En defined as:
Eo = (0000 + S Vn.
V nt
Σ = B1181 + S1
V18
n
where So, S are the covariance matrix of multi-dimennional 60: S/.
D. Plots for Synthetic Experiments in Section 5.2
In Figure 4, we provide the plots of estimator values for bias comparison between our method and the method in (Ji et al., 2023).
E. Examples of Quadratic Causal Estimands
We provide examples of causal quantities involving quadratic functions.
Neymanian confidence interval. As discussed in Section 5.3.1, the classic Neymanian confidence interval (Splawa- Neyman, 1 1923) depends on the variance of the potential outcome difference, that is, Var(Yi(1) - Y₁(0)). The variance can rewritten as the partially-identiffbbe term (E{()(1)) (()))²) with a quadratic function minus the identifiable term (1) {²
Testing null effect. The null hypothesis of no treatment effect: Y,(0) Y₁(1) for any unit is equivalently to (((Y,(1) 0. The term ([(()())- and the null hypothesis of no effect can be rejected if the PI set for this quantity is significantly above zero.
is partially identifiable involving quadratic objective function,
Correlation index of potential outcomes (Fan et al., 2023). As discussed in Section 5.3.2, the correlation between potential outcomes quantifies the linear dependence between Y,(O) and Y((1): ): large positive correlation indicates that the potential outcomes are positively associated, suggesting that the treatment effect is relatively homogencous across individuals. This correlation is only partially identifiable, primarily because the covariance between the potential outcomes, Co) (1)) = ([) - E( which involves a quadratic objective function, is only partially identifiable.
27

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Covariatc-Aware Transport for Causal Bounds
0.50
20
100
300
23
100
(a) Linear location model
(h) Quadratic location model
(c) Scale model
1.75
1.50
1.2
2.55
1.00
0.50
32-
0.2
300
0.0
26
120
(d) Linear location model
(c) Quadratic location model
(f) Scale model
Figure 4. Values of estimators for V. (averaged over 800 repetitions) for well-specified model and two mis-specified models. The first row: m = n = 500, the second row: m = n 1500. The models presented here correspond one-to-one with those Figure 3. Specifically, Vee stands for the true value of Ve.
Covariance of treatment effects of different potential outcomes. For 2-dimensional potential outcomes, the covariance
between treatment effects of different dimensions of the outcomes Cov(r, +?) with -! = yz (1) xx) j =
1,2, Y; (Y("), )) involves quadratic function of the potential outcome vectors and is only partially identifiable. By investigating the lower and upper bounds of the covariance, can detect positively or negatively associated treatment effects. There are many relevant applications in economics, e.g. the effects of promotion scheme on the sale of complementary goods shall be positively correlated.
F. Causal OT Relaxation
In this section, we propose Causal OT relaxation (Definition F.1) of the target COT. We show that the population optimal objective value of the Causal OT relaxation improves over Vip (closer to Ve compared to Vip), and we provide explicit upper bound of the error VV. Veausal in Proposition F.3. For limitations of this Causal OT relaxation of Vg, please refer to Section 6.
Definition F.1 (Causal OT lower bound Veausal (n)). For oo), let
causal (7) argmin Ex #Elloaal
(YY()), Y()) + 12 2 2(1)
where the set of joint coupling v consistent with joint distributions of outcome and associated mirror covariate in treatment and control groups, as well as obeying additional causal constraints, is defined as
Heansal = { P(Y²2 x z²) ( = (Y TY(), PY())2, 2) = TY(I)Z(I>> ZY(0))Z00)22(1) = TY(0))
Then the Causal OT lower bound Veausal (7) is defined as
Veausal (n) = E (n) [h (o) Y())).
Further, this section, we impose the following assumption to justify Definition F.1 and our analysis.
28

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Covariatc-Aware Transport for Causal Bounds
Assumption F.2 (Uniqueness of Caassal (1), m.) Assume that (1) is uniquely defined, and the mo in Table is also uniquely defined.
Compared lg. Hecssal imposes an additional constraint the coupling satisfy the causal condition that Y(0) | 2()), 2(1) has the same distribution Y()) Z()). In other words, (()) is independent of Z(1) given Z()). This assumption is reasonable since Y (0) and (1) belong to different "worlds", and any dependence between them should operate exclusively through (0)..
The following result shows that (7)) interpolates the lower bounds Vi and Ve as ranges from to co, particularly dominating (i)(t) for any 2 0.
Proposition F.3 (Interpolation between lower bounds). Under Assp. 3.2, 3.3, F2, for any T 2 0, we have
(i). (Monotonicity) Vip(m) S Veausal (1) & Ve. and Veausal (1) is non-decreasing and continuous with respect to
(i)) (Interpolation) + Vcausa Ve-
(iii). (Convergence rate) Further assume that there exists Lz > O, such that
W1 (P110)2220)) Pr(o))=n)) I I.z ||20 21l12
VzD, 21 E 2,
(Condition A)
or
V20, E 2, Wi (Py(j)Z=20) Pr(z))=E)) S Lz 100 21l12
(Condition B)
then have
0 S V. Venusal (1) S (L
1
where La = sup B.
[o h(
[ele V1ll2
Proof of Proposition F3. (i). Since Teaaaa C Npp then Vip(n) S (y) Similar to Proposition 3.6, TI, C TTaamal since any coupling such that 2()) (1) a.s. satisfies the causal constraints in [leassal thus Veausa(() S Vo.
(ii). This follows from lim Vip (1) = V. (Proposition 3.6) and (n) < Veausal (1) < Ve from (i).
(iii.. We will prove under (Condition B), and the proof under (Condition A) similar by swapping Y()) and Y(1). By the optimality of Taussal (1) and II, S Heassal we have Veausal (17) + 1)Bm (v) [[] 2 s Ve, which gives
E Fa (1) [1|2(0) 2(1) Sn 'VV -V- - Venmsa(t)).
(26)
Lemma F.4 says
Ve Veausal (1) S L v Exg (n) [1| Z
Combining Eq. (26) and Lemma F.4, we arrive at
Ve Veausal (n) S Ln -- - -Vausal (1)).
which further yields the desired result V. Veausai (1) < (In
Lemma F.4 (Upper bound of V. Vemusat). Under the assumptions of Proposition F.3 and the additional assumption in (ii))
V. Veausal()) (1) S L, By R (m) [I] - 2(0) 13)
29

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Covariatc-Aware Transport for Causal Bounds
Proof of Lemma F.4. For notation simplicity, in this proof, ( =g) to denote d cansal,, (0)0a)))(()2200)===,)))))) (1))
and caessal 20, 21) to denote dr d ((0) (2) (())) 21). Then
Ve Veausal (17)
Emg [nn (0), (1)) | [ - - [ (Erimee(m) (n) (N( 2 2(1) z(0]]
- / E (0), Y(1)) ( = zu] - (/ E (m) (M()()), ( = 20 2() = 21 2 causaal (21 22)) dP 2(0) (20)
$ f Ex; (0). Y()) |Z( 20l - (n) (O), 2 20. 2(1) 21] d d 20) dP 2 (20)
= / Exx (O)) Z(0) = 00] - E Brinna (1) (O) Z = 20, Z(1) = 21] diTaausal 21).
(27)
Let [I( 21) = {TT P())): (0) Pr(o) 22-1 TY()) Py(1))z211+ }. then
Bag (Y)()) 7(0) _0] F,z (O), Y(1) ((1) Z(0) =a]
min =EIl(E0.00)
E. (O),
Since π Taansal satisfies the causal constraint, that is, under ग Tamsal (1) (()) _ ((1) ((1) and (()) Z()) = Y()) ((),, we have
Exiaaalt (O), Y(1) 0 = 20. Z(1 = =1]
min =(II(Fo.FL)
EN (O)
By Assumption 3.3, h is locally Lipschitz, and thus on the compact domain y, there constant L,, such that
h(y, yo) hy. vi) s La llye
Vill2
Vy. Vo. V1 V.
(28)
Now we prove the result using the similar technique proving the triangle inequality of Wasserstein distances. Let T1 PP()2) be the optimal coupling of the OT problem:
min =E
E. (0)
and let T2 E P()²) be the optimal coupling of
Wi (Py(n)zza)) Py(1) z=w))
By the gluing lemma in Chapter 1 of (Villani et al., 2009), there is coupling ma E p(Y?), where we denote y ((y(), y(2), ())), (a) e V), such that
(2) (1) (2))) = =1,,
(2) (3) 122-
Therefore, have
Ex [NYY (0), <Er, (y [(oo
min =E
<tx My + Ln ( [13
min
Ex (N)Y Y())) T., W. (Py(s)zzm>> Pr(j) 2=w)
min
E, [N (1)) + +LLL 1120 21112
30

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Covariatc-Aware Transport for Causal Bounds
where the first inequality is due ( y ( [20; =0); the second inequality is due (28); the equation is due the
construction of my by gluing wi; n2; the last inequality is due to (Condition B). Let L I, Iz,, get
min E (0), (())] TE 20)
Ev (0), ( L || =1 ll2 =x11()
min
Similarly for the other direction, we can get
mim
Ex [b)
E. (O) (()) 2 -L - =1 21122 xx.1()).)..
min
20)
Plug the upper and lower bounds back to Eq. (27),
Ve (V) S / L||z0 - 21 cissal 21)
LLE [1] [(0)]22) s LV (iinniti [Il - ( [g)
where the Cauchy-Schwarz inequality in the second inequality.
31


Reviews:
[{"review_id": "6LyAuXSr22", "reviewer": "Reviewer_9P7F", "summary": "This paper tackles the challenge of partial identification (PI) in causal inference, where causal estimands depending on the joint distribution of potential outcomes are not fully identifiable. While incorporating covariate information can tighten PI bounds, solving the corresponding Conditional Optimal Transport (COT) problem is computationally demanding and statistically unstable. The authors propose a **mirror relaxation** of COT that transforms it into a standard Optimal Transport (OT) problem with an added penalty term encouraging covariate consistency. This approach—termed **interpolating OT (Vip(η))**—creates a family of bounds interpolating between unconditional OT (Vu) and exact COT (Vc). The paper proves that the bounds become tighter with larger penalty, and the approach is consistent, computationally tractable, and robust. Theoretical results include convergence rates and interpolation properties. Experiments on synthetic and real data (including the STAR dataset) demonstrate improved performance over existing COT-based methods in both accuracy and efficiency.\n\n## Update after the rebuttal:\n\nThanks for your response. It is an interesting topic. However, I have consistently regarded this work as a brief extension of the paper[1] by additionally incorporating covariate elements into the analysis. This addition does not appear to increase the technical complexity. A more precise explanation from the authors regarding the core technical challenges of this work, as compared to prior literature, would be instrumental in convincing me of the paper's contributions. Besides, the experimental part is also weak.\n\nThanks for your response. I increase my score to 3.\n\n[1] Bridging multiple worlds: multi-marginal optimal transport for causal partial-identification problem, Zijun Gao, Shu Ge, Jian Qian", "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": "We address potential limitations. \n\n> The penalty tuning.\n\nWe provide a data-driven selection method for Q1, which works well in Fig 3, 4.\n\n> Efficiency loss relative to COT.\n\nNo direct estimator of COT has been established (see Q2). Although there is a gap of Vip and Vc, we maintain consistency and explicit convergence rate based on a finite sample, which is not known for COT. \n\n> Multi-valued or continuous treatments.\n\nVip can be extended to multi-valued treatment using multi-marginal OT. \nContinuous treatments are not applicable directly for classic OT framework.  \n\n> 1. Penalty:\n\nUnder-penalizing may get an estimator to be below the COT value. Over-penalizing leads to a slow convergence on the $\\eta$ in Thm 4.3, (thus minimizing empirical PI width is not ideal). Notably, $V_{ip}(\\eta)$ with arbitrary $\\eta$ already improves over Vu by using covariate, Vc has no known direct estimator.\nWe use the elbow method to select $\\eta$ as in [6], which is common in unsupervised setups, such as determining the number of principal components in PCA.\n\n> 2. Loss to COT:\n\nExp 3.8 shows the gap between $Vip(\\eta)$ and Vc for Gaussian model. In general, we do not have a bound. In Sec 6, we discuss a variant to address this issue.\nWe aim to utilize the covariate to improve over Vu. (Prop. 3.6, Sec 5). COT has no direct estimator, Vip possesses an explicit finite-sample convergence rate (Thm 4.3). \n\n> 3. High Dim:\n\nThe convergence of OT value is known to depend on the dimension, like a rate of $O(n^{-2/d})$ for squared Wasserstein [7], which is sharp [4]. Note that, our convergence rate in Thm 4.3 is aligned with this result. \nIf we have a reliable parametric distribution of Y given Z, the Vip estimator is compatible with the models. But we should note the risk of model misspecification as in Fig 3. (b) (e). \nIf no parametric model is available, a variable selection method for covariate would help. See Q10.\n\n> 4. Overfitting:\n\nFig 3 and Fig 4 show that the elbow method is stable as the sample size goes from 500 to 1500.\n\n> 5. Extension:\n\nFor multi-level, the Vip estimator can be directly extended using the multi-marginal OT [5]. \nFor continuous value, our causal bound is out of the scope of a standard OT framework.\n\n> 6. Sensitivity to h:\n\nThe cost function h in the causal estimand is typically chosen by the user, e.g. Sec 5.3. The Vc is a minimum of a linear functional of h, an error of order $O(a)$ will cause an error of order at most $O(a)$.\n\n> 7. Real-World:\n\nVip is the OT problem with objective h with encouraging the alignment of Z in the two groups by regularization. Regularization is popular in classic statistics and machine learning approaches, e.g. LASSO. \n\n> 8. Semi-Parametric\n\nSemi-para estimates identifiable causal quantities but we do partially identified ones.\nBounds based on variance inequalities are not consistent but ours is consistent.\nFan (2023) considers inference for PI with moment equalities. \nThey model the joint distribution of Y using copula, which is suited for univariate outcomes, but our approach can handle vectors. They use Bernstein copulas, but we have no restrictions on the coupling.\nThey require estimating conditional CDFs, we don’t.\n\n> 9. Alters:\n\nE.g. [3]. But our mirror method has pros: (i) the mirror relaxation $\\leq$ COT, and for causal bound, an underestimation beats overestimation since a lower bound still forms a valid causal bound. (ii) we guarantee an improvement over Vu. \n\n> 10. Var-Select:\n\n$\n\\sup_{\\eta} \\inf_{\\pi} \\int h(Y(0), Y(1)) + \\sum_{i} \\eta_i |Z^{(i)}(0) - Z^{(i)}(1)|  d\\pi - C \\sum_{i} |\\eta_i|.\n$\n\nWe use the L1 norm of the $\\eta$ vector to encourage sparsity. \n\n> 11. Data Perturb:\n\nThe robustness property of $V_{\\ip}$ against data perturbation inherits from the stability of the OT with respect to its marginals. When the objective function $h$ is Lipschitz, then a distributional shift of order $O(a)$ measured in 1-Wasserstein distance induces a bias of at most order $O(a^{2/15})$ on the OT map used in computing $Vip(\\eta)$, also the estimator [8].  \n\n> 12. Hidden Confound:\n\nIn this case, we can relax the OT. Suppose that the bias of observed conditionsal law is measured by a distance D and smaller than a constant, then a relaxed OT problem (or robust OT) can be used [1,2].\n\n\n[1] \"Optimal transport with relaxed marginal constraints.\" \n\n[2] \"On robust optimal transport: Computational complexity and barycenter computation.\" \n\n[3] \"Consistent optimal transport with empirical conditional measures.\"\n\n[4] \"Sharp convergence rates for empirical optimal transport with smooth costs.\"\n\n[5] \"Bridging multiple worlds: multi-marginal optimal transport for causal partial-identification problem.\" \n\n[6] “Determining the number of clusters/segments in hierarchical clustering/segmentation algorithms.” \n\n[7] “On the rate of convergence in Wasserstein distance of the empirical measure.\" \n\n[8] \"Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space.\"", "claims_and_evidence": "The paper claims:\n- A novel relaxation of COT leads to computationally efficient and statistically consistent PI estimation.\n- The proposed interpolated bounds (Vip(η)) tighten PI intervals compared to unconditional OT, and converge to exact COT as η → ∞.\n- The method achieves improved empirical performance in both synthetic and real-world settings.\n\nThese claims are well-supported by:\n- Rigorous theoretical development, including interpolation guarantees (Proposition 3.6) and finite-sample convergence bounds (Theorem 4.3).\n- Extensive synthetic experiments comparing their method to DualBounds, with results showing tighter intervals and more accurate estimates.\n- Real data application demonstrating narrower confidence intervals and better correlation estimation than baselines.", "methods_and_evaluation_criteria": "The method is mathematically elegant and well-motivated:\n- The use of mirror covariates enables re-formulating COT into a penalty-regularized OT problem.\n- The Vip(η) formulation interpolates smoothly between OT and COT bounds.\n- The estimator is simple to implement using standard OT solvers (e.g. Sinkhorn, LP).\n\nEvaluation is comprehensive:\n- Includes both synthetic setups (with known ground truth) and real-world datasets.\n- Comparison with DualBounds under various model types (linear, quadratic, scale) and nuisance estimators (ridge, KNN).", "theoretical_claims": "The paper provides strong theoretical contributions:\n- Interpolation result (Proposition 3.6) establishes that Vip(η) bridges Vu and Vc.\n- Theorem 4.3 shows convergence rates under quadratic costs using Brenier maps and convexity theory.\n- Extensions to α-mixing and non-i.i.d. settings (Theorem 4.7) are thoughtful and relevant.\n\nAssumptions (smoothness, compact support) are clearly stated and reasonable in causal inference contexts.", "experimental_designs_or_analyses": "Experiments are well-designed:\n- Synthetic data covers three structural models (linear, quadratic, scale), highlighting method robustness.\n- Real data (STAR dataset) shows practical impact by tightening Neyman variance bounds and PI on correlation.\n- Metrics are interpretable (e.g., PI length, L1 error to oracle, variance estimator, sample size equivalence).\n- Use of existing baselines (DualBounds) provides fair comparison.\n\nPlots and tables clearly support conclusions.", "relation_to_broader_scientific_literature": "The paper makes a novel contribution at the intersection of optimal transport and causal inference:\n- It builds on prior work on partial identification (e.g., Ji et al., 2023; Chemseddine et al., 2024) and causal OT.\n- It sidesteps the estimation pitfalls of COT by introducing a penalty-based relaxation grounded in OT theory.\n- The authors relate their method to copula bounds, GAN-based OT, and convex transport maps, integrating multiple threads of literature.", "essential_references_not_discussed": "No essential omissions were identified. The paper discusses both econometric and ML-focused OT methods, and thoroughly compares with relevant baselines (e.g., DualBounds). Related work on copulas, semi-parametric bounds, and Brenier-based methods are well-cited.", "other_strengths_and_weaknesses": "**Strengths:**\n- Elegant theoretical formulation with interpolation between known extremes.\n- Statistically grounded and computationally feasible.\n- Strong empirical performance, including on real data.\n- Avoids reliance on unstable nuisance estimation.\n\n**Weaknesses:**\n- The penalty parameter η still requires tuning.\n- No analysis of worst-case efficiency loss relative to true COT.\n- Extension to multi-valued or continuous treatments is not discussed.", "other_comments_or_suggestions": "NA", "questions_for_authors": "1. **Penalty Parameter Selection (η):**  \n   Your method critically depends on the penalty parameter η for interpolation between Vu and Vc. However, no practical strategy is provided for choosing η. Can you suggest a data-driven or theoretically justified selection method (e.g., minimizing empirical PI width with statistical guarantees)? What are the consequences of over- or under-penalizing?\n\n2. **Efficiency Loss Compared to COT:**  \n   While you prove that Vip(η) interpolates between Vu and Vc, you do not quantify how close Vip(η) is to Vc in practice. Can you provide theoretical bounds (e.g., in terms of η, dimension, sample size) or empirical results that quantify the potential efficiency loss?\n\n3. **Scalability to High Dimensions:**  \n   OT-based methods are known to struggle in high-dimensional spaces. Have you evaluated how your method behaves as the dimension of covariates (Z) increases, particularly in terms of sample complexity and computation?\n\n4. **Tuning η vs. Overfitting in Finite Samples:**  \n   Since η indirectly governs the extent of covariate adjustment, can tuning it on the same sample introduce overfitting (e.g., inadvertently fitting to noise)? How robust is your plug-in estimator to this issue?\n\n5. **Extension to Multi-valued or Continuous Treatments:**  \n   Your framework is based on binary treatment. Can the mirror-relaxation idea extend to multi-valued or continuous treatments? If not directly, what conceptual or computational barriers would arise?\n\n6. **Sensitivity to Misspecified Cost Function (h):**  \n   Your theoretical results assume a known and fixed cost function. In practice, h may itself be misspecified or learned from data. How sensitive is your approach to the choice or error in h?\n\n7. **Real-World Deployment and Interpretability:**  \n   In applied domains (e.g., medicine, economics), interpretability and ease of use are key. Can you comment on how practitioners should interpret the Vip(η) bounds, and whether the required penalization intuition is accessible to non-theorists?\n\n8. **Comparison to Semi-Parametric Bounds (e.g., Fan et al., 2023):**  \n   How does your method compare (in tightness or assumptions) to other approaches that derive bounds using semi-parametric methods or variance inequalities? Can the two be combined?\n\n9. **Alternative Relaxations Besides Mirror Covariates:**  \n   Mirror covariates are introduced as a practical workaround to COT. Did you explore or benchmark other forms of COT relaxations (e.g., entropic regularization, conditional adversarial OT)? Why is your choice preferable?\n\n10. **Potential for Variable Selection:**  \n   In the discussion, you mention future directions for covariate selection. Could your current framework be extended to perform automatic covariate selection (e.g., via L1 regularization or adaptive penalization)?\n\n11. **Stability under Data Perturbations:**  \n   Have you examined how stable the Vip(η) estimator is to data perturbations (e.g., small changes in treatment assignment or outcome)? Robustness to noise is important for PI estimation in observational data.\n\n12. **Robustness to Hidden Confounding:**  \n   Your results assume unconfoundedness (or randomization). In practice, hidden confounding is inevitable. Can the method be adapted or extended to provide valid bounds under partial confounding assumptions (e.g., Rosenbaum sensitivity models)?", "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": "554cQZL0VT", "reviewer": "Reviewer_hw3E", "summary": "This paper studies partial identification intervals for the Rubin causal model, which is an important problem in the causal inference literature. The problem arises from the fact that we can never observe the counterfactual. Indeed, while the treatment effect (obtained when choosing h(Y(1), Y(0)) =  Y(1) - Y(0) ) is still identifiable, the expected term E h(Y(1), Y(0)) is generally not identifiable for general choices of h.\n\nNevertheless, we can find lower bounds for this term. Two natural lower bounds have been discussed in the literature. 1) the optimal lower bound that requires to solve a conditional OT problem, and 2) a strong relaxation where we ignore the dependency on the covariates Z, resulting in an OT problem. This paper proposes a very elegant approach for improving upon 2) while still only relying on the solution of a normal OT problem and not of a conditional OT problem. \n\nThe problem is relevant and the paper is well written and very well executed. I recommend the acceptance of the paper. To support an award such as a spotlight or oral presentation, I would however expect a stronger motivation for the broader impact of this paper.", "questions": "", "limitations": "", "rating": 5, "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": "We highly appreciate the positive comments of the reviewer and are happy to answer any questions if needed.", "claims_and_evidence": "Yes, all claims are well supported.", "methods_and_evaluation_criteria": "Yes, the paper evaluates their method on both synthetic data as well as real world data.", "theoretical_claims": "The paper provides an informative finite sample analysis in Section 4. I have not checked the proofs in depth, but the results match expected rates and the proofs are well written and organised.", "experimental_designs_or_analyses": "The experimental design is well executed.", "relation_to_broader_scientific_literature": "I have published in the causal inference domain.", "essential_references_not_discussed": "I think the authors do a good job in discussing the literature.", "other_strengths_and_weaknesses": "-", "other_comments_or_suggestions": "I think this is a nice paper and the authors found a good balance between theory and experimental results. I don't know of any suggestion that would strictly improve the current document.", "questions_for_authors": "-", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rSPTahSCMh", "reviewer": "Reviewer_1LE1", "summary": "This paper leverages the conditional optimal transport (COT) to derive or tighten the partial identification (PI) bounds for some causal estimands. Since the COT is not easy to compute in practice, the authors propose a relaxation based on mirror covariates, leading to a optimization problem whose objective function interpolates between the unconditional and conditional optimal transport problems for the PI bounds. The convergence properties of the plug-in estimator are also studied. Finally, the authors also compare their estimator with the existing one that also utilizes the optimal transport technique.", "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": "We greatly appreciate the reviewer's summary and questions. In the following, we address each question individually.\n\n > In Assp 3.3, why is it important to assume that $\\nabla_y h(y, \\cdot)$ is injective for all $y \\in \\mathcal{Y}$ ?\n\nThis is because we define the $V_{ip}(\\eta)$ by the expectation of $h$ under a coupling $\\pi$ that is computed through our mirror-relaxed OT problem. To make the $\\pi$ well-defined, a standard assumption of to let the objective to have an injective gradient everywhere, e.g., a quadratic function.\n\n> In Prop 4.1, do we need the ratio $\\frac{n}{m}$ to converge to a positive constant? \n\nFor Prop 4.1, as long as $m,n \\rightarrow \\infty$, the estimator is consistent. The convergence rate may depend on the ratio $\\frac{n}{m}$, but we do not invetigate in this direction. Alternatively, in Thm 4.1, we let the convergence rate to depend on $\\min (m,n)$.\n\n>  For Table 3, can the authors somehow obtain the true value of $\\rho$ in this example? \n\nUnfortunately, in table 3, we are analyzing a real dataset and the true value of $\\rho$ is unidentifiable. We thank the reviewer for their suggestion on making a simulation such that $\\rho > 0$ and the lower bound is meaningful. As a response to this, we can actually twist the synthetic experiments in Fig 3 to set the estimand to be $\\rho > 0$, since estimating $\\rho$ is essentially estimating for quadratic $h$. In the real dataset, the sample appears to be noisy and the correlation between covariate and outcome is not as significant as that in the synthetic dataset (this is why the lower bound is negative, indicating that we cannot reject the possibility of negative $\\rho$); In Fig 3, we have larger sample, the estimation error is relatively small, and we set the correlation to be significantly positive, therefore, in that case, the lower bound will be larger than zero. \n\n> One important point that the author may consider addressing is to design a valid statistical inference procedure on the partial identification bounds.\n\nThe OT estimation may struggle when the dimension of covaraite plus outcome is larger than three, thus a useful inference procedure would be challanging to construct. However, when the dimension is smaller than three, there are some CLT limiting results that can be used to build an inference procedure. We will leave this for future research.\n\n> Can the authors generalize the results to convex cost functions?\n\nThis is also in our scope of the future research. In the present version, the results depends on the convergence rate of the Brenier's map, which are studied under quadratic loss function. An extension to convex cost functions would be very nontrivial, and we are investigating  alternative ways to reduce the effect of large $\\eta$ on the magnitude of the estimator. \n\n> For Figure 3 (c,f), I wonder why the $L_1$ error of $\\hat V_{ip}(\\eta)$ will eventually go up as $\\eta$ increases.\n\nThis is because even though $V_{ip}(\\eta) \\leq V_{c}$ in the population level, the estimator $\\hat V_{ip}(\\eta)$ may overestimate $V_{ip}(\\eta)$ to be larger than $V_{c}$ based on a finite sample. Therefore, for the $\\eta$ such that $\\hat V_{ip}(\\eta)$ is larger than $V_{c}$, the $L_1$ error increases since $\\hat V_{ip}(\\eta)$ is increasing with respect to $\\eta$.\n\n> On Line 727: did you define the operation $\\circ$ somewhere in the paper?\n\nSorry for the confusion, $\\circ$ denotes the composition of mapping, which will be added in the revised version.", "claims_and_evidence": "All the claims and theory in this paper are clear and supported by rigorous proofs or empirical experiments.", "methods_and_evaluation_criteria": "The proposed methods for using the conditional optimal transport on the partial identification bounds make sense to me. In addition, the simulation studies and experiments are illuminating. \n\nI only have concerns on a minor point: for Assumption 3.3, why is it important to assume that $\\nabla_y h(y,\\cdot)$ is injective for all $y\\in \\mathcal{Y}$? Are there any example causal estimands where this Assumption does not hold?", "theoretical_claims": "Yes, I basically checked all the proofs for the theoretical claims in this paper. All of them are rigorous and clear. \n\nThere is only a minor question on Proposition 4.1 and its proof: as $n,m\\to\\infty$, do we need the ratio $\\frac{n}{m}$ to converge to a positive constant? If $\\frac{n}{m}\\to 0$ or $\\frac{n}{m}\\to \\infty$ as $n,m\\to\\infty$, it might not affect the consistency but I suspect that it will affect the convergence rate.", "experimental_designs_or_analyses": "Yes, I checked all the experimental designs and analyses carefully. The results are solid. However, I have a question or comment on Table 3 that the authors may consider: Can the authors somehow obtain the true value of $\\rho$ in this example? If not, I would suggest running some simulation where the authors know how to compute the true $\\rho$. Then, suppose that $\\rho \\in (0,1)$ and the proposed estimator is applied on the data to compute the lower bounds for $\\rho$. As $\\eta$ grows, we would expect that the lower bounds are bigger than 0 as well. In this case, the partial identification lower bounds are meaningful.", "relation_to_broader_scientific_literature": "This paper improves upon a paper by Ji et al. (2023) by utilizing the techniques related to conditional optimal transport.\n\n\nJi, W., Lei, L., and Spector, A. Model-agnostic covariate-assisted inference on partially identified causal effects. arXiv preprint arXiv:2310.08115, 2023.", "essential_references_not_discussed": "Not that I am aware of. However, for the contents between Lines 639 and 646, it seems that the discussions are duplicated.", "other_strengths_and_weaknesses": "One important point that the author may consider addressing is to design a valid statistical inference procedure on the partial identification bounds. This can help further advance the impacts of this paper.", "other_comments_or_suggestions": "1. For the description of Figure 1, it would be clearer to state that $P(Z=1)=P(Z=0)=\\frac{1}{2}$.\n\n2. Second column of Line 217: Can the authors generalize the results to convex cost functions?\n\n3. Second column of Line 231: a typo \"certein\" should be \"certain\".\n\n4. For Figures 3 and 4, since the authors did Monte Carlo experiments, it would be better to plot the standard error of each curve in the plots as shared regions as well.\n\n5. For Figure 3 (c,f), I wonder why the $L_1$ error of $\\hat{V}_{ip}(\\eta)$ will eventually go up as $\\eta$ increases.\n\n6. On Line 727: did you define the operation $\\circ$ somewhere in the paper?", "questions_for_authors": "See my comments above.", "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": "8sFXsN5RIf", "reviewer": "Reviewer_CWjX", "summary": "The paper investigates the problem of tightening partial identification (PI) bounds in causal inference by incorporating covariate information through a conditional optimal transport (COT) framework. The authors propose a novel relaxation that reduces COT to standard optimal transport (OT), improving computational feasibility while maintaining the benefits of covariate adjustment. The approach enables a data-driven estimation of the PI set using existing OT solvers. Theoretical analysis includes convergence guarantees and an exploration of asymptotic properties. Empirical validation is performed using synthetic and real-world data, demonstrating superior performance over existing methods in terms of bound tightening and estimation accuracy.", "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": "We highly appreciate the positive comments of the reviewer and are happy to answer any question if needed.", "claims_and_evidence": "All the claims made in the submission are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed method is able to estimate partial identification bound with covariate adjustment.", "theoretical_claims": "The theoretical claims seem correct.", "experimental_designs_or_analyses": "NA", "relation_to_broader_scientific_literature": "NA", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "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": "YlwbRIi8aD", "reviewer": "Reviewer_UMQY", "summary": "The manuscript introduces a novel method for bounding treatment effects using covariate information, reframing the problem as an optimal transport task. Specifically, the authors propose adding a penalty term to the standard optimization objective that encourages covariates to have similar distributions in both treatment arms. They show that varying the weight of this penalty effectively interpolates between unconditional and conditional approaches. They study the statistical and computational aspects of their algorithm, which can be solved using linear programming, and present experimental benchmarks.\n\n## update after rebuttal\n\nI thank the authors for resolving a confusion on my part about their target estimand. I still have some reservations about the use of Neyman confidence intervals as a benchmark. In any event, I will maintain my score of 4.", "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": "We highly appreciate the reviewers' summar and comments.\n\n> Several other PI methods have been published in recent years that do not necessarily rely on optimal transport theory, ...\n\nThe other PI methods deal with the case where there is unobserved confounder/leaky IV, such that the observed marginal distributions of outcomes do not equal the true ones, thus needing PI. In contrast, we are interested in estimating a functional of the joint coupling of the outcomes, which is also partially identifiable and thus we seek PI through OT (a coupling theory). As a result, our PI and the other PI methods that the reviewer lists deal with different cases of partial identifiability, thus are not very relevant and cannot be compared in the our setting.\n\n> It appears from the code that this is fixed at 1. Unless I'm missing something?\n\nIn the code, the input $d_Z$ for function *vip_estimate* can be set to any integer, for computing the true value of Vc or Vip, in the simulation result, we make the examples using $d_Z = 1$.\n\n> The Neyman CI approach is not really an apples-to-apples comparison.\n\nThe Neyman CI approach is basically about the variance estimation of the outcomes from the sample, which is a standard estimand in causal inference. We use this as an example to show that the quadratic objective h is a practical one. Specifically, in [1], the tightest Neyman variance estimator without covariate information is the same as the optimal objective value of the associated OT problem, and our approach incorporates the covariate information in this formulation on top of their work.\n\n> Some aspects of this manuscript are clearly quite thoroughly researched ... However, this is not the case when it comes to partial identification (even including Appx A1) …\n\nOur PI and the other PI methods that the reviewer lists deal with different cases of partial identifiability. \n\nIn the reviewer’s list, due to hidden confounder / leaky instrumental variables, the observed marginal data distributions are different from the target marginal distribution.\n(i) hidden confounder:\nhttps://ojs.aaai.org/index.php/AAAI/article/view/17437\nhttps://proceedings.mlr.press/v162/zhang22ab.html\nhttps://proceedings.mlr.press/v213/padh23a.html\n(ii) leaky instrumental variable:\nhttps://proceedings.mlr.press/v235/jiang24b.html\nhttps://openreview.net/forum?id=OaJLMx2nwS\n\nIn contrast, our PI is due to the unobserved joint coupling distribution of the outcomes in different treatment groups (i.e. *cross-world dependence*). Therefore, we utilize the framework of optimal transport to analyze the possible couplings of the observed cross-world outcomes. As a result, our PI approach and the other PI approaches the reviewer lists target different types of partial identifiability. In future work, we will investigate the combination of our PI and the other PIs, i.e., dealing with scenarios involving two sources of partial identifiabilities.\n\nIn the literature related to the PI relevant to our approach (PI on joint couplings), we have discussed several papers in the Appendix A. In the revised version, we will discuss the difference between our PI and the other PI that is more relevant to the reviewer’s list (e.g. [2]).\n\n\n> I'm intrigued by this target estimand ... between an ATE and a CATE …\n\nOur target estimand is different from ATE and CATE in that, in our randomized experiment setting, both ATE and CATE are identifiable ($E[Y(1) - Y(0)]$), while our target estimand $E[(Y(0) -  Y(1))^2]$ is partially identifiable because it not only relies on the marginal distribution of $Y(0), Y(1)$, but more importantly depends on their unobserved joint distribution. Therefore, the nature of our target estimand and ATE (CATE) are different, thus our PI targets on the partial identifiability of coupling of outcomes, while other PI, particularly ATE, CATE, may focus on the the partial identifiability resulted by hidden confounders.\n\n> (a) whether the user is supposed to know the true parents/children of Z and (b) which of the following graphs are fair game ...\n\nWe set $X$ to be treatment, $Y$ to be outcome, and $Z$ to be covariate.\n\nIn view of a causal diagram, our randomized experiment setting is: (3) $X \\rightarrow Y \\leftarrow Z$. Our approach can be extended to the case (1) $Z \\rightarrow X \\rightarrow Y \\leftarrow Z$, i.e., allowing the covariates to influence the treatment assignment, by incorporating the propensity scores. We leave this direction for future research. \n\nAccording to the DAG, we have: Z has no parents, and Y is the children of Z. Actually, for the vector variable Z, we allow an arbitrary structure inside the vector Z. In the paper,, we assume there are no latent confounders (Assp. 3.1).\n\n\n[1] Aronow, Peter M., Donald P. Green, and Donald KK Lee. \"Sharp bounds on the variance in randomized experiments.\" (2014): 850-871.\n\n[2] Imbens, Guido W., and Donald B. Rubin. Causal inference in statistics, social, and biomedical sciences. Cambridge university press, 2015.", "claims_and_evidence": "The primary claims of the paper are: \n\n-under the stated assumptions, the interpolating OT lower bound exists and is unique;\n\n-the interpolating OT lower bound is a monotonically increasing function of the penalty parameter $\\eta$ and interpolates between unconditional ($\\eta=0$) and conditional ($\\eta= \\infty$) OT lower bounds;\n\n-the convergence rate of the estimator for this parameter is upper bounded by a well defined function of the sample size;\n\n-the method performs well in experiments with simulated and real-data.\n\nThe first three claims, which are all theoretical in nature, are well articulated and generally convincing (although I have some questions about structural assumptions, see below). I find the experiments less compelling, as the only methods included in the benchmark are dualbounds (a relatively recent, seemingly unpublished method) and Neyman confidence intervals (which are not conditional and not intended to be partial identification intervals). Several other PI methods have been published in recent years that do not necessarily rely on optimal transport theory, and I would be curious to see how they stack up (more on this below).", "methods_and_evaluation_criteria": "The real and simulated datasets make sense, although I have a couple of comments/questions:\n\n-The use of $d_Z$ in the description of the DGP (§5.2) suggests that multiple covariates are in play, but it appears from the code that this is fixed at 1. Unless I'm missing something?\n\n-The Neyman CI approach is not really an apples-to-apples comparison, even to the unconditional PI. This conflates uncertainty from finite samples with uncertainty from the structure of the DGP itself. I would consider dropping this after other PI methods are incorporated.", "theoretical_claims": "The mathematical reasoning appears clear and sound, although I confess I did not closely examine the proofs.", "experimental_designs_or_analyses": "The experimental design seems sound, but see my comments above regarding relevant (and irrelevant) benchmarks.", "relation_to_broader_scientific_literature": "This topic is of general interest to the causal inference community, and the method could have applications in econometrics and/or healthcare. Presentation would be aided by a running example of $W, Z, Y$ variables that could help ground the discussion.", "essential_references_not_discussed": "Some aspects of this manuscript are clearly quite thoroughly researched, featuring strong engagement with the contemporary literature. However, this is not the case when it comes to partial identification (even including Appx A1). Several methods have been proposed in recent years for bounding treatment effects under minimal assumptions, and I was surprised to see none featured in the benchmark experiments. For example:\n\n-https://ojs.aaai.org/index.php/AAAI/article/view/17437\n\n-https://proceedings.mlr.press/v162/zhang22ab.html\n\n-https://proceedings.mlr.press/v213/padh23a.html\n\n-https://proceedings.mlr.press/v235/jiang24b.html\n\n-https://openreview.net/forum?id=OaJLMx2nwS\n\nThese may not all be equally relevant, but in any event the text should explain why and include at least some of these in the experiments. (With respect to IV models, it's not clear to me whether $Z$ may be considered a \"leaky\" instrument? More on this below.)", "other_strengths_and_weaknesses": "The paper is very clear and well-written. The mathematical results are rigorous and sound. I'm intrigued by this target estimand, which appears novel as far as I can tell. It's somewhere in between an ATE and a CATE, with $\\eta$ controlling the degree of interpolation. Here's one way of stating this: $\\tau(z, \\eta) := \\mathbb E[Y(1) - Y(0) \\mid z + \\mathcal N(0, \\eta^{-2})]$. I know the present method doesn't commit us to any particular parametric assumptions the way this does, but am I correct in saying this formulation captures the spirit of the proposal? That is, $\\tau(z, \\eta)$ approaches the ATE as $\\eta \\rightarrow 0$, the CATE as $\\eta \\rightarrow \\infty$, and something in between for all intermediate values. Of course, the goal here is *partial* identification rather than point identification, but I still find it helpful to have some idea just what the target estimand is that we're bounding.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "My main question pertains to what structural assumptions (if any) are in play regarding these covariates $Z$?\nI will use graphical notation to spell out my questions, although I appreciate from the text that the authors are approaching this from the potential outcomes tradition. \nBasically, I would like to know (a) whether the user is supposed to know the true parents/children of $Z$; and (b) which of the following graphs are fair game for this method: \n\n(1) $Z \\rightarrow X \\rightarrow Y \\leftarrow Z$\n\n(2) $X \\rightarrow Z \\rightarrow Y \\leftarrow X$\n\n(3) $X \\rightarrow Y \\leftarrow Z$\n\n(4) Anything else.\n\nAs an additional question, I would like to know precisely what assumptions are made regarding latent confounders? Apologies if this is stated somewhere in the text, but I could not find it.", "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": {"ethical_review_concerns": "N/A"}}]

Author rebuttal:

Meta-review: The authors introduce a novel framework for causal inference with inherent partial idenfitiability of the joint distribution of the outcomes that incorporates covariate important (conditional optimal transport) and is computationally feasible by relaxation. Their relaxed COT offers a natural way to find a tighter lower bound for estimands that depend on the joint distribution of the outcomes. The reviewers appreciate the presentation of the paper, novelty of the method, the rigorous theoretical results that strengthen the motivation of the method, and the experimental evaluation. 

Some minor points can still be improved in the final version of the paper however: 
Please include proper and readable y-axis labels and (properly rendered) labels in the plots of the figures that match minimum publication standards. Also it would help the impact of the paper (for less familiar readers) if you could explain a bit more in which downstream tasks / applications (with examples or references) estimands beyond the ATE/CATE (which most people know) - for which in your case you don't need to run your method since they're identifiable - in the sense of the general $h$ would be crucial. 
Figure 1 could also be clearer - where is the reader supposed to see the numbers in the caption? it seems like an illustration but then you make it seem like an experiment with actual values - even though I get the intuition that the authors want to convey with the example, and the idea of COT but I couldn't see it visualized in this figure (e.g. the point would be to show differing lower bounds but they are not visualized). 
Finally, perhaps the authors can more directly/visibly demonstrate how their bounds improve upon marginal OT bounds (like in Gao '24 or Torous '24 - unless they also give the exact some values as V_ip(0) in terms of how they solve the marginal OT optimization problem?) - in particular, having them more directly noted in the paper/plots/tables (instead of expecting all readers that skim the paper to see that setting eta= 0 is equivalent to V_u) so that the gains and impact are more tangible.
Decision: Accept (poster)