Lookahead Counterfactual Fairness
Authors: Zhiqun Zuo
Tian Xie
Xuwei Tan
Xueru Zhang
Mohammad Mahdi Khalili
Venue: Accepted by TMLR ()

Abstract: As machine learning (ML) algorithms are used in applications that involve humans, concerns have arisen that these algorithms may be biased against certain social groups. $\textit{Counterfactual fairness}$ (CF) is a fairness notion proposed in Kusner et al. (2017) that measures the unfairness of ML predictions; it requires that the prediction perceived by an individual in the real world has the same marginal distribution as it would be in a counterfactual world, in which the individual belongs to a different group. Although CF ensures fair ML predictions, it fails to consider the downstream effects of ML predictions on individuals. Since humans are strategic and often adapt their behaviors in response to the ML system, predictions that satisfy CF may not lead to a fair future outcome for the individuals. In this paper, we introduce $\textit{lookahead counterfactual fairness}$ (LCF),  a fairness notion accounting for the downstream effects of ML models which requires the individual $\textit{future status}$ to be counterfactually fair. We theoretically identify conditions under which LCF can be satisfied and propose an algorithm based on the theorems. We also extend the concept to path-dependent fairness. Experiments on both synthetic and real data validate the proposed method.

Published in Transactions on Machine Learning Research (12/2024)
Lookahead Counterfactual Fairness
Zhiqun Zuo
zuo. mao.1670sse edu
Deportment of Computer Science and Engineering The Ohio State University
Tian Xie
nie. 1ie.1379@osu. edu
Department Computer Science and Engineering The Ohio State University
Xuwei Tan
tan, 1 edu
Department of Computer Science and Engineering The Ohio State University
Xueru Zhang
zhang. 1 12807 edu
Department of Computer Science and Engineering The Ohio State University
Mohammad Mahdi Khalili
khalili, 1 17@osu. edu
Department of Computer Science and Engineering The Ohio State University
Reviewed OpenReview: https: //penreveee net/ forum? id- (tddkknggpkTt88rreefeeeee htt.://gggiiiii/ooooo//////////a 200 2o 20Zuo 5D (2apooo qun Zuo1)
Abstract
As machine learning (ML) algorithms used in applications that involve humans, concerns have arisen that these algorithms may be biased against certain social groups. Counterfac- tual fairness (CF) is fairness notion proposed in Kusner et al. (2017) that measures the unfairness of ML predictions; it requires that the prediction perceived by an individual in the real world has the same marginal distribution as would be a counterfactual world, in which the individual belongs a different group. Although CF ensures fair ML predictions, it fails to consider the downstreem effects of ML predictions are strategic and often adapt their behaviors in response to the ML system, predictions that satisfy CF may not lead to fair future outcome for the individuals. In this paper, we introduce lookahead counterfactual fairness (LCF), a fairness notion accounting for the downstream effects ML models which requires the individual future status to be counter- factually fair. We theoretically identify conditions under which LCF can be satisfied and propose an algorithm based on the thcorems. We also extend the concept to path-dependent fairness. Experiments both synthetic and real data validate the proposed method"
individuals. Since humans
1 Introduction
The integration of machine learning (ML) into high-stakes domains (e.g., lending, hiring, college admissions, healtheare) has the potential to enhance traditional human-driven processes. However, it may introduce the risk of perpetuating biases and unfair treatment of protected groups. For instance, the violence risk
assessment tool SAVRY has been shown to discriminate against males and foreigners (Tolan el. al. 2019);
Amazon's previous hiring system exhibited gender bias (Dastin 2018 the accuracy of a computer- clinical diagnostic system varies significantly across paticnts from different racial groups (Dancshjou et al.
The code for this paper is available in https://gstthbb.com wttp:/ccoooooo///eeee..

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2021
Numerous fairness notions have been proposed in the literature to address unfairness issues, including wnwwareness fairness that prevents the explicit use of demographic attributes in the decision-making process, parity-based fairness that equalizes certain statistics (e.g.. accuracy, true/false positive rate) across different groups (Haddt et al.) 2016b; Khalili et al. 2023 2021b a
Abroshan et al. 2024h, preference-basee fairness
that ensures a group of individuals, as a whole, regard the results or consequences they receive from the MT.
system more favorably than those received by another group (Zafar et al. 2017 Do et al. 2022). Unlike
these notions that overlook the underlying causal structures among different variables Kusner et al. (2017)
introduced the concept of counterfactual fairness (CF), which requires that an individual should receive
consistent treatment distribution in a counterfactual world where their sensitive attributes differs. Since
then many approaches have been developed to train ML models that satisfy CF (Chiappa 2019 Zuo et al.
2022
Wu et al. 2019 Xu et. al. 2019 Ma et al. 2023 Zuo el al. 2023 Abroshan et al.) 2022).
However, CF is primarily studied in static settings without considering the downstream impacts ML. deci- sions may have on individuals. Bocause humans in practice often adapt their behaviors in response to the ML system, their future status may be significantly impacted by ML. decisions (Miller et al.] 2020 Shavil
et al.
2020 Hardt et al.) 2016a). For example, individuals receiving approvals in loan applications may have more resources and be better equipped to improve their future creditworthiness (Zhang et al.. 2020).
Content recommended in digital platforms can steer consumer behavior and reshape their preferences (Dean
& Morgenstern 2022 Carroll et al. 2022). As a result, a model that satisfies CF in a static setting without accounting for such downstream effects may lead to unexpected adverse outcomes.
Although the downstream impacts of fair ML have also been studied in prior works (Henzinger al. 2023a
Xie & Zhang 2024a Ge et al. 2021 Henzinger et al.) 2023b Liu et al. 2018 Zhang et al.) 2020 Xie et al.
2024), the impact of counterfactually fair decisions remain relatively unexplored. The most related work
to this study is (Hu & Zhang, 2022). which considers sequential interactions between individuals and an ML system over time and their goal is to ensure ML decisions satisfy path-specific counterfactual fairness constraint throughout the sequential interactions. However, Hu & Zhang (2022) still focuses on the fairnoss of ML decisions but not the fairness of the individual's 's actual stalus. Indeed, it has been well-evidenced that ML decisions satisfying certain fairness constraints during model deployment may reshape the population and unintentionally exacerbate the group disparity (Liu et al. 2018 Zhang et al. 2019 2020). A prime example is Lāu et al. (018)), which studied the lending problem and showed that the lending decisions satisfying statistical parity or equal opportunity fairness (Hardt et al. 201 may actually cause harm to disadvantaged groups by lowering their future credit scores, resulting in amplified group disparity. [Tang] ct al. 2023) considered sequential interactions between ML decisions and individuals, where they studied the impact of counterfactual fair predictions on statistical fairness but their goal is still to ensure parity-based fairness at the group level.
In this work, we focus counterfactual fairness evaluated over individual future status (label), which ac- counts for the downstream effects of ML decisions individuals. We aim to examine under what conditions and by what algorithms the disparity between individual future status in factual and counterfactual worlds can be mitigated after deploying ML decisions. To this end, we first introduce a new fairness notion called "lookahead counterfactual fairness (LCF).' Unlike the original counterfactual fairness proposed by Kusner let al. (0117) that requires the ML predictions received by individuals to be the same as those in the coun- terfactual world, LCF takes one step further by enforcing the individual future status (after responding to ML predictions) to be the same.
Given the definition of LCF, we then develop algorithms that learn ML models under LCF, To model the effects of ML decisions on individuals, we focus on scenarios where individuals subject to certain ML decisions adapt their behaviors strategically by increasing their chances of receiving favorable decisions; this can be mathematically formulated as modifying their features toward the direction of the gradient of the decision function [Rosenfeld et al., 2020 Xie & Zhang 2024b). We first theoretically identify conditions under which an ML model can satisfy LCF, and then develop an algorithm for training ML models under LCF, We also extend the algorithm and theorems to path-dependent LCF, which only considers unfairness incurred by the causal effect from the sensitive attribute to the outcome along certain paths.
Our contributions can be summarized as follows:
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Published in Transactions on Machine Learning Research (12/2024)
We propose lookahead couteeractual fairness (LCF), novel fairness notion that evaluates coun- terfactual fairness over individual future status (i.e., actual labels after responding to MI. systems). Unlike the original CF notion that focuses on current ML predictions, LCF accounts for the subse- quent impacts of ML decisions and aims to ensure fairness over individual actual future status. We also extend the definition to path-dependent L.F.
For scenarios where individuals respond to ML models by changing features toward the direction of the gradient of decision functions, theoretically identify conditions under which an ML model can satisfy LCF. We further develop an algorithm for training ML models under LCF.
We conduct extensive experiments on both synthetic and real data to validate the proposed algo- rithm. Results show that compared conventional counterfactual fair predictors, our method can improve disparity with respect to the individual actual future status.
2
Related Work
Causal fairness has been explored in many aspects in recent years' research. Kilbertus et al. (017)) point out that no observational criterion can distinguish scenarios determined by different causal mechanisms but have the same observational distribution. They propose the definition of unresolved discrimination and proxy discrimination based on the intuition that some of the paths from the sensitive attribute to the prediction can be acceptable. Nabi & Shpitser (2018) argue that a fair causal inference on the outcome can be obtained by solving a constrained optimization problem. These notions are based on defining constraints on the interventional distributions.
Counterfactual fairness (Kusner et al. 2017) requires the prediction on the target variable to have the same distribution in the factual world and counterfactual world. Many extensions of traditional statistical fairness notions such Fair Average Causal Effect (FACE) (can be regarded
comteefactual demographic parity) (Khademi et al. 2019), Fair on Average Causal Effect on the Treated (FACT) (can be regarded as
counterfactual equalized odds) (Khademi et al. 2019), and CAPT fairness (counterfactual individual fairness) [Ehyaei et al., 2024 have been proposed. Path-specific counterfactual fairness (Chiappa 2019) considered the path-specific causal effect. However, the notions are focused on fairness in static settings and do not consider the future effect. Most recent work about counterfactual fairness about achieving counterfactual fairness in different applications, such as graph data (Wang al.) 2024a (b), medical LLMs (Poulain et al. 2024), or software debuging (Xiao et al., 2024). Some literature try to use counterfactual fairness for explanations [Goethals et al. 2024). Connecting counterfactual fairness with group fairness notions (Anthis & Veitch 2024) or exploring counterfactual fairness with partial knowledge about the causal model (Shao
et al.] 2024 Pinto et al.) 2024 Duong et al. 2024 Zhou et al. 2024) are also receiving much attention.
Machado et al. (024) propose an idea of interpretable counteriactual fairness by deriving counterfactuals with optimal transports (De Lara et al., 2024). While extending the definition of counterfactual fairness to include the downstream effects has boen less focused on.
Several studies in the literature consider the downstream effect fairness of ML predictions. There are two kinds of objectives in the study of downstream effects: ensuring fair predictions in the future fair true status in the future. The two most related works to our paper are Hu & Zhang (2022) and Tang et al. 22023) Hu & Zhang (2022) consider the problem of ensuring ML predictions satisly path-specific counterfactur fairness over time after interactions between individuals and an ML. system. Tang et al. (2023) study the impact on the future true status of the ML predictions. Even though they considered the impact of a counterfactually fair predictor, their goal is to ensure parity-based fairness. Therefore, the current works lack the consideration of ensuring counterfactual fairness the true label after the individual responds to the current ML prediction. Our paper is aimed at solving this problem.
3 Problem Formulation
Consider supervised learning problem with a training dataset consisting of triples (A, Y). where A A is sensitive attribute distinguishing individuals from multiple groups (egg. race, gender),
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x= X2, E & is d-dimensional feature vector, and Y y S R is the target variable in- dicating individual's underlying status (e.g., Y in lending identifies an applicant's 's ability to repay the loan, Y in healthcare may represent patients' insulin spike level). The goal is to learn a predictor from training data that can predict Y given inputs A and X. Let y denote the output of the predictor.
We assume (A, Y, associated with a structural causal model (SCM) (Pearl al. 2000) M UV F), where V YY) represents observable variables, U includes unobservable (exogenous) variables that are not caused by any variable in V, and F = f. is set of d + functions called structural equations that determines how each observable variable is constructed. More precisely, we have the following structural equations,
X; fippa, Unú)), vi (1,-..,. ,d),
A = falpas, Upar):
= fy (pay, Upar),
(1)
where pa, S V, Aad S V and pay S V are observable variables that are the parents of X,, A, and Y, respectively. Upa S U are unobservable variables that are the parents of X,. Similarly, we denote unobservable variables Upas CU and Upay E U as the parents of and Y, respectively.
3.1 Background: counterfactuals
If the probability density functions of unobserved variables are known, we can leverage the structural equa- tions in SCM to find the marginal distribution of any observable variable V; E V and even study how intervening certain observable variables impacts other variables. Specifically, the intervention on vari- able V/ is equivalent Given new structural equation Vi = e and other unchanged structural equations, we can find out how the distribution of other observable variables changes we change value v.
replacing structural equation V, = -f(pp;; Up.)) with equation V; v for some v.
In addition to understanding the impact of an intervention, SCM can further facilitate counterfactual inference, which aims to answer the question "what would be the value of Y if Z had laken value = in the presence of evidence O = o (both Y and 7 are two observable variables)? The answer to this question is denoted by (U) with U following conditional distribution of Pr(U = u|O = o}. Given U = u and structural equations F, the counterfactual value of Y can be computed by replacing the structural equation of 7 with % z and replacing U with in the rest of the structural equations. Such counterfactual is typically denoted by Yz (w). Given evidence O ==a o, the distribution of counterfactual value (z)=(U) can be calculated as follows
(U) =
= =]== Σ Pr(Y = PT(U = u/0 = o}.
(2)
Example 3.1 (Law school success). Consider two groups of college students distinguished by gender A E {0, 1} whose first-year average (FYA) in college denoted by Y. The FYA of each student causally related to (observable) grade-point average (GPA) before entering college XG, entrance exam score (LSAT)
X1, and gender A. Suppose there are two unobservable variables U (UA, Uxv), e.g. Uxy may be interpreted as the student's knowledge. Consider the following structural equations:
A UA
XG ba + w A + UXY,
X,. = br. + a A
Y by + w A + UXY;
where (ba, wa, br, w&, by, w are know parameters of the causal model. Given observation XG - 1, = 0. the counterfactual value can be calculated with an adduuctionaactiinn-preiiiiiii 20166: (i) abduction that finds posterior distribution Pr(U = ul X = A, = 0}. Here, we have Uxy = 1-bc and U A = 0 with probability 1; (ii) action that performs intervention = by replacing structural equations
procedure Glymour et al.
2Given structural equations equation 1 and the marginal distribution of U, Pr(U = a. o) can be calculated using the Pr(U=e. O=e] also he calculated. Varinee- Technique and the Jacobian factor. As a result, Pr(U = x(O (o Pr[O=o)
Variables
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of A; (fii) prediction that computes distribution of YU- given X6 1, =0 using new structural
equations and the posterior. We have:
YA-1(U)= = +--- w + bg with probability
3.2 Counterfactual Fairness
Counterfactual Fairness (CF) was first proposed by Kusner et al. (2017): it requires that for an individual with (X = A = a), the prediction Y in the factual world should be the same as that in the counterfactual world in which the individual belongs to a different group. Mathemalically, CF is defined as follows: Va, à e X, E X.ye V.
Pr (
Pr (P)((P_a(U)= y| ==
While the CF notion has been widely used the literature, does not account for the downstream impacts of ML. prediction y individuals in factual and counterfactual worlds. To illustrate the importance of considering such impacts, provide example below.
Example 3.2. Consider automatic lending where an ML model is used to decide whether to issue loan to an applicant based on credit score X and sensitive attribute A. As highlighted in Liu et al. (2018). issuing loans to unqualified people who camnot repay the loan may hurt them by worsening their future credit scores. Assume an applicant in the factual world is qualified for the loan and does not default. But in a counterfactual world where the applicant belongs to another group, he/she not qualified. Under counterfactually fair predictions, both individuals the factual and counterfactual worlds should receive the loan with the same probability. Suppose both are issued loan, then the one in the counterfactual world would have worse credit score in the future. Thus, it is crucial to consider the downstream effects when learning a fair ML model.
3.3 Characterize downstream effects
Motivated by Example 3.2 this work studies CF in a dynamic setting where the deployed ML decisions may affect individual behavior and change their future features and statuses. Formally, let X' and Y' denote an individual's future feature vector and status, respectively. We use an individual response r to capture the impact of ML prediction Y on individuals, as defined below.
Definition 3.1 (Individual response). An individual response r U x + u x is a map from the current exogenous variables U E U4 endogenous variables V V, and prediction y y to the future exogenous variables U" and endogenous variables V".
One way to tackle the issue in Example 3.2 is to explicitly consider the individual response and impose fairness constraint on future status Y' instead of the prediction V. We call such fairness notion the Lookahead Counterfactual Fairness (LCF) and present it in Section
4 Lookahead Counterfactual Fairness
We consider the fairness over the individual's future outcome Y', Given struc- tural causal model M = (U, F), individual response r, and data (,Y,,), we define lookahead counterfactual fairness below.
U1
A
Definition 4.1. We say an MT model satisfies lookahead counterfactual fairness (LCF) under response if the following holds Va, A. X,y e):
Y
Up
Pr Pr = y X A = - = -)
(3)
Figure Causal graph in
LCF implics that the subsequent consequence of ML decisions for given Example 4.1 individual in the factual world should be the same that in the counterfactual world where the individual

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belongs to other demographic groups. Note that CF may contradict LCF: even under counterfactually fair predictor, individuals in the factual and counterfactual worlds may end up with very different. future statuses. We show this with an example below.
Example 4.1. Consider the causal graph in Figure 1 and the structural functions as follows:
X fx(Ui) U1,
== fy X, A) = U/ + A,
U1 = r(U1, - -
U12 r(U2, 8) U2 - U2 8,
X' = fx(UU) = -U
Y' = fr X', = -
Based Kusner al. (017)) predictor that only uses U1 and U2 imput is counterfactually fair Therefore, Y U2) satisfies CF. Let U1 and U2 be uniformly distributed over [-1, Note that the response r(U1, 8) and r(U2, Y) imply that individuals make efforts to change feature vectors through changing the unobservable variables, which results higher ÿ the future. It is easy to see that a CF prodictor h(U,, = U1 + U2 minimizes the MSE loss B((Y - Y)3 f-1, and Pr(A- = 1] = 0.5. However, since Vu, -V, 8 1, we have:
Pr (r y/ = x, a) 8(y
2),
Pr (r y = A, = a) 5(y - à 2).
where 8(y) = 1 y and 8(y) otherwisc. It shows that although the docisions in the factual and counterfactual worlds are the same, the future staluses Y' still different and Definition 4.1 does not hold.
Theorem 4.1 below identifies more general scenarios under which LCF can be violated with a CF predictor.
Theorem 4.1 (Violation of LCF under CF predictors). Consider causal model M (U, V. and individual response in the following form:
U' U'=)
If the response r is function and the status Y in factual and counterfactual worlds have different distribu- tions, i.e.,
y Pr y/
imposing any arbitrary model Y that satisfies CF will violate LCF, i.c.,
Pr(Y)eaaU)) ylX x, Pr (U) y| x, a).
5 Learning under LCF
This section introduces algorithm for learning predictor under LCF. In particular, we focus special case with the causal model and the individual response defined below.
Given sets of unobservable variables U = (U,,... Ua, Uy} and observable variables (X,, XJ, A,Y we consider cansal model with the following structural functions:
X; A), Y=
Xv,
(4)
where f. is an invertible function" and fy invertible w.r.t. Uy. After receiving the ML prediction Y, the individual's future features X' and status y' change accordingly. Specifically, consider scenarios where individual unobservable variables U change based the following
U = r;U,; -
Vi (1,......d Uy= = ry - v
(5)
Note that U1 and U2 be generated for cach sample 4). See Soction 4.1 of Kusner al 2017 for details. 'Several works causal inference consider invertible structural function, e.g., bijective models introduced in Nasr- Cosr-Fsfhhany 2023).

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and the future attributes X and status Y' also change accordingly, i.e.
X = [iUU,, =
U.).
(6)
The above scenario implies that individuals respond to ML model by strategically moving features toward the direction that increases their chances of receiving favorable decisions, step size dj > 0 controls the magnitude of data change and can be interpreted as the effort budget individuals have on changing their data. Note that. this type of response has been widely studied in stralegic classífication literature (Rosenfeld) et al. 2020
Hardt et al. 2016a). The above process with d = is visualized in Figure
Our goal is to train an ML model under LCF constraint. Before presenting our method, we first define the notion of counter-
factual random variables.
Definition 5.1 (Counterfactual random variable). Let x and a be realizations of random variables X and A, and o a. We say X = X (U) and Y : Yaaea(U) are the counterfactual random variables associated with (z, a) if U follows the condi- tional distribution Pr[U]X = z, A a as given by the causal model M. The realizations of X, Y are denoted by ø and ý.
A
Y
U
The following theorem constructs a predictor 9 that satisfies LCF, i.c., deploying the prodictor g in Theorem 5.1 ensures the future status Y" is counterfactually fair.
Figure 2: A causal graph and individual responses with two features X2, The black arrows represent the connections de- scribed in structural functions. The red ar- rows represent the response process. The green dash arrows are the potential connec- tion to prediction Y.
Theorem 5.1 (Predictor with perfect LCF). Consider causal model M = (U,V, F), where U = {Ux, )y}, Ux = [U1, U2.,U,,,UUU, V = {A,X,Y), X = X2,, Xal"" and the structural equations are given by,
X=aoUx BA,
Y= ===TX T + +Uy
[wi, W2, .., wal", and + denotes the element wise where a |01, ... oa)² 3 = 13 32, .... BalT, w
production. Then, the following predictor satisfies LCF,
, U) -1 pat +p hNU),
2 with := a[llwaall++pp + and Ý is the counterfactual random variable associated with Y. Here, p2 and ps and function l((.) are arbilrary and can trained lo improve prediction performance.
where P1 = T
Proof Skelch. For any given x, a, we can find the conditional distribution of U. For a sample u drawn from the distribution, we can compute x, H. y and ÿ. Then we have
g(ū u) pIO² + P2ù p3 h(u).
From this we can compute the gradient of w.r.t. ux, uy. With response function we can get u'x and dyy With the structural functions, can know y' and V.. So, we have
(g(y, ux) Ōglū. ux, uy)
- y'1 y T
Oy
Bocause we know that ag(v.ux.ur) dy
2p1y = Ty, we have
1 v'1 by 1 y8
With law of total probability, we have LCF satisfied.

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The above theorem implies that 9 should be constructed based on the counterfactual random variable Y and U. Even though U is unobserved, it can be obtained from the inverse of structural equations. Quantity T in Theorem 5.1 depends on the step size 7) in individual response, and parameters a, 7, w in structural we can achicve perfect LCF.
T functions. When p1 2'
Tt worth noting that Definition 4.1 can be a very strong constraint and imposing (A-a(U) and a (U) to have the same distribution may degrade the performance of the predictor significantly. To tackle this, we may consider weaker version of LCF.
Definition 5.2 (Relaxed LCF). We say Relaxed LCF holds if V(a, AP a + ā, E EX V we have:
Pr
(U)a < (Y=a(UU) - Ya-a(U)I] x 2, A =1
(9)
Definition 5.2 implics that after individuals respond to ML model, the difference between the future status y in factual and counterfactual worlds should be smaller than the difference between original status Y in factual and counterfactual worlds. In other words, it means that the disparity between factual and counterfactual worlds must decrease over time. In Section 7 we empirically show that constraint in equation g is weaker than the constraint in equation 3 and can lead to a better prodiction performance.
Corollary 5.1 (Relaxed LCF with prodictor equation 8). Consider the same causal model defined in Theorem 5.1 and the prediclor defined in equation Relazed LCF holds if pi E (0,
Proof Sketch. When p1 (O,T), from
Og(y, ux, ux, uy) By
0g ux, uxx
){
W 3'1 4
Og
have
2p1 V v/ by -
(ū y)l.
T
Therefore // - 1/'l 18 y/. With law of total probability, the Relaxed LCF is satisfied.
Apart from relaxing Pi in predictor as shown in equation g we can also relax the form of the predictor satisfy Relaxed LCF, as shown in Theorem 5.2
Theorem 5.2 (Predictor under Relaxed LCF). Consider the same causal model defined in Theorem 5.1 A predictor g( U) salisfies Relazed LCF if has the following three properties:
(i) g(ū, u) is strictly conver in y-
(W) g(ū, u) can be expressed as g(ū, u) = g1 (i) + 92()..
(1 The derivative of g(ū u) w.r.t. z is K Lipschite continuous in á with K < "[ [ i.e., 0g(ūn u) Oÿ
0g(ūz, u)
< K l0 T2l
Proof Skelch. When g(ū, u) saties property (ii), we can prove that.
@g(u, uy)
Øglū. ux, uy
y' = vj - y + T
Oy
still holds. Because of properties (i), we have
@g((y, uy)
Ög(üs ux ex,ux) uy)
(U v)
0.
Oy
Oy
Therefore, when
0o(u.xx .x) By
0ol0.ux.ey)
2 yl. // 3/1 1y 511. which guaranteed by property
By
(iii).
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Theorems 5.1 and 5.2 provide insights designing algorithms to train a predictor with perfect Relaxed
LCF. Specifically, given training data D {(x19) (y ())))(())) )) we first estimate the structural equations.
Then, we choose a parameterized predictor that satisfies the conditions in Theorem 5.1 or 5.2 An example is shown in Algorithm| 1 which finds an optimal predictor in the form ( u) = p1ÿ + P2ú + Ps + ho(u) under LCF, where P1 20( [++)
the training parameter for function h, and p2, Ps are two other training parameters. Under Algorithm 1 can find the optimal values for p2, P3, 0 using training data D. If we only want to satisfy Relaxed LCF (Definition 5.2 can be training parameter with 0 < P1 <T
Algorithm Training predictor with perfect LCF
Input:
Training data D = {(x11) y a (0))))) 1==1 response parameter 1).
1: Estimate the structural equations using DD to determine parameters a, 3, we, and 7
2: For each data point (xx(), (0 (0),,, draw samples {u())b)) ا== from conditional distribution Pr[U]X
x09)) A = } and generate counterfactual g(öli) associated with ((li based on structural equations
3: Compute + 2n( [=oo|l?+>*)
Solve the following optimization problem,
p2, Ps, = arg min Σ
/ (9 (1061,,0)) ,00
P2.Ds.
1=1 ==1
where
(1015, w(ll) (Ho)) 2
+ P2g + +p + ho(u),
i is a parameter for function h, and is a loss function. Output: P2, Ps.
It worth noting that the results in Theorems 5.1 and 5.2 for linear causal models. When the causal model is non-linear, it is hard to construct model satisfying perfect LCF in Definition 4.1 Nonetheless, we can still show that it is possible to satisfy Relaxed LCF (Definition (.2) for certain non-lincar causal models. Theorem 5.3 below focuses on a special case when X is not linearly dependent on and Ux and it identifies the condition under which Relaxed LCF can be guaranteed.
Theorem 5.3. Consider a bijective causal model M = (U, V,,, where U is scalar exogenous variable, = (A, X, Y), and the structural equations X = fx(A, E R = (x) E IR can be written in the form of
J(U [ mo(A))
for some function f, where uo (A) is arbitrary function of A. Define function I(s) f(s) df(e) as the mulliplication of f(s) and its derivalive. IJ [(() has the following properlies:
f(s) is monotonic and strictly concave;
If f(s) 2 j(s') then [(s) f(s))'(''), Va
r(s) 2 and there erists constont M >> 0 such that I(s) is M M-Lipschitz continuous.
Then, the following predictor satisfies Relazed LCF,
g U) p1 p + h(U),
where Pı E (0, , P2 are learnable parameters, Y the counterfactual random variable associated with Y, and h(U) can be an arbitrary monotonic function that increasing (resp. decreasing) when f(s) is increasing (resp. decreasing).

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Published in Transactions on Machine Learning Research (12/2024)
Proof Sketch. For sample u drawn from the conditional distribution of U given X x, a, we compute y', D', and get
V j(1()), y = f(02), x' ='= y' I(d4).
The specific formulas of @1 22 d3 and @4 can be seen in the full proof. With the mean value theorem, we know that
ly 51 fca)loo Φ21.
where E (01,02) and
w 1- f'22) 0s1= The three properties ensure that
where 0g E (Øs, 21).
f(a) > f(e2). 101 22 2 10s - Φal-
Therefore, 13' 1/1 1 yl- With law of total probability, have the Relaxed LCF satisfiod.
Theorems 5.2 and 5.3 show that designing predictor under Relaxed LCF highly depends on the form of causal structure and structural equations. To wrap up this section, we would like to identify conditions under which Relaxed LCF holds in a causal graph that X is determined by the product of Ux and A. Theorem 5.4. Consider a non-linear causal model M (U,V.F), where U = {Ux, Uy Ux [U1,U2. U2. ..., Ua]", V {A, x, X, X = [X1, X2,.., Xa]T; sume that the structural functions are given by,
021 a binary sensitive allribute. As-
x= = 2 xx B),
== + +Uv,
(10)
3 [8 "a, and & denotes the element wise production. A predictor g(Y) satisfics Relared LCF 9 and the causal model have the following three properties.
where a .......
aal 02,..
(i) The value domain of A satisfies 0102 > 0.
(8) g(ū) is strictly conver.
(ti) The derivate of g(0) is K-Lipschitz continuous with K
n| [ i.e,
0g(ūn) 0g(ū2) Oy
91 521
Proof Sketch. For sample drawn from the conditional distribution of U given X x. A a, compute the y' and y' and get
1 8'1 ly ÿ A|.
The definition of can be seen in the full proof. Because property (i) and property (ii),
(u A 0. Property (iii) ensures that |A| 2 yyy 5)- Therefore, b 1/ 1/ vl-
Although the structural equation associated with Y is still linear in X and Uy emphasize that such linear assumption has been very common in the literature due to the complex nature of strategic classification Zhang et al. ) Liu et al. 2020):: Bechavod et al. (2022). For instance, Bechavod et al. (2022) assumed
the actual status of individuals Is is Y = 3,,, linear function of features X. Zhang et. al. (2022 assumed
that X itself may be non-linear in some underlying traits of the individuals, but the relationship between
and P(Y 1|X) still linear. Indoed, due the individual's strategic response, conducting the theoretical analysis accounting for such responses can be highly challenging. Nonetheless, it is worthwhile extending LCF to non-linear settings and leave this for future works.
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6 Path- nath-dependent LCF
An extension of counterfactual fairness called path-dependent fairness has been introduced in Kusner et al. (2017). In this section, also want to introduce extension of LCF called path-dependent LCF. We will also modify Algorithm 1 to satisfy path-dependent T.CF.
We start by introducing the notion of path-dependent counterfactuals. In causal model associated with a causal graph G, we denote PG.. as a set of unfair paths from sensitive attribute A to Y. We define Xps, the set of features that are not present in any of the unfair paths. Under observation X = A a, we call Xpp tapg (U) path-dependent counterfactua ual random variable for Y, and its distribution can be calculated follows:
Pr(Yata.Xpg temg (U) = y|x = x, A a} = Σ Pr(YAsa.XTg, temg (u) = Pr(U = ul = x, A a}.
For simplicity, we use YPD and URD to represent a path-dependent counterfactual and the corresponding realization. That is, YPD = Yp- =apg, (U) where follows Pr(U|X X x, A = a). We consider the
same kind of causal model described in Section 5 the future attributes X' and outcome Y' are determined by equation 5 and equation We formally define the path-dependent LCF in the following definition.
Definition 6.1. We say an ML model satisfies path-dependent lookahead counterfactual fairness w.r.t. the unfair path set Poi if the following holds Va, à E A, X E X E::
Pr 9 VataXpg ppe (U) x x, A a) Pr 9 (P t2p (U) y/ X x, A = =a)=a)
Then we have the following theorem.
Theorem 6.1. Consider cousal model and structural equations defined in Theorem 5.1 If we denote the features unfair path XPas and remaining features XPs can re-write structural equations
Xpo
UXPOA + +BPo A apss
XP
aPiA OUXpg, + +BP A,
T Y upes XPos twp XPe, vUy
Then, the following predictor satisfies path-dependen LCF,
u(YPD.U) = PI8ED -pcU))-p-p00
PDYPD ps h(U),
where P1 = T with
T allluoo, C Crps + 1wPs @ Caps 1(2
p2 and pa are learnable parameters to improve prediction performance and is an arbitary function.
Proof Skelch. Consider a sample u, we can compute Trg TP EP D and ÿpn- Then can gel similar
equation about ly' øp like Theorem 5.1 Then we can get lg' ppDl from the form of g. With law of total probability, we have the path-dependent LCF satisfied.
7 Experiment
We conduct experiments both synthetic and real data to validate the proposed method.
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7.1 Synthetic Data
We generate the synthetic data based on the causal model described in Theorem 5.1 where set d = 10 and generated 1000 data points. We assume Ux and Uy follow the uniform distribution over [0, and the sensitive attribute A E {0, I is a Bernoulli random variable with Pr(AA = 0} 0.5. Then, we generate X and Y using the structural functions described in Theorem 5.1 Based on the causal model, the conditional distribution of Ux and Uy given = x. = a are as follows,
Ux o x Bu) ): a
Uy| = =x Uniform(0, 1)-
(11)
Basclincs. We used two basclines for comparison: (i) Unfair predictor (UF) is a linear model without lairness constraint imposed. It. lakes feature X input and predicts Y. (ii) Counterfactual fair predictor (CF) only takes the unobservable variables U as the imput and was proposed by Kusner et al. (017)).
train predictor in the form of Fq. S In our experiment, h(u) is linear function. To train 9 u). we follows Algorithm [ with = 100. set at 20%/20%/ 222%2220% 200 ratio randomly and repeat the
Implementation Details. To find predictor satisfying Definition 4.1
We split the dataset into the training/ /aaaaaaaaaaaaeteeeeeeeeeee experiment 5 times. We use the validation set to find the optimal number of training epochs and the learning rate. Based on our observation, Adam optimization with learning rate equal 10-3 and 2000 epochs gives us the best performance.
Metrics. We use three metrics to evaluate the methods. To evaluate the performance, we use the mean squared error (MSE). Given a dataset (x((),, (0)), (0) Jー=1 for each x00) and a we generate m 100 values of (+ from the posterior distribution. MSE can be estimated as follows
MSE Σ
y gobll
(12)
mn
where (LLl) is the prediction for data (x (9) u (..)). Note that for the UF baseline, the prediction does not depend on y(b,) Therefore, g(9)5l docs not change by j for the UF prodictor. To evaluate fairness, wo define a metric called average future causal effect (AFCE),
AFCE
y M0D -/00
It the average difference between the factual and counterfactual future outcomes. To compare YY- Y with YY' Y under different algorithms, we use the unfairness improvement ratio (UIR) defined below. The larger UIR implies higher improvement in disparity.
Σ Σ
UIR
1
100%.
D DE j=
GIL) J(bbl
Results. Table 1 illustrates the results when we set = 10 and pl = T. The results show that our method can achieve perfect LCF with pl T. Note that in our experiment, the range of Y is [0, and our method and UF can achieve similar MSE. Moreover, our method achieves better performance than the CF method because Y includes useful predictive information and using it in our predictor can improve performance and decrease the disparity simultaneously. Because both CF and UF do not take into account future outcome Y', Y-' -Y" is similar YY - YI, leading to UIR O. Based on Corollary 5.1 the value of pi can impact the strength of fairness. We examine the tradeoff between accuracy and fairness by changing
Z.
TTe exact values for parameters a. 8, 10 and 7 can be found in the Appendix
6 Check Section 4.1 of Kusner al. 2017 for details on why equation 12
empirical estimate of MSF
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Table I: Results on Synthetic Data: comparison with two baselines, unfair predictor (UF) and counterfactual fair predictor (CF), in terms of accuracy (MSE) and lookahead counterfactual fairness (AFCE, UIR).
Method
MSE
AFCE
UIR
UF
0.036 = 0.003
1.296 = 0.000
0% = 0
CF
0.520 + 0.045
1.296 + 0.000
0% + 0
Ours (p1 = T/2)
0.064 + 0.001
0.000 + 0.0016
100% +
1.2
n=
0.020
n=
10
0=
0.015
0.8
AFCE
0.6
0.010
0.005
0.2
0.0
0.07
0.08
0.09
0.10
0.11
0.50
0.55
0.60
0.65
MSE
(a) Accuracy-fairuess trade-off of predictor Eq. on synthetic data: we vary from T 512 to T under dif- ferent 77. When p1 I, attain perfect LCF.
(b) Accuracy-fairness trade-off on the law school dataset: we vary pi from 5I2 to I under different T 4/. When pi = g: we allain perfect LCF.
T the value of p1 from 512 to I under different Figure 3a shows the MSE as function of AFCE. The results show that when 77 1 we can casily control the accuracy-fairece trade-off in our algorithm by adjusting pr. When becomes large, we can get high LCF improvement while maintaining a low MSE. To show how our method impacts a specific individual, we choose the first data point our test dataset and plot the distribution of factual future status Y' and counterfactual future status Y' for this specific data point under different methods. Figure ! illustrates such distributions. It can be seen in the most left plot that there is obvious gap between factual Y and counterfactual Y. Both UF and CF not decrease this gap for future outcome Y'. However, with our method,
observe that the distributions of Y' and Y' become closer to each other. When P1 = F (the most right plot in Figure 4 the two distributions become the same in the factual and counterfactual worlds.
7.2 Real Data: The Law School Success Dataset
We further measure the performance of our proposed method using the Law School Admission Dataset Wightman 1998). In this experiment, the objective is to forecast the first-year average grades (FYA) students in law school using their undergraduate GPA and LSSA scores.
GPA
Race
Dataset. The dataset consists of 21,791 records. Each record is character- ized by attributes: Sex (S), Race (R), UGPA (G), LSAT (L), and FYA (F). Both Sex and Race are categorical in nature. The Sex attribute be either male or female, while Race can be Amerindian, Asian, Black, Hispanic, Mexican, Puerto Rican, White, or other. The UGPA contimuous variable ranging from to 4. LSAT is an integer-based attribute with a range [0, 60). FYA, which the target variable for prodiction, a real number ranging from -4 (it has been normalized). In this study, we consider S as the sensitive attribute, while R, G. and I. are treated features.
LSAT
K
See
PYA
Figure 5: Causal model for the Law School Dataset.
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Cureen
UF
CF
LCP(p F)
LCF(p)
2.6
1.5
0.27
0.5
1.3
1.00
10.
denetty
0.70
1.0
1144
0.90
factioal
0.25
11222
000
1.00
1000
0.000
1000
0100
2.5
6
111
2
Figure 4: Density plot for Y' and Y in synthetic data. For chosen data point, we sampled a batch of U under the conditional distribution of it and plot the distribution of Y' and Y
Causal Model. We adopt the causal model as presented Kusner et al. 2017); which can be visualized Figure In this causal graph, K represents an unobserved variable, which can be interpreted as knowledge. Thus, the model suggests that students' grades (UGPA, LSAT, FYA) are influenced by their sex, race, and underlying knowledge. We assume that the prior distribution for K follows normal distribution, denoted as (O0 1). We adopt the same structural equations as Kusner et al.
2017
G N(w K + w + wgS + ba, aG); L Poisson (exp (w { K + wi R + w bz,)),
N(w K + + R 51..)
Implementation Details. Note that race is an immutable characteristic. Therefore, we assume that the individuals only adjust their knowledge K in response to the prediction model Ý. That is K' = K + xÝ In contrast to synthetic data, the parameters of structural equations are unknown, and we have to use the training dataset estimate them. Following the approach Kusner al. (2017), we assume that G and F and w # H w respectively, Note that L is an integer, and it follows Poisson distribution with the parameter exp(w K + wf br}. Using the Markov chain Monte Carlo (MCMC) method Geyer (1922), )
adhere Gaussian distributions centered at w K + u
estimate the parameters and the conditional distribution of K given (R, G, L). For cach given data, we sampled m 500 different k's from this conditional distribution. We partitioned the data into training, validation, and test sets with 20%% 20% ratio.
Table 2: Results on Law School Data: comparison with two baselines, unfair predictor (UF) and counterfac- tual fair predictor (CF), in terms of accuracy (MSE) and lookahead counterfactual fairness (AFCE, UIR).
Method
MSE
AFCE
UIR
UF CF
0.393 + 0.046
0.026
0.003
0% + 0
0.496 + 0.051
0.026 + 0.003
0% + 0
Ours (pi T/4)
0.493 + 0.049
0.013 = 0.002
50% +-4 0
Ours (P1 = T/2)
0.529 = 0.049
0.000 = 0.000
100% + 0
Results. Table illustrates the results with 7) = 10 and P1 4 and pı where T = 1/ The results show that our method achieves a similar MSE as the CF predictor. However, can improve AFCE significantly compared to the baselines. Figure T shows the distribution of Y and y' for the first data point in the test set in the factual and counterfactual worlds. Under the UF and CF predictor, the disparity between factual and factual Y' remains similar to the disparity between factual and counterfactual Y. On the other hand, the disparity between factual and counterfactual Y' under our a algorithms gels better for both Pi = T/2 and P1 = T/4. Figure [3b demonstrates that for the law school dataset, the trade-off between MSE and AFCE can be adjusted by changing hyperparameter P1- Figure 0 show the factual and counterfactual distributions in real data experiment. It can be seen that our method is the only way that can decrease the gap between Y and 5 in obvious way.
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Current
LCP
LCP(P 3
2.5
Amalty
dusetty
fectial coutter
fertral
0.5
0.0
0.0
072
022
1.0
12
1.5
270
y
Figure 6: Density plot for F' and F' in law school data. For chosen data point, we sampled K from the conditional distribution of K and plot the distribution of F' and F..
8
Conclusion
This work studied the impact of ML decisions individuals' fulure status using counterfactual inference framework. We observed that imposing the CF predictor may not decrease the group disparity in individuals' future status. We thus introduced the lookahead counterfactual fairness (LCF) notion, which takes into account the downstream effects ML models and requires the individual future status to be counterfactually fair. We proposed a method to train an MT. model under LCF and evaluated the method through empirical studies on synthetic and real data.
Acknowledgements
This material is based upon work supported by the U.S. National Science Foundation under award I1S- 2202699, IIS-2416895, IIS-2301599, and CMMI-2301601, and by OSU President's Research Excellence Ac- celerator Grant, and grants from the Ohio State University's Translational Data Analytios Institute and College of Engineering Strategic Research Initiative.
References
Loan Prediction Problem Dataset kaggle.com.
https://www.kage www.:kggll..c//aaaaaaaeeeeeeeeee comdaatsets/
altmiastellatttoo/oommpeecccoood wwrpriion-roooeemddddaeeeee dataset. [Accessed 20-10-2024).
Mahed Abroshan, Mohammad Mahdi Khalili, and Andrew Elliott. Counterfactual fairness in synthetic data
generation. In NewrIPS Workshop on Synthetic Data for Empowering ML Research, 2022.
Mahed Abroshan, Andrew Elliott, and Mohammad Mahdi Khalili. Imposing fairness constraints in synthetic data generation. In International Conference on Arlificial Intelligence and Stalistics, pp. 2269 2277. PMLR, 2024.
Jacy Anthis and Victor Veitch. Causal context connects counterfactual fairness to robust prediction and group fairness. Advances in Newral Information Processing Systems, 36, 2024.
Yahav Bechavod, Chara Podimata, Steven Wu, and Juba Ziani. Information discrepancy in strategic learning. In International Conference Machine Learning, pp. 1691-1777 2022.
Micah D Carroll, Anca Dragan, Stuart Russell, and Dylan Moddfeld-Menell Estimating and penalizing induced preference shifts in recommender systems. In International Conference on Machine Learning, pp. 2686 2708. PMLR, 2022.
Silvia Chiappa. Path-specific counterfactual fairness. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 7801 7808, 2019.
Roxana. Dancshjou, Kailas Vodrahalli, Weixin Liang, Roberto A Novoa, Mclissa Jenkins, Veronica Rotem- berg, Justin Ko, Susan M Swetter, Elizabeth Bailey, Olivier Gevaert, al. Disparities in dermatology ai: Assessments using diverse clinical images. arXiv preprint arxim:111 211::111.0000 211.0000000 2021.
15

---

Published in Transactions on Machine Learning Research (12/2024)
Jeffrey Dastin. Amazon scraps secret ai recruiting tool that showed bias against women. http://reut.rs. 2MXzkly 2018.
Lucas De Lara, Alberto González-Sanz, Nicholas Asher, Laurent Risser, and Jean-Michel Loubes. Transport- based counterfactual models. Journal of Machine Learning Research, 25( (5(136):1-59) 2024.
Sarah Dean and Jamie Morgenstern. Preference dynamics under personalized recommendations. In Proceed- ings of the 23rd ACM Conference on Economics and Computation, pp. 795 816, 2022.
Virginie Do, Sam Corbett-Davies Jamal Atif, and Nicolas Usunier. Online certification of preference- based fairness for personalized recommender systems. In Proceedings of the AAAT Conference on Artificial Intelligence, volume 36, pp. 6532 6540, 2022.
Tri Dung Duong, Qian Li, and Guandong Xu. Achieving counterfactual fairness with imperfect structural causal model. Erpert Systems with Applications, 240:122411 2024.
Ahmad-Reza Ehyaei, Kiarash Mohammadi, Amir-Hossein Karimi, Samira Samadi, and Golnoosh Farnadi. Causal adversarial perturbations for individual fairness and robustness in heterogencous data spaces. In Proceedings of the AAAT Conference on Artificial Intelligence, volume 38, pp. 11847-11855, 2024.
Yinggiang Ge, Shuchang Liu, Ruoyuan Gao, Yikun Xian, Yunqi Li, Xiangyu Zhao, Changhua Pei, Fei Sun, Junfeng Ge, Wenwu Ou, et al. Towards long-term fairness in recommendation. In Proceedings of the 14th ACM international conference on web search and data mining, pp. 445 453, 2021.
Charles J Geyer. Practical markov chain monte carlo. Statistical science, pp.
1992.
Madelyn Glymour, Judea Pearl, and Nicholas P Jewell. Causal inference in statistics: A primer. John Wiley & Sons, 2016.
Sofie Goethals, David Martens, and Toon Calders. Precof: counterfactual explanations for fairness. Machine Learning, 3111:3331,33314,, 2024.
Moritz Hardt, Nimrod Megiddo, Christos Papadimitriou, and Mary Wootters. Strategic classification. Tn Proceedings of the 2016 ACM conference on innovations in theoretical computer science, pp. 111 122, 2016a.
Moritz Hardt, Eric Price, and Nati Srebro. Equality of opportunity in supervised learning. Advances in neural information processing systems, 29:3315 1-3315-3323,, 2016b.
Thomas Henzinger, Mahyar Karimi, Konstantin Kueffner, and Kanshik Mallik. Runtime monitoring of dynamic fairness properties. In Proceadings of the 2023 ACM Conference on Fairness, Accoundabilily, and Transporency, pp. 604 614, 2023a.
Thomas Henzinger, Mahyar Karimi, Konstantin Kueffner, and Kaushik Mallik. Monitoring algorithmic fairness. Xive preprint orXiv:2305.15979, 2023b.
Yaowei Hu and Lu Zhang, Achieving long-term term fairness in sequential decision making. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 9549-9557, 2022.
Aria Khademi, Sanghack Lee, David Foley, and Vasant Honavar. Fairness in algorithmic decision making: An excursion through the lens of causality. In The World Wide Web Conference, pp. 2907-2914, 2019.
Mohammad Mahdi Khalili, Xueru Zhang, and Mahed Abroshan. Fair sequential selection using supervised learning models. Advances in Neural Information Processing Systems, 34:28144 28155, 2021a.
Mohammad Mahdi Khalili, Xueru Zhang, Mahed Abroshan, and Somayeh Sojoudi. Improving fairness and privacy in selection problems. In Proceedings of the AAAI Conference Artificial Intelligence, volume 35, pp. 8092-818188 1090- 2021b.
Mohammad Mahdi Khalili, Xueru Zhang, and Mahed Abroshan. Loss balancing for fair supervised learning. In International Conference on Machine Learning, pp.
PMT.R, 2023.
16

---

Published in Transactions on Machine Learning Research (12/2024)
Niki Kilbertus, Mateo Rojas Carulla, Giambattista Parascandolo, Moritz Hardt, Dominik Janzing, and Bernhard Schölkopl. Avoiding discrimination through causal reasoning. Advances in neural information processing systems, 30, 2017.
Matt Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva. Counterfactual fairness. Advances in neural information processing systems, 30, 2017.
Lydia T Liu, Sarah Dean, Esther Rolf, Max Sinchowitz, and Moritz Hardt. Delayed impact of fair machine learning. In International Conference on Machine Learning, pp. 3150-3158, PMLR, 2018.
Lydia T Liu, Ashia Wilson Nika Haghtalab, Adam Tauman Kalai, Christian Borgs, and Jennifer Chayes.
The disparate equilibria of algorithmic decision making when individuals invest rationally. In Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency, pp. 381 391, 2020.
Jing Ma, Rnocheng Guo, Aidong Zhang, and Jundong Li. Learning for counterfactual fairness from obser- vational data. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 1620-1630, 2023.
Agathe Fernandes Machado, Arthur Charpentier, and Ewen Gallic. Sequential conditional transport on probabilistic graphs for interpretable counterfactual fairness. arXiv preprint Xrx:2:2408.co 03425, 2024.
John Miller, Smitha Milli, and Moritz Hardt. Strategic classification is causal modeling in disguise. In Internaiional Conference on Machine Learning, pp. 6917 6926. PMLR, 2020.
Razich Nabi and Ilya Shpitser. Fair inference on outcomes. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
Arash Nasr-Esfahany, Mohammad Alizadeh, and Devavrat Shah. Counterfactual identifiability of bijective causal models. arXiv preprint Ekir:2302.02222 2023.
Judea Pearl al. Models, reasoning and inference. Cambridge, UK: CambrriggeUnivvessttyyyP 19(2). 2000.
Mariana Pinto, Andre V Carreiro, Pedro Madeira, Alberto Lopez, and Hugo Gamboa. The matrix reloaded: Towards counterfactual group fairness in machine learning, Journal of Dala-centric Machine Learning Research, 2024.
Raphael Poulain, Hamed Fayyaz, and Rahmatollah Beheshti. Aligning (medical) lims for (counterfactull) fairness. arXie preprint Xixv:2248.. 12055, 2024.
Nir Rosenfeld, Anna Hilgard, Sai Srivatsa Ravindranath, and David C Parkes. From predictions to decisions: Using lookahead regularization. Advances in Neural Information Processing Systems, 33:4115-4126, 2020.
Pongyang Shao, Le Wu, Kun Zhang, Defu Lian, Richang Hong, Yong Li, and Meng Wang. Average uscr-side counterfactual fairness for collaborative filtering. ACM Transuctions on Information Systems, 42(5): 26, 2024.
Yonadav Shavit, Benjamin Edelman, and Brian Axelrod. Causal strategic linear regression. In International Conference on Machine Learning, pp. 8676-8686. PMLR, 2020.
Zeyu Tang, Yatong Chen, Yang Liu, and Kun Zhang. Tier balancing: Towards dynamic fairness over underlying causal factors. arXiv preprint arXiv:2301.08987 2023.
Songül Tolan, Marius Miron, Emilia Gómez, and Carlos Castillo. Why machine learning may lead to unfairness: Evidence from risk assessment for juvenile justice in catalonia. In Proceedings of the Seventeenth International Conference on Artificial Intelligence and Law, pp. 83 92, 2019.
Zichong Wang, Zhibo Chu, Ronald Blanco, Zhong Chen, Shu-Ching Chen, and Wenbin Zhang. Advanc- ing graph counterfactual fairness through fair representation learning. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 40-58. Springer, 2024a.
17

---

Published in Transactions on Machine Learning Research (12/2024)
Zichong Wang, Meikang Qiu, Min Chen, Malek Ben Salem, Xin Yao, and Wenbin Zhang. Toward fair graph neural networks via real counterfactual samples. Knowledge and Information Systems, 66(11):6617-664) 2024b.
Linda F Wightman. Lsac national longitudinal bar passage study. Isac research report series. 1998.
Yongkai Wu, Lu Zhang, and Xintao Wu. Counterfactual fairness: Unidentification, bound and algorithm. In Proceedings of the twenty-eighth international joint conference on Artificial Intelligence, 2019.
Ying Xiao, Jie M Zhang, Yepang Liu, Mohammad Reza Mousavi, Sieen Liu, and Dingyuan Xue. Mirrorfair: Fixing fairness bugs in machine learning software via counterfactusl predictions. Proceedings of the ACM on Software Engineering. (1FSS)):122121433 2024.
Tian Xie and Xueru Zhang. Automating data annotation under strategic human agents: Risks and potential solutions. arXiv preprint arXiv:2405.08027 2024a.
Tian Xie and Xueru Zhang. Non-linear welfare-aware strategic learning. arXiv preprinl orXiu:2/05.01810 2024b.
Tian Xie, Zhiqun Zuo, Mohammad Mahdi Khalili, and Xueru Zhang. Learning under imitative strategic behavior with unforeseeable outcomes. Transactions on Machine Learning Research, 2024. ISSN 2835- 8856. URL
Depeng Xu, Shuhan Yuan, and Xintao Wu. Achieving differential privacy and fairness in logistic regression. In Companion Proceedings of The 2019 World Wide Web Conference, pp. 594-599, 2019.
Muhammad Bilal Zafar, Isabel Valera, Manuel Gomez Rodriguez, and Krishna P Gunmadi. Fairness be- yond disparate treatment & disparate impact: Learning classification without disparate mistreatment. In Proceedings of the 26th international conference world wide web, pp. 1171-1180, 2017.
Xueru Zhang, Mohammadmahdi Khaliligarekani, Cem Tekin, et al. Group retention when using machine learning in sequential decision making: the interplay between user dynamics and fairness. Advances in Newral Information Processing Systems, 32:15269 15278, 2019.
Xueru Zhang, Ruibo Tu, Yang Liu, Mingyan Liu, Hedvig Kjellstrom, Kun Zhang, and Cheng Zhang. How do fair decisions fare in long-term qualification? Advances in Newral Information Processing Systems,, 33: 18457-18463 2020.
Xueru Zhang, Mohammad Mahdi Khalili, Kun Jin, Parinaz Naghizadeh, and Mingyan Liu. Fairness inter- ventions as (Dis) Incentives for strategic manipulation. In Proceedings of the 39th International Conference on Machine Learning, pp. 19239-262666 2022.
Zeyu Zhou, Ruqi Bai, and David Inouye. Improving practical counterfactual fairness with limited causal knowledge. In ICLR 2024 Workshop on Navigating and Addressing Data Problems for Foundation Models, 2024.
Aoqi Zuo, Susan Wei, Tongliang Liu, Bo Han, Kun Zhang, and Mingming Gong. Counterfactual fairness with partially known causal graph. Advances in Neural Information Processing Systems, 35:1238 15:1238-1252. 2022.
Zhiqun Zuo, Mohammad Mahdi Khalili, and Xucru Zhang. Counterfactually fair representation, Advances in Neural Information Processing Systems, 36:12124 12140, 2023.
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A Proofs
A.1 Proof of Theorem 5.1 and Theorem 5.2
Proof. For any given a, we can find the conditional distribution Ux|X = A A= = a and Uv = x, A =a based on causal model M. Consider sample [ux, uy] drawn from this conditional distribution. For this sample, we have,
f a ① ux 3ā,
ÿ = w To + ruy.
Since ÿ is also function of ux, uy, utilizing that
dy
0y Duy
w ①
%
Oux
the gradient of s ux, uy) w.r.t. ux, uy
Ōg(ū, ux uy)
Öglū, ux,
V Vax9 9
w a
Oux
Oij
V 9 ux,
Öglū, ux, uy
Oÿ 7.
Ouy
The response function r is defined as
uix ux + Vaxg.
uy = uy + Vuyg.
y can be calculated using response r follows,
y' = y + nnu" a
0glū, Ogly.ux, ux uy)
+ n mlw @ al 8glū. ux uy)
Oglū. ux, ux) Ouy + my Iglu. ux ux. uy)
+ 1
(13)
Oux
Oj
Similarly, we can calculate counterfactual value yf follows,
y' = ÿ + vpp"" Agly, ux ux.uy) Tux
Ogly, ux uy)
Ogly, ux, uy)
+ 177 2 8g(v. ux ux. ur)
+ || @
+ +77
(14)
8y
Ouy
Qy
Note that the following hold for g.
0g(ū. ux, uy)
Ög(v, ux uy)
(15)
Oux
Oux Og(u, ux, ur)
Og(ū, ux, ux)
(16)
Duy
Ouy
Thus,
(ggly, (Oylu,x,,,y ux, Oy
Ig(ū. uxx uy)
1 11 = ÿ - y + + (Iww 3 0a(2+
(17)
Given above equation, now can prove Theorem 5.1 Corollary
5.1
and Theorem
5.2
For g in Theorem 5.1 and Corollary 5.1 we have,
g ux, uy) 21 P2ů + p3 + h(u).
(18)
The partial derivative of can be computed as
Ög(ū, ux, uy) 2p1 p2-
(19)
Oğ
19

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We denote 7 (1lw @ a +7²). From the theorem, we know that p1 2. Therefore, we have
1 + 5)1 0.
v' - 1- = 1
Since, for any realization of the above equation holds, we conclude that the following holds,
( A Pr(V (U) y| -=
When P1 E (0, T), we have that
1 - 5/ 10 + P. (y 101 = 0.
Because
2p1
(y 0)
( 4)
T
and
2p1
(y 5)
2 yl
we have 18/ -'1 < ly - 5). With law of total probability, we have
Pr(([V) (U) Y (a (U)I < (U)
=
For g in Theorem 5.2 since g ux, uy) is strictly convex in Ø, have,
@gg() ux, uy) Öglū. ux ux, uy)
( v)
>
Öy
Oÿ
Note that derivative of g ux, uy) with respect ÿ is K-Lipschitz continuous in g.
Ögly, ux, uy)
Øg(ū, ux)
2/y - 7 w O all +2)
Oy
0ÿ
we have that
1n (11w o al13 +22) (
(8g(u (0 ur)
0g(ū. ux ux, uy) 0ÿ
2 - yl
dy
Therefore,
w 8/1 < 1y -
So we have
Pr([[VA (V) - va)x(U)) < (V)+a(U) - VA- (U))(( =x, 4= =1
A.2 Theorem 5.2 for non-binary A
Let {a} U { a|ي .... ر be set of all possible values for A. Let y be the counterfactual random variable associated with abl given observation LCF, where y satisfies the properties in Theorem
and A a. Then, g LyY +... y/mm) satisfies
5.2
20

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Proof. For any given 2, a, we assume the set of counterfactual is {ā'"), _.a), ,d.,,,.. al.... Consider sample [ux, wy] drawn from the condition distribution of Ux|X A and Ur X x, A a, with a
predictor (Hil)) +... u), use the same way in A.1 we can get
1 s1/ - اار y mlll o aall + x2) ( [ v(()) u)
ag(y + 700
+ 818 1] + gl++1) 1), 5//m) u)
agll) + ... 7lm
By + 8 7 8 B
with j j m. When > اااق we have
bull glml
< xy + 7. + ... .لاي)) + 7j++1) glml
and when y < أااق
b0ll + +....... -(mml y 7i"h + ..... أااار + ااأ++و 1). 9/mml
Because g is strictly convex and Lipschitz continuous, we have
(gاl - v) ( [Og( (Oy(gl) + ....... t...g/ml...... w) 0illl .... jlml
Ogly + 7/ + + ا + (ل15+t1 ... 3gmmm, u)
< 0,
By + 8(L ... للاق 1] + اا++اق ... Hlmml
and Wllw aall Xllwoal|++)) + (Dy(il)) + :lmll u) 0yl1 + ..... glml
0gly + T(0l + ... + أاو 1] + 8ل.+ j/mll .n) By 8 + 3 + 8 mm)
<21gb]] y/.
Therefore,
1041 1/1 < =اrاا yl
So we proved that, for any j E 21,
Pr(()a=aaU)) Y ( <U) a x, a) =1
A.3 Proof of Theorem 5K3
Proof. We start from the case when f is increasing. For any given æ, a, we can find the conditional distribution U|X xr A = based on the causal model M. Consider a sample u drawn from this conditional distribution. For this sample, we have
y= f uo(a)), -= f + va(ū))
So, the gradient of u(..u) p18² + p2 h(u) w.r.t. is
dg(ي.u) u)
2p10 dū
dh(u)
+
du
du
du
Therefore, y' can be calculated using response as follows,
u y y (u+ +
uo(aa) vo(a) + +mp11 y dÿ
dh(u)
+ n du
du
Similarly, we have the future counterfactual status as
V' o'=ī v==j(u (' un(ā) 21p y du
dy
dh(u) + n du
When y =- G. it is obvious that y' y/=8 '/' since d = y du
When y > G, bocause f is increasing, we have
uo (a) > uo (ä).
21

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Because h(U) is increasing, we have dh(w) dÿ
du 2 0 and
dh(u)
2npr1 dy
dh(u)
2np1 v
n du
2 0,
du 11 du
2 0.
du
We further denote
Φ + (ā)
Φ = u + uo(a)
Φ + (a + 2np11 1/ dy
dh(u)
+ du
du
dÿ
dh(u)
24 + uo(a) + 2pp
+ du
du
We already have
Φ1 02, Φ1 S Ø3.
02 S Φ1.
There are three cases for the relationship between Φ1, Φ2, Φ3 and Φ1-
Case 1: Φ1 < 02 Øs S Φ1. In this case, from the mean value theorem,
y Øن = (f(z)) f(()) =
f (c:) I02 dıl.
where e₁ 02), and
-~ F((a)) (c2) 0al.
where 02 (0a, d4). Because
(i dÿ
dy du
101 03s = =(uola) - =o(ā)) + 2n11
du
and
dğ du
dy du
S
we have
104 dal < un(a) un( 1022 $11 Because f is strictly concave, f' is strictly decreasing, and we have,
[((a)) > (f)))
Therefore,
1 -
3/ < y - -Vl.
Case 2: Φ1 S og < <Φ² S S1 In this case,
lg - و = j(b2) - I(bi) = f(b2) - j((b) - f(0s) - I(01),
y ー = f((r) f(ob) f(0.) f(02) -f(02) flos).
From the mean value theorem,
[f(ds) = 21l
22

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where e1 (01,0s) and
Ø21,
where 02 E (122, 04). Because I is decreasing, we have
[f(a) (o)).
Bscause
164 - 02l = 2np18' 1 dÿ
+ r dh((u)
du
du
dy
+ "l dl((u)
103 - Φ1 = 20p11
du
du
we have that 10s 103-990 163--11 10_ Φ21. Therefore, 18/ - 3 by 3/1.
Casc 3: Φ1 < d2 S Φe S Ø3. In this case, we have
I -- --01=f((0s) f(0o).
From the mean value theorem,
ly 01 =1= 1 ~
where e₁ E (ø1, 22), and
1/' - 1-1 = 1 (o2) 103 - 2sl,
where 02 ds). Because
102 Φ1 = ua(a) - uo(ā);
(u dy du
i di
I0s dal = (uo(ä) - ua(a)) + 2np1
du
and
y zu 1
$ M,
(u + uo)) - (u + uo vo))
when pı I nM
10b3 dal S (uo(a) uo(a)) 2 (uo(a) ua)) S (a)a)) -mn(a) 102 -b1l
Because (ci) > (o()) we have //' -i''l < by 51.
In conclusion, we prove that lV/ 8'1 < by ø for every sample With law of total probability, we have
Pr (((111ta(T) - a (U) < (V)+a(U) - Y (U)) x = x, A 1=A=00 a) = 1.
For the case when f decreasing, we can consider -Y = =(U + uo then we have
Pr ( (0) <1 -Ya- - (- (- --YY- x, A aa)=
which is to say
Pr (f(1xtma(U) - Yhra(U)) < (U)=a(U)- - Ya-a(U))3 x A A= = a) = 1.
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A.4 Proof of Theorem 5.4
From the causal functions defined in Theorem 5.4 given any ar, a, can find the conditional distri- bution Ux|X= = I, A = and Uy|X = A = Similar to the proof of Theorem 5.2 we have
Proof.
- a(a uux- +
ÿ TT i + yuy.
Because
0y dux
Oy duy
w ② ac,
7.
the gradient of g(i) w.r.t ux, uy are
V V = Ög(ū)
Sw a,
0g(ū)
V g 09 7
The response function is defined y' can be calculated using the response r as follows,
"x ux + F. "y uy + V g.
v/ y + 7 (aāllu laollw@al al12 +22) Og(ū) 0y
In the counterfactual world,
V = U O all + +2²) 0g(y) 8y
So we have,
Og(U) Oūj
0g(z)
y - 5/1 5 naaäl w O all? + 3))
Oy
We denote that A n[aāā w @ all2 + 7 M
Pglv)
Because A is binary attribute, we have
aă = 0102 > O.
From the property of that ( is strictly convex, we have
0g(ū)
0g(v) By
(u 1))
< 0.
By
Note that the derivative of (()) is K K-Lipschitz continuous,
Og(ū)
Ag(v)
2
Og
Oy
Ta(ā|) w 2 aal3 722)
we have
0g(U) Oğ
Ag(u), Oy
n(aā|l w o aal2 +
2 - 51.
Therefore, for every u sampled from the conditional distribution, 11/ -/1l 1/1 < b/j - yl. So we proved
Pr([[V/<a (U) - Yaaa (U)) < (YA=a(U) YA-a(U)) = a) = 1.
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A.5 Proof of Theorem 6.1
Proof. For any given x, can find the consitional distribution Ux|X = x, A and Uy = x, = based on causal model M. Consider sample [ux, uy] drawn from this conditional distribution. For this sample, we have,
irG. aPo + uXrex T UPD WPAA TPeΛ + wpy IPe + +uy.
SPoN a
T
So, the gradient of g@pPD, uXPea"
" "Xps DA
(yy w.r.t. uXPOs "Xps .y are
0glüPD. UXPes"
"Xp: 02
uy)
dg(üPD, uXren"
"Kpg uy)
Vaxras
3 OPoa O OPPa
QuXPos
0j
ŌglürD, uXros"
"Xps," ur)
ŌglürD, SebiDD,xxppp "XPos" uxps,
ur
V Vuxpg
Owps + aapa'
GA
auxps
Oÿ
Öglū, UXpga" duy
u "Xpe, uy)
Öglū uXroa" u ux
uy)
7.
OyPD
The response function is defined
4'x ux + Vuxg. uix = uy + Vur 9-
y can be calculated using response follows,
Öglürd, "Xpg, )
Öglürd.ux uXrox"
- .uy
s y + nwFer a
+ mwp T
DuxPas
Ouxp,
ÖglüpD, UXPgA
uXpe ,u)
0glüPD, uXPes uXpg, .y)
+ 17 uPes 3 aopoo
+1 upps
C C
jÿPD
DA
DğPD
0glüPD. UXFo,"
uy. uy)
ÖglūPD, UXPe, ur
+ m.y
m/?
Ouy
Similarly, we can calculate path-dependent counterfactual value VPD as follows,
Ogly, uXPex uxp5 .uy)
agly, uxyy, "xpg GA
uyy)
TPD =PD + mwper a
T + mwpp
Ouxros
@uxpg,
Og(u. uXroa"
uxps, (r)
Ogly: uXrox
+ WPen 2 aepo
+1 WPp
1 Capp
Oy
Oy
Ogly. UXpos
uy, uy)
Ogly, uXpo,
uy, uy)
+ 177
+ +17
Ouy
Öy
Thus,
VDD - 8/1 =
(g(y, UXpo, "Xpg, ar) 8glürD, MXPa, uxpe, .r)
ĀPD y t n (Ilup, o 0ape 11 +1wps, @ aos, BB x²)
8y
ØorD
Denote P1 2n(luPE Corea 11 10pg pops - Since the partial gradient of g(PP. UXPes uXrs, uy)
w.r.t. UPD is 2p10PD + P2, we know that Ig Frol Since for any realization of u, the equation holds, can conclude that the path-dependent LCF holds.
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A.6 Proof of Theorem 4.1
Proof. Since Y is determined by U, we denote the causal function from U to A as
Y = [(U,A) A).
Suppose the conditional distribution U given = 2; a could denoted as Pc(U), we have
Pr Pr(Ya=a(U)IX-= = x, 4= a) d)
Σ
Pa(u).
Because
Pr(Ya-a(U)- y/ a= # Pr(Ya-a(U)) = y/X x, --
we have,
Σ
Pa(u)
Σ
Pa(u).
(20)
WE (E(a)(()))=g))
=E(M)((x,))=))
Because the predictor satisfies CF,
U = r(U,
U A r(U,
the future outcome could be written
Pr(Y 2, -=
Σ
Pa(u).
=)(ul))r(u.))e))=o))
From Eq 20 have
Σ
Pa(u) #
Σ
Pa(u),
v((((((u)))a)=-))
xK(((((.)))))==)))
which is to say
Pr((\_a = x,A= # Pr((Ya-a(U) = y/X = x, A=
B
Parameters for Synthetic Data Simulation
When generating the synthetic data, we used
0.37454002
0.02058449
0.61185289
0.95071481
0.96990985
0.13949386
0. 0.73199394
0.83244264
0.29214465
0.59865848
0.21 2331131911
0.36636184
0.15601864
0.18182497
0.45606998
0. 0.15599452
0.18340451
0.78517596
;
7 0.60754485
0.05808361
0.30424224
0.19967378
0.86617615
0.52475643
0.51423444
0.60111501
0.43194502
0.59241457
0.70807258
0..2112914
(0.04645041
These values generated randomly.
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C Empirical Evaluation of Theorem 5.2
In this section, we use the same synthetic dataset generated in Section 7.1 to validate Theorem 5.2) We keep all the experimental settings as the same as Section 7.1 but use different predictor g(ū. u). The form of g u) is
g(0.u) u) = pıo + P2ū + h(u),
with = -. It obvious that satisfies property (i) and (ii) in Theorem 5.2. Because in the synthetic causal model, y is always larger than 0, property (ii) is also satisfied.
Table 3: Results on Synthetic Data for Theorem 5.2 comparison with two baselines, unfair predictor (UF) and counterfactual fair predictor (CF), in terms of accuracy (MSE) and lookahead counterfactual fairness (AFCE, UIR).
Method
MSE
AFCE
UIR
UF
0.036
0.003
1.296 + 0.000
0% + 0
0.520
0.045
1.296 + 0.000
0% + 0
Ours
0.064 0.001
0.930 0.001
28.2% + 0
Table 8 displays the results our method compared to the baselines. Except for our method, UF and CF baselines cannot improve LCF. When the properties in Theorem 5.2 are all satisfied, our method can guarantce an improvement of LCF.
D Empirical Evaluation of Theorem 5.3
We generate a synthetic dataset with the structural function:
Y = (aU' + t
The domain U (0, 1). a sampled from a uniform distribution and set as 0.5987 this experiment. In this case, f(s) = sš. Therefore, property (i), (ii) are satisfied. Since > e, we have M - 3. We use predictor
g = pI82 + P2 + h(u),
and choose u 2 Table
displays the results. Our method achieves large improvement in LCF,
Table 4: Results on Synthetic Data for Theorem 5.3 comparison with two baselines, unfair predictor (UF) and counterfactual fair predictor (CF), in terms of accuracy (MSE) and lookahead counterfactual fairness (AFCE, UTR).
Method
MSE
AFCE
UIR
UF
0.012 + 0.001
1.084 + 0.000
0% + 0
CF
0.329 +:2 0.019
0.932 + 0.007
14.0% + 0.65%
Ours
5.298 + 1.704
0.124 + 0.086
88.6% + 7.93%
It should be noticed that although CF predictor improves AFCE, it not contradictory Theorem 4.1 because Theorem 4.1 is for LCF not relaxted T.CF.
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E Empirical Evaluation of Theorem 5.4
We generate synthetic dataset following the structural functions 10 We choose = - 2 and d d= 10.
We generated 1000 data samples. The parameters used in the structural functions are displayed as follows.
0.37454012
0.02058449
0.61185289
0.95071431
0.96990985
0.139493866
0.73199394
0.83244264
0.2921 4465
0.59865848
0.21233911
0.36636184
0.15601864
0.18182497
0.45606998
8
7 0.60754485
0. 0.15599452
0.18340451
0.78517596
0.05808361
0.30424224
0.19967378
0.8661761 7615
0.52475643
0.51423444
0.60111501
0.43194502
0.59241457
0.70807258
0.29122914
0.046445041
To construct a predictor g(ÿ) satisfies property (ii) and (iii) described in Theorem 5.4 we set
Table 5: Results on Synthetic Data for Theorem 5.4
comparison with two baselines, unfair predictor (UF)
and counterfactual fair predictor (CF), in terms of accuracy (MSE) and lookahead counterfactual fairness (AFCE, UIR).
Method
MSE
AFCE
UIR
UF
0.036 = 0.002
17.480 + 0.494
0% +
CF
1.400 + 0.098
11.193 + 1.019
35.9% 11.6%
Ours
1.068 = 0.432
0.000 = 0.000
100% = 0
M = P1ú² P2û + P3,
with P1 2n(ayag [[ Table displays the experiment results. In this case, CF improved LCF. However, know that there is guarantee that CF predictor will always improve L.F. And our method, not only can provide the theoretical guarantee, but also achieves better MSE-LCF trade-off.
F Empirical Evaluation On Real-world (Loan) Dataset
We measure the performance of our proposed method using Loan Prediction Problem Datasct (kag). In this experiment, the objective is to forecast the Loan Amount, (Y) of individuals using their Gender (A), income (X1), co-applicant income (X2), married status (X3) and area of the owned property (X4).
The cansal model behind the dataset that there exists an exogenous variable U that represents the hidden financial status of the person. The structural functions are given as
X1 = u Xab
X₂ A U + X 02):
Y W(nAA+ u u +
+ Xaby,
X3 and X4 have no parent nodes. We use the same implementation what we in the experiments for the Law School Success datasct. Table shows the results of our method compared to the basclines. Again, our method can achieve perfect LCF by setting Pı as I. Compared to CF predictor, our method has only slightly larger MSE, but our LCF is greatly improved.
28

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Table 6: Results on Loan Prediction Datset: comparison with two basclines, unfair predictor (UF) and counterfactual fair predictor (CF), in terms of accuracy (MSF) and lookahead counterfactual fairness (AFCE, CIR).
Method
MSE (x10*)
AFCE
UIR
UF
1.352 + 0.835
11.751 + 0.848
3.49% + 7.19%
CF
2.596 +: 0.255
11.792 + 0.790
0% + 0 50% + 3.35%
Ours (pi = T/4)
2.646 0.540
5.896
0.395
Ours (P1 T/2)
2.733 0.197
0.001
0.000
100% +
29


Reviews:
[{"review_id": "qot9uvdNQE", "reviewer": "Reviewer_47kL", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "S1. The paper introduces LCF, which considers fairness in long-term effects. This addresses a known limitation in existing fairness frameworks, where ML predictions satisfying fairness in a static sense can still lead to unintended future disparities. \n\nS2. The authors provide a solid theoretical foundation for LCF, identifying conditions under which ML models can satisfy this fairness notion.\n\nS3. The extension to path-dependent LCF is another valuable contribution, adding flexibility to the application of LCF by focusing on specific causal paths that may influence unfair outcomes.\n\nW1. While the paper focuses on linear causal models and provides comprehensive results for these cases, many real-world systems involve complex, non-linear interactions. This is an important area that should have received more attention.\n\nW2. The theoretical guarantees of LCF rely on assumptions (e.g., invertibility of certain functions) that may not always hold in practical settings. \n\nW3. The experiments primarily focus on a synthetic dataset and the Law School dataset. Incorporating more diverse datasets would better demonstrate the proposed fairness framework, particularly since the density plot comparisons for the Law School dataset show minimal differences across methods.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors address a critical gap in the fairness literature by proposing a fairness notion, Lookahead Counterfactual Fairness (LCF), which accounts for downstream effects of machine learning models on individuals' future outcomes. This work broadens the scope of fairness by integrating long-term consequences into the fairness paradigm. Furthermore, the authors propose a learning method to achieve LCF.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. It would be helpful to include an overview or intuitive explanation for the proof of each theorem, making the theoretical results more accessible.\n\n2. Incorporating more diverse datasets from domains such as healthcare or financial lending could better illustrate the advantages of the proposed methods, particularly in cases where CF underperforms compared to LCF.\n\n3. The predictor g in Eq. 8 is not clear to me. Specifically, why does the input of the predictor include $\\check{Y}$ and $U$, given that $U$ is not an observed variable? How can it be used as input for the predictor?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "uPu3xjQjky", "reviewer": "Reviewer_wQHn", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\nLookahead counterfactual fairness seems strictly better than counterfactual fairness, since we should care more about the outcomes of decisions than the decisions themselves. And since counterfactual fairness is popular, this paper may be of interest to a lot of people in the field.\n\nThe paper seems technically sound, though I haven't checked the details.\n\n## Weaknesses\n\nThis paper inherits many of the substantial weaknesses of counterfactual fairness:\n* the completely implausible assumption that a structural causal model for the system being studied is available\n* the dubious notion of counterfactuals for immutable characteristics like race and gender\n\nThese weaknesses don't rule this paper out, since the original counterfactual fairness paper shares them and is highly influential. If we accept the counterfactual fairness framework as sound, there are still questions about this paper (see below) that I would want answered before I can recommend it for acceptance.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces lookahead counterfactual fairness (LCF), which requires parity over future outcomes between the factual and counterfactual worlds for a decision to be considered “fair”. The paper characterizes when such parity is possible under simple structural causal models, and how to find a LCF-fair predictor.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "## Critical\n\n*Strategic response vs decision consequences*\n\nThere are two downstream effects that could be relevant for the fairness of a decision: the consequences of the decisions (eg default and further financial harm as the consequence of issuing a loan that can’t be repaid), and the strategic response to the existence of the decision making system (eg changing behavior to increase the chance of being issued a loan). The former is pretty interesting; it seems reasonable to make decisions that produce similar consequences for the factual and counterfactual decision subjects. However, this paper mostly focuses on the latter effect from Sec 4 onwards. This has some value as an extension of the strategic classification literature, but to me is much less interesting than a notion of fairness that focuses on the first-order consequences of the decisions on people.\n\nWhy was the decision made to focus on strategic response? Can the machinery developed in this paper be repurposed to ensure LCF with respect to decisions' consequences?\n\nRelatedly, why is the future outcome the same as the predicted outcome? This makes sense if we only care about strategic responses, but the paper is motivated by the need to consider the consequences of the decisions. In the lending context, surely the future outcome of interest is some sort of financial wellbeing measure? Whether the applicant would be creditworthy for a subsequent loan is of course related to their future financial state, but it’s pretty tangentially related compared to something like income or wealth.\n\nIf the paper is going to be motivated by the value of considering the consequences of decisions on people, then I think it needs to be changed to consider those consequences more directly. As it stands the paper is really about strategic classification, which is fine (though much less interesting to me) but not made clear in the early sections.\n\n## Smaller edits/clarifications\n\n> “But in a counterfactual world where the applicant belongs to another group, he/she is not qualified.” \n\nThis needs more explanation. It’s contested as to whether it even makes sense to reason about counterfactuals for immutable characteristics. But even granting that such a thing makes sense, why is this person no longer qualified to receive the loan? Is the mechanism “counterfactually change person to disadvantaged group” -> “person faces education/employment/housing discrimination” -> “person has worse income/wealth” -> “person no longer qualifies”? But if this is what we mean by a counterfactual, why would it be desirable to equalize decisions between counterfactuals as required by CF? For example, it seems obviously undesirable to refuse government support to someone because in the counterfactual world where they hadn’t faced discrimination they would have a higher income, making them ineligible.", "broader_impact_concerns": "Since we never have access to a reasonable structural causal model, counterfactual fairness approaches tend not to be applied to real decisions, limiting concerns about broader impact. To the extent that this paper shifts the discussion towards the consequences of decisions, its broader impact could be positive.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "LYGoVGtZv2", "reviewer": "Reviewer_zW8N", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n- Overall, I think the general notion of lookahead counterfactual fairness is interesting and has strong real-world motivation.\n- The paper is well written with good examples to illustrate the intuitions behind the proposed notion.\n- The extension to path-specific lookahead counterfactual fairness increases the utility of the approach for discrimination following indirect causal paths in future outcomes.\n\n## Weaknesses\n\n- The paper lacks contextualization with other notions of causal fairness. I believe it would be beneficial to include a related works section and compare the proposed definition with other fairness notions and their goals such as Kusner et al (2017), Hu and Zhang (2021), etc.\n- It is not clear to me why Theorem 3.1 holds. What is the intuition behind why a counterfactual fairness constraint violates the LCF definition? There is no proof of this theorem provided in the appendix.\n- In the empirical evaluation, the two datasets are quite different in the produced result. In the synthetic data case, it is evident that the CF predictor induces a large disparity between the factual and counterfactual future outcomes. However, in the law school dataset experiments, the CF already seems to perform quite well since the distributions are very close. In this scenario, why is LCF a better constraint than CF for future outcomes? I am not completely convinced that the proposed algorithm is desirable as opposed to CF from the current experimental results.\n- The proofs are hard to read as there are many errors and inconsistencies. I don’t believe the gradients are computed correctly at several places in the proof.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work proposes the notion of lookahead counterfactual fairness (LCF), a new fairness notion that builds upon counterfactual fairness with a specific constraint of requiring fairness in future outcomes. The idea is motivated by the situation where predicted outcomes may have downstream effects for future outcomes (e.g., bank loans may decrease credit scores that can determine future loan approvals). The authors define conditions under which a predictor satisfies LCF and perform experiments on synthetic and real-world datasets to demonstrate the effectiveness of the proposed notion in ensuring counterfactually fair future outcomes.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "## Requested Changes\n\nCritical\n\n- It would be good to see experiments on more real-world datasets where the CF predictor produces a large disparity in the future outcome, but the LCF predictor has a clear advantage.\n- How is the value of $p_1$ derived? The authors define it w.r.t the parameters of the structural equations, but there is no intuition or derivation of this value. This is important since this value has a large influence on achieving the proposed notion of LCF.\n- I strongly suggest the authors revise all the proofs (and include the proof for Theorem 3.1). There are several errors in the provided proofs in the appendix\n    \n    In A.1:\n    \n    1. I am not sure that the gradients in Eq 13 and 14 are computed correctly. How exactly is this computed?\n    2. The response function $r$ is not defined and made clear for each proof.\n    3. In Eq. 21, the derivative of g w.r.t $\\widehat{y}$ is not computed correctly. Instead of $2p_1\\widehat{y}$, it should be $2p_1\\widehat{y} + p_2$.\n    4. Corollary 4.2 should be Corollary 4.1\n    5. How do Eq. 19 and 21 imply Eq. 22? There seems to be a big leap in this step.\n    \n    Generally:\n    1. It would be helpful to fully define the predictor g in each of the proofs before computing the gradients.\n    \n\nNice to have\n\n- The notion of lookahead counterfactual fairness is similar to the idea of long-term fairness from Hu and Zhang (2021). It would be good to elaborate on the relationship between the two approaches. It seems to me that Hu and Zhang focus on *long-term fairness* in the sequential decision-making regime. However, the formulation is not with respect to counterfactual fairness.\n- I think a general proof sketch of each theorem would be beneficial to get an idea of why the assumptions are important to prove the result.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "pb3uLLFr6R", "reviewer": "Reviewer_KqmR", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\nI think the paper identifies a natural form of counterfactual fairness requirement that cannot be met through previous formulations and that require new methods to be satisfied.\n\nThe proposed method seems to achieve a measurably higher LCF in the real dataset they consider, compared to previous baselines. \n\nThe model of how the prediction affects downstream outcome seems natural and captures the widely studied strategic classification setting.\n\n**Weaknesses**\nThe proof for Thm 3.1 is missing. And the statement is unclear. It currently seems to imply that under the conditions stated, any predictor that satisfies CF violates LCF. But it seems like it should be that there is a predictor satisfying CF but violating LCF. \n\nThe notion of relaxed LCF (Def 4.2) does not seem to be well justified. It seems like it could favor predictors that violate CF by a lot without improving on LCF. One of the methods satisfying LCF (given by Cor 4.1) is empirically evaluated, but justification or evaluation for the other methods (Thm 4.2, 4.3, 4.4) are not provided.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper formulates Lookahead Counterfactual Fairness (LCF) as the objective of achieving counterfactual fairness relative to a downstream outcome of a prediction. They show that achieving counterfactual fairness of a prediction (CF) does not guarantee LCF. Methods for achieving CF typically rely on making predictions based on unobserved features that are not caused by any of the observed features. Even if such features that are predictive for the prediction task exist, they might be causally linked in the causal model of the downstream outcome. Therefore, they cannot be directly used to build LCF predictors. \n\nThe paper proposes new methods for achieving LCF by modeling the causal relationships in the form of structural equations and finds predictors satisfying LCF for linear causal models. The causal relationships model a setting where the prediction influences the downstream outcome by causing individuals to strategically respond to the prediction to maximize their chances of receiving a favorable prediction. \n\nThe paper implements and evaluates an algorithm that entails estimating parameters of the proposed predictor in synthetic and a real dataset.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "[Critical] Clarification on the statement of Thm 3.1 and its proof: Is the statement that under the conditions of the theorem, any predictor that satisfies CF violates LCF. Or is it that there is a predictor satisfying CF but violating LCF?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes look ahead counterfactual fairness,  a fairness notion accounting for the downstream effects of machine models which requires the individual to be counterfactually fair. The conditions under which the proposed fairness motion can be satisfied is derived theoretically. An algorithm is also proposed to achieve the proposed fairness notion. The experiments on both synthetic and real data demonstrate the effectiveness of the proposed method. The paper had some flaws but were fixed in the rebuttal phase. The proposed idea is interesting and has the potential to inspire new works on causal fairness. I thus recommend acceptance of this paper.
Decision: Accept (poster)