A Local Method for Satisfying Interventional Fairness with Partially Known Causal Graphs
Authors: Haoxuan Li
Yue Liu
Zhi Geng
Kun Zhang
Venue: NeurIPS 2024 poster (2024)
Keywords: Interventional fairness
PDAG
MPDAG
Causal effect
Primary area: fairness

Abstract: Developing fair automated machine learning algorithms is critical in making safe and trustworthy decisions. Many causality-based fairness notions have been proposed to address the above issues by quantifying the causal connections between sensitive attributes and decisions, and when the true causal graph is fully known, certain algorithms that achieve interventional fairness have been proposed. However, when the true causal graph is unknown, it is still challenging to effectively and efficiently exploit partially directed acyclic graphs (PDAGs) to achieve interventional fairness. To exploit the PDAGs for achieving interventional fairness, previous methods have been built on variable selection or causal effect identification, but limited to reduced prediction accuracy or strong assumptions. In this paper, we propose a general min-max optimization framework that can achieve interventional fairness with promising prediction accuracy and can be extended to maximally oriented PDAGs (MPDAGs) with added background knowledge. Specifically, we first estimate all possible treatment effects of sensitive attributes on a given prediction model from all possible adjustment sets of sensitive attributes via an efficient local approach. Next, we propose to alternatively update the prediction model and possible estimated causal effects, where the prediction model is trained via a min-max loss to control the worst-case fairness violations. Extensive experiments on synthetic and real-world datasets verify the superiority of our methods. To benefit the research community, we have released our project at https://github.com/haoxuanli-pku/NeurIPS24-Interventional-Fairness-with-PDAGs.

A Local Method for Satisfying Interventional Fairness with Partially Known Causal Graphs
Haoxuan Li Yue Liu
Zhi Geng"
Kun Zhang 2,5
Peking University Mohamed bin Zayed University of Artificial Intelligence 3 Renmin University of China *Beijing Technology and Business University CCrnneiee Mellon University
hxli@stu. pku. edu. en, liuyue statsăruc. edw.. cn zhigeng@btbu. edu. en, kunz1 @cmu. edu
Abstract
Developing fair automated machine learning algorithms is critical in making safe and trustworthy decisions. Many causality-base fairness notions have been pro- posed to address the above issues by quantifying the causal connections between sensitive attributes and decisions, and when the true causal graph is fully known, certain algorithms that achieve interventional fairness have been proposed. How- ever, when the true causal graph is unknown, it is still challenging to effectively and efficiently exploit partially directed acyclic graphs (PDAGs) to achieve inter- ventional fairness. To exploit the PDAGs for achieving interventional fairness, previous methods have been built on variable selection or causal effect identifi- cation, but limited to reduced prediction accuracy strong assumptions. In this paper, we propose a general min-max optimization framework that can achieve interventional fairness with promising prediction accuracy and can be extended to maximally oriented PDAGs (MPDAGs) with added background knowledge Specifically, first estimate all possible treatment effects of sensitive attributes on a given prediction model from all possible adjustment sets of sensitive at- tributes via an efficient local approach. Next, we propose to alternatively update the prediction model and possible estimated causal effects, where the prediction model is trained via min-max loss to control the worst-case fairness violations. Extensive experiments on synthetic and real-world datasets verify the superior- ity of our methods. To benefit the research community, we have reAleased our project at https://gthub..om/ how.ou/uacoomaaaaaaaiiiiaaaaaaii http////wwuuuupni////////////// I -wweettiooa--aainsssssss with-PDAGs
Introduction
Making automated machine learning algorithms fair is critical to producing safe and trustworthy
decisions with different sensitive attributes [Brennan et al. 2009 Dieterich et al., 2016 Agarwal
let al. 2018 Chen et 2018 Chouldechova et al. 2018 Hoffman et al. 2018 Yurochkin et al.) 2019
Li al. 2023). To achieve fair predictions, association- based and causality- faimess notions
have been proposed. Specifically, the former requires statistical independence between the sensitive
attribute and predicted outcome (Dwork al. 2012 Hardt al. 2016 Chouldechoval 2017 Jin
ct al. 2024a b), whereas the later investigates causal effect of the sensitive attribute on the predicted outcome, requiring that the predicted outcome be the same across the real-world without intervention and the counterfactual world with intervention on sensitive attribute Zhang and Bareinboim 2018
Zhang al. 2017a 2018a b Khademi et al. 2019 Galhotra et al. 2022
Yue Liu is the corresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

Despite many algorithms have been developed to achieve causality-based fairness. most of them require the true causal directed acyelic graph (DAG) fully known [Kusner et al. 2017 Nabi and Shpitter 2018 Chiappa 2019 Chikahara et al. 2021 Nevertheless, true causal DAGs and structural equations are usually not directly available in practice [Colombo et al. 2014]. Moreover, without strong assumptions, e.g., linearity (Shimizu et al. [006] and additive noise [Hoyer et al.) 2008, Peters (ct al., 2014], the true causal DAG may not be recoverable from only the observed data, which raises a great challenge to achieve causality-based fairness based on partially DAGs (PDAGs).
To achieve maassality-bsseee fairness under partially known causal graphs, recent approaches can be broadly classified into two categories: variable selection methods (Zuo et al., [022] and causal effect identification methods [Perkovic| [2200 Zuo et al. 2024). Specifically, the variable selection method first adopts causal discovery algorithms to obtain Markov equivalence class of DAGs that encode the same set of conditional independencies from the data, and then classifies the covariate variables into three categories: definite non-descendants, possible descendants, and definite descendants of the sensitive attributes. By noting that prediction model would be counterfactually fair if the prediction model is function of the non-descendants of sensitive attributes [Kusner et al. 2017]. these methods proposed to use only the definite non-descendants or further incorporate possible descendants to make fair predictions. Despite can theoretically guarantee interventional fairness, disregarding the descendants results in a notable decline in performance. Another category of methods proposed to identify causal effects of sensitive attributes the outcome variable directly from the maximally oriented PDAGs (MPDAGs), but relies on strong assumptions for identification.
In this paper, we aim to effectively and efficiently achieve interventional fairness with partially known causal graphs. Different from the previous variable selection and causal effect identification methods, we exploit all variables to ensure relatively high prediction accuracy, as well as does not need to rely on additional strong assumptions for identification. Specifically, we propose a novel local method to partially identify the causal effects of sensitive attributes on the predictor for satisfying the interventional fairness, which does not require a global search of all possible DAGs, but can estimate all possible causal effects using the obtained CPDAG. Inspired by the IDA framework (Maathuis et al., 2009]. we first propose a local algorithm to obtain possible parental sets of the sensitive attributes on the PDAGs, from which we estimate all possible propensities for various cases. We then calculate all possible violations of intervention fairness using all possible propensities. Next, we propose to alternatively update the prediction model and the corresponding estimation of the possible causal effects, where the prediction model is trained via a min-max loss to control the worst-case fairness violations. The validity of our method also holds for MPDAGs with added background knowledge.
The contributions of this paper are summarized as follows:
We propose a general min-max optimization framework to achieve interventional fairness, which enables use all variables to achieve relatively high prediction accuracy, and can be extended to MPDAGs with added background knowledge.
Based on the proposed framework, we provide an efficient algorithm to estimate all possible causal effects of sensitive attribute on predictions for MPDAGs.
We further provide a joint learning approach that alternatively updates the prediction model and the corresponding estimation of the possible causal effects, where the prediction model is trained via a min-max loss to control the worst-case fairness violations.
We conduct extensive experiments on synthetic and real-world datasets to demonstrate the effec- tiveness of our methods in achieving interventional fairness with promising accuracy.
2 Preliminaries
2.1 DAGs, PDAGs, CPDAGs, and MPDAGs
In graph G = (V. E). where V and E represent the node set and edge set in G. we say G is directed, undirected, or partially directed if all edges in the graph are directed, undirected, or a mixture of directed and undirected edges, respectively. The skeleton of G is an undirected graph obtained by removing all arrowheads from G. Given a graph G, an X, is called a parent of X, and X, is called a child of X; if X; X, in G. Also, X; is a sibling of X, if X; X, g. If X; and X, are connected by an edge, they are adjacent. The notation pa( (X,, 9), ch (X,, 9). sib( (0,X, and adj( (X,, G) respectively represent sets of parents, children, siblings, and adjacent vertices of X, in
2

---

G. A graph is termed complete if all distinct vertices are adjacent. A path is a sequence of distinct vertices (X...... .Xx,) where any two consecutive vertices are adjacent. A path is called partially directed from Xk, to Xx; if XR. +- X Xk.+1 does not occur in 9 for any = 1. ....... jj 1. A partially directed path is directed (undirected) if all edges on the path are directed (undirected). A vertex X; is an ancestor of X, and Xj is descendant of X; if there is a directed path from X; to X, or X; = X,- A directed (undirected) cycle is a directed (undirected) path from a vertex to itself. Particularly, a cycle with the number of edges equal to three is called a triangle.
In a directed acyelic graph (DAG), all edges are directed and there no directed cycle. A partially directed acyclic graph (PDAG) may contain both directed and undirected edges without directed cycles. Two DAGs are Markov equivalent if they induce the same set of conditional independence relations [Pear] 1988). A Markov equivalence class, denoted by [9], contains all DAGs equivalent to G. A Markov equivalence class can be uniquely represented by a partially directed graph called completely partially directed acyclic graph (CPDAG) g", in which two vertices are adjacent if and only if they are adjacent in G. and a directed edge occurs if and only it appears in all DAGs in [9] [Andersson et al., 1997 Chickering 2002a] A CPDAG G can be refined to maximally oriented partially directed acyclic graph (maximal PDAG or MPDAG) H by giving background knowledge
Ba consisting of some directed causal relationships between variables in the form X; - X; (Hauser
and Bühlmann, 2012 Eigenmann et al. 2017 Wang et al. 2017 Rothenhäusler et al. 2018 [Meek]
[1995] proved that, with series of orientation rules called Meek's rules, some undirected edges may
be further directed (see Algorithm 4 in Appendix for details), and the resulting graph is an MPDAG. Both DAG and a CPDAG can be viewed special cases of an MPDAG, where the background knowledge is fully known and unknown, respectively.
2.2 Structural Causal Model
We follow Pearl 2009 to define the structural causal model (SCM) as triplet (V, F. to describe the causal relationships between variables. Specifically, V is a set of observable endogenous variables, U is set of latent independent background variables that cannot be caused by any variable in V, and F a set of functions (Ji,..-;Jivi one for each V; E V, such that V, = Si (pa,, U.), where pa; S V [Vi} and U, E U. Notably, the set of equations F induces a directed graph over the variables, here assumed to be a DAG, where the directed causes of Vi represents its parent set. A causal DAG model consists of a DAG G and joint distribution over V such that the distribution can be factorized as P(")...., "I) = II,evv (aa(ppp))))) [Perković et al. 2017].
2.3 Interventional Fairness
Given DAG 9 and two distinet variables X and Y, the causal effect of X Y can be interpreted by the post-intervention distribution of Y intervening on X via do operator [Pearl 1995| 2009.. Formally, given a distribution P(U) over the background variables U, an intervention do x) that force variable X to take certain value x is defined as the substitution of the structural equation X = fz (paz, U2) with X- x, and the post-interventional density of Y is denoted as f(Y y do(X = x)). However, if we only know a CPDAG G or an MPDAG H. the causal effect of X on Y may not be identifiable from observational data (Perković et al. 2015 2017
Wu et al. 2019a b).
Build on the do operator, interventional fairness criterion [Kilbertus et al. 2017 Salimi et al. 2019 requires that given the covariates, intervening the value of the sensitive attribute does not affect the output distribution of the output predictor. Formally, let A, Y, and X denote sensitive attributes, outcomes of interest, and other covariates, and Y be predictor produced by a learning algorithm as Fl a prediction of Y. Without loss of generality we say the predictor Ý is interventionally fair with respect to the sensitive attributes if it satisfies the following condition:
Definition 2.1 (Interventional fairness [Kilhertus et al. 2017). We say the prediction Ý is interven- tionally fair with respect to the sensitive attributes A if the following holds:
p(Ý do(A = a)) = P(Ý = y do(A = =d)).
for all possible values of y and any value that A can take.
Note that our proposed method can be naturally extend to interventional faimess with admissible attributes Xod C X (Salimi et al. 2019], defined as p(Ý = y do(A = a), do(Xad xaa)) P(ý do(A = a'), do(Xad kad)). by observing that (A) and do(Xed) are symmetric and including X.d into A.
3

---

3 A General Min-Max Optimization Framework
3.1 Motivation and Method Overview
Given only observational data, the underlying true causal DAG may not be recoverable without strong assumptions such as linearity [Shimizu et al. 2006] or additive noise [Hoyer et al. 2008 Peters et al.
2014 Instead, we use causal discovery algorithms (Spirtes and Glymour
et al.
2006
Zhang and Hyvärinen 2009 Peters et al. 2014] to obtain a CPDAG that contains that true
causal DAG. To exploit the obtained CPDAG for achieving interventional fairesss, previous methods
have been built on variable selection [Zoo et al. 2022] or causal effect identification [Perkovic] 22020]
Zuo et al. 2024). but limited reduced prediction accuracy or strong assumption for identification.
Differing from the above work, we propose a novel local method to partially identify the causal effects of sensitive attributes the predictor for satisfying the interventional fairness. Interestingly, our approach does not require a global search of all possible DAGs, but can estimate all possible causal effects using the obtained CPDAG. In the following, we first theoretically state the necessary and sufficient condition for discriminating a set to be possible parent set of the sensitive attribute for propose a local method for finding all possible
CPDAG and MPDAG, respectively, from which parent sets of sensitive attributes and estimating the corresponding propensities (Sec. (32).. We then calculate the all possible degrees of intervention fairness being violated using all possible estimated propensities (Sec. (.3). Finally, we further propose a min-max joint learning approach to make the predictor satisfy intervention fairness by controlling for worst-case faimess violation (Sec. 3.4)
3.2 Finding Possible Parental Sets and Estimating Propensities
Given CPDAG obtained from observational data, since enumerating all DAGs is infeasible when the
size of the Markov equivalence class is large (He et al. 2015 /uo et al. 2022 we propose to adopt a novel framework called IDA [Maathuis et al. | parental sets of the sensitive attribute. This provides a more efficient solution since enumerating possible parental sets only requires the local structure around the sensitive attribute. We further extend the above theoretical results to MPDAGs with background knowledge added, and estimate possible propensities by regressing sensitive attribute on each possible parental set.
and He 2020 to only enumerate possible
Definition 3.1 (v-structure). For three distinct vertices X,, X, and Xx. if X; + X2 + Xa and X; is not adjacent to X2 in G. then the triplet (X;, Xj, Xx)) is called a v-structure collided on X,.
Pearl 2009 have shown that two DAGs are equivalent if and only if they have the same skeleton and the same v-structures. Given a CPDAG G* contains all DAGs equivalent to G and a sensitive attribute A, the local structure around A can be divided into three cases: parents pa( 9*) + A, children [Allddrenpa(AA,.. G*) + A. and siblings sib( A. G*) A with undirected edges. Let S(A) be a subset of sib(A, G*), we denote ()(A))+ as DAG that is obtained from CPDAG G by changing all undirected edges {z -7-A4 1A-A, VZ E S(4) into the directed edges (Z + A, YZ E S(A)) as parents, and all of other undirected edges sZ - A, AZ & S(A) into the directed edges with opposite direction {Z< + 4 & S(A)) as children. We say S(A) + A is a possible parental set of the sensitive attribute A for g*, if there exists DAG G in the equivalence class G* with the same directed edges adjacent to A as 9 (S(A)) Then a sufficient and necessary condition for determining whether a set S(A) sib(A, is a possible parent set of the sensitive attribute A is shown in below.
Lemma 3.2 (Maathuis et al. ).09)). Given CPDAG G*, set S(A) C sib(A, G*) is a possible parent set of the sensitive attribute A, and only if there is no more v-structure in 9 ((AA) than G
From the above lemma, given any S(A) S sib(A, we can determine whether S(A) is possible parental set from a local way. In particular, let the induced subgraph of G = (V, E)) over V C V be the subgraph g" = (V', EE') by restricting the edges E on the set of vertices V", where the edge set E' contains all edges with both endpoints in V" Then Lemma 3.2 is equivalent to check whether the induced subgraph of G* over S(A) is complete, i.e., all vertices in the induced subgraph of G* over S(A) are adjacent. This is because if there are two vertices X; and X, in S(A) that are not adjacent, then by Definition 3.1 a v-structure is formed X; + A Xj-
For MPDAG, a key difference compared with CPDAG is the possible generation of a directed trian- gular cycle (e.g., A + X; + X, + A) Fang and He 2020 when incorporating the background

---

A local algorithm for finding possible adjustment sets and estimating corresponding propensity model parameters of the sensitive attribute further using direct causal information.
Algorithm 1:
Input:
Sensitive attribute A, CPDAG g", and consistent direct causal information set Ba-
Construct the MPDAG H. from 9 and Ba using Meek's rules (see Algotirhm for details);
2 Set SA = / and m7 1;
for each ((m) C sib A, H) such that orienting Sim) + A and A + sib( A, H)))s()))) does not introduce any v-structure collided on A or any directed triangle containing A do for number of steps for training the possible propensity model on ((m) do
Sample batch of units {(am. Tm (stm)) ) R===
Update 3(m) by descending along the gradient V 3lo) l( (
end
SA t SA U (pa( [A (()) and m +- m +
end
Output: A set SA of possible adjustment sets (m)) and propensity model parameters ((m))
knowledge and using Meek's rule for orienting undirected edges adjacent to the sensitive attribute A. Motivated by such difference, we entend the theoretical results on CPDAGs to MPDAGs for deter- mining possible parental sets of A, which can also be implemented via a local way.
Definition 3.3 (Direct triangle structure). For three distinct vertices X, and X if if; - X,
X₁ + X,, then the triplet (X,, Xj, Xx) is called a direct triangle structure.
Lemma 3.4 (Fang and He (2020)). Given an MPDAG H. set S(A) sib(A, H) is possible parent set of A. if and only if there is no more direct triangle structure and v-structure in HS(A)-+A than't H.
we can conclude that for a given MPDAG H. it is equivalent to checking whether the induced subgraph of HH over S(A) is complete, as well there does not exist S S(A) and (pa( A, H) (A)) such that - S, otherwise a direct triangle structrue formed
From Lemma 3.4
C' E A - C + S A. This provides efficient way to locally find the possible parental sets of A. Without loss of generality, we denote the set of possible parental sets of A with a total number M as
(A), SA = pal A, H) (S( pa), (A, H) US (A)),...,P((A,, H) U (A))
Next, to estimate p( = yldo(A = a)) in the interventional faimess notion, for each possi- ble parental set pa(A, H)) U ((m))()) (A) with m = 1..... M, we propose to first estimate the cor- Specifically, we use the covariates X re- stricted on PA(A, H. U ((()) (A), denoted as X pla((A, ())(A,)))(())A)) (A), and train the propensity model g(Xl pa(A, (U) (A)i ((m) for estimating propensity P(A pa(A, H. ((m) by minimizing
responding propensity P(A pa(A, H) U (((m)())
((3(m))
1 N Ar log g(X (atpsg//P(PAAPMAa) (a(A.))))())())) + (1 A;) (1 g(X (a(XXpa(AAA)))) pa(4.
Σ
where ((m)) is the propensity model parameter and e!! g(zi|pa(AA, (m) estimated propensity of unit corresponding to the possible parent set pa(A, A, H) U S ((m) (A) for
(A): {(m)) is the
i = I,..., N and
I,.... M. Since CPDAGs can be viewed as special cases of MPDAGs,
without loss of generality, we summarized the proposed local algorithm on MPDAGs in Alg. (see Alg. in Appendix for implementing the proposed local algorithm CPDAGs).
3.3 Estimating and Bounding Interventional Fairness
We then aim to estimate and bound all possible causal effects of sensitive attribute on the predictor
Y. Note that each parental set pa( A, H) ((m) (A) a valid back-door adjustment set in the
back-door adjustment formula [Pear 1995 2009, we have the following identification results.
5

---

Lemma 3.5. With observational data, for m E {1..... M) if & pa(A, H) US (A) then the post-intervention distribution can be calculated from the observational data by:
p( yldo(A== a)) / P Y y = pa A, H) S (A)) dP (pa(A, H. ())
P Y po US
(A))
dP (pa(A, H) (s(
P P(A al pa AA, H) U S (m)) (A))
where P (A (A, H. US (A)) is estimated via g pal (a) )(m)) in Sec. 3.2
)l))
Proof of Lemma 3.5.
p( = yldo(A (A = a))
| P Y yydo( A a). pa(A, H) US( (A) dP (pa( (A, H) (A))
P y = -y/A= = a, p, (, H) (S(()) dP (pa(A, H) U (A)
P y = y. p H. US(
dP pa((A, (A, H) U US(()(()) (A)).
PAA U)
where the first and the third equations are from the conditional probability formula.
From Lemma 3.5 for each possible parental set pa(A, H) U S ((m) ((m) with m 1...., , M, given estimated propensities e!m) in Sec. 3.2 we can train treatment effect estimation model h(z ((m)) = -(()) to estimate p(L = y do(A = 1)) P(ř = y do(A = 0)) by minimizing
(((mm)) 1
As((;;;)) 0) (1 A:) h (zzz ? p/m)
{(m)
where f(a;; 0) = Y, is an outcome predictor parameterized by 0, and l( ((m))= =,!mm p!()) aims to
evaluate the interventional fairness violation of that learned predictor.
3.4 Min-Max Joint Learning Approach
We now aim to train a predictor satisfy interventional fairness. Since the parental set of the sensitive attribute in the true DAG is unknown, we propose min-max learning approach to control for the worst-case interventional faimess violations of the predictor. Specifically, given all possible causal
effects +f of the sensitive attribute A on the predictor Ý in Section 3.3 the prediction model Y = [(x; 0) is trained by minimizing the average prediction error with the worst-case violations of interventional faimess as a penalty term
1 N f(x;0))² + y. max.
min E(0: $00
((M))
(Y
1
(m)
N
N
1=1
s.t.
7 (m)
5C+ + f/m)) i 1
(m)
N, m
M,
7, (m)
> CC (m)
N, m
M,
(()) 2 0, I,..., N, ,,...,, M,
'It is worth noting that this assumption always holds, since the training of the predictor ý cannot affect the origin sensitive attribute A.

---

end Algorithm 2: A min-max optimization approach alternatively updating possible counterfactual treatment effect models and prediction model controlling the worse-case fairness violations.
Input: Sensitive attribute A, outcome of interest Y, and other observable attributes X, possible adjustment sets ((m) and propensity model parameters (f(m) from Alg. while stopping criteria is not satisfied do for m = L.... .M do for number of steps for training the possible counterfactual treatment effect model do
4
Sample a batch of units {(amy -m, Vmm, )===
Update ((()) by descending along the gradient V BCany ( ((((m);
Compute possible counterfactual treatment effects 7. f.(m) h ((zi (p(m)); end
for number of steps for truining the prediction model do
Sample batch of units {(a 3ı, f))
10
Update 9 by descending along the gradient of min-max loss Vol(O; (()) ..... B(A));
11
12
end
13 end
which is convex optimization problem when p!!!)
h(z;; ((m)) is linear. It is equivalent to
N
N
min E(0)
N Σ (Y f(z;0)) + max
Σ (-C-A(m)) ) + -
N
1=1
where y and À are syper-parameeter for trade-off between prediction accuracy and interventional fairness. Since achieving strict interventional faimesss for all individuals, i.e., having zero causal effects of sensitive attribute the predictor, is usually unrealistic and would come the cost of much prediction accuracy, we introduce a slack variable !!!")) for each individual and a pre-specified
threshold C, which penalizes the loss when the estimated causal effect (=((m) C. Note that when implementing the proposed min-max optimization approach, the treatment effect estimation models for evaluating the fairness violations in Section 3.3 and the prediction model controlling for worse-case fairness violations in Section 3.4 sh.4lhould be updated alternatively, which can be viewed as an iterative process of interventional fairness evaluation and policy improvement of the prediction model. We summarized the whole min-max optimization algorithm in Alg, 2
Experiments
In this section, both synthetic and real-world experiments conducted evaluate the prediction accuracy and fairness of our approach. The root mean squared error (RMSE) between Y and Ý is used measure the prediction performance, and the RMSE between Ý [do a) and Yldo(A a') is used to measure the violation of the interventional fairness, named "unfairness".
Baselines. We consider six baseline prediction models: (1) Full uses all attributes, (2) Unaware uses all attributes except the sensitive attribute, (3) Oracle uses all attributes that are non-descendants of the sensitive attribute given the ground-truth DAG, (4) FairRelax uses all definite con-descendants and possible descendants of the sensitive attribute in a CPDAG (or an MPDAG), (5) Fair uses all definite non-descendants of the sensitive attribute in a CPDAG (or MPDAG), and (6) e-IFair uses all attributes and implement the constrained optimization problem.
Synthetic Study. Synthetic data are generated from linear structural equation model based ground truth DAG. Specificallyy we first randomly generate a DAG with d nodes and 2d directed edges according the Erdōs- Rényi (ER) model with d {10, 20, 30, 40) in experiment. Following the previous studies [Zuo et al., 2022 2024], the path coefficients Byk of directed edges X, + X sampled from uniform distribution 0[-22 -0.5] U 00.5. 2))- The data generated using noise
X1 = Exjepa Xx) 35 + G, = 1,..., where pa (Xx) represents the parent nodes of Xt.
7) with 7 E 2.5), and n is the sample size, which is set to 1,000 in our
7

---

Table 1: Average RMSE and unfairness for synthetic datasets on the held-out test set.
Noise 15
NODE
10, DDGE 20
NODE
20,
EDGE
40
NODE
30.
EDGE
60
NODE 40,
EDGE
80
Method
RMSE 4
Unfaimess
RMSE 1
Unfainess
RMSB 4
Unfairness
RMSE 4
Unfairness J
Oracle
0.757 +: 0.349
0.000 a 0.000
0.579 a: 0.245
0.000 4: 0.000
00,571 +: 0.194
0,000 a: 0.000
0.578 a: 0,200
0.000 d: 0.000
Full
0.576 + 0.218
0.195
0.232
0.494
0.133
0,095 +: 0.128
0.542 +: 0.196
0.063
0,088
0.538
0,185
0.067
0.113
Unaware
0.587 4: 0.219
0.150 = 0.208
0,498
0.134
0.058 1: 0.095
0.544 4: 0.196
0.050
0.076
0.540 1: 0.183
0.043
2: 0.066
FairRelax
0.653 +: 0.255
0.142 = 0.201
0.586
=: 0.217
0.055
4: 0.092
0.603
4: 0.241
0.045
0.068
0.611 4: 0.254
0.041
:: 0.068
Fair
0,747
0:200
0.627
0:2223
0,050
0.074
0.661 0.263
0.043
0.067
0.630
0.2292
0.038 0.059
e-IFair
0.644 = 0.262
0.137
= 0.187
0.570
0.215
0.056
1: 0.080
0.589 4 0.239
0.048
0.065
0.609
2: 0.241
0.040
2: 0.063
Ours
0.623 + 0.210
0.119 + 0.175 0.561
0.126
0.049 + 0.073
0.597 + 0.185
0.037
0.054
0.606 + 0.178
0.036 + 0.054
Noise = 2.5
NODE-
10, EDGE 20
NODE
20, EDGE 40
NODE 30,
EDGE 60
NODE 40, EDGE -80
Method
RMSE
Uufaimess
RMSE
Unfaireess
RMSB J
Unfairness
RMSE
Unfairness
Oracle
0.729 1: 0.344
0.000 a: 0.000
0.874
0.625
0.000 1: 0.000
0.801 +: 0.497
0.000 0.000
0.820 1: 0.472
0.000 :: 0.000
Full
0.667 + 0.274
0.185 + 0.189
0.761
0.440
0.150 + 0.425
0,736 + 0,417
0.075
0,087
0.729 :: 0.334
0.110 +: 0,188
Unaware
0.674 0.276
0.094
0.772
0.457
0.062
0.417
0.043
0.733
0.041 :: 0.079
FairRelax
0.738 +: 0.283
0.059
= 0.077
0.898 0.600
0,050 4: 0.119
0.831 + 0.487
0.830
0.040
0.791
+: 0.410
0.040 +: 0.079
Fair
0.774 + 0.274
0.052
0.067
0.937
0.642
0.046 + 0.118
0.891 + 0.550
0.029
0.039
0.816 0.411
0.089 + 0.079
e-IFair
2: 0.275
0.082
0.872
0,016
0.418
0.045
0.789
0.039 :: 0.078
Qurs
0.719 + 0.280
0.049 + 0.073|
0.857
0.466
0.045 + 0.090
0.823 + 0.413
0.023 0.031
0.788
+ 0.334
0.038 + 0.070
1.0
0.14
0.16
0.12
0.04
0.04
0.5
0.00
0.01 0.56 5:32
0.36
0.20
0.00
2.00
0.06
0.05
0.32
5.35
0.20
0.5
0.
0.4
0.
3.2
1.5
2.5
0.00
3.0
0A
0.0
12
2.5
2.
typerparmmater
Hyper pa mometer
Hyper parameter
(a) RMSE
(b) Unfaimess
(c) RMSE
(d) Unfairness
Figure 1: Performance under varying hyper- parerr-arameterss C and X on RMSE and unfairness.
experiment. Next, we use the PC algorithm in the causal-learn package to learn a CPDAG. Then we randomly select two nodes as the outcome Y and the sensitive attribute A, respectively. We sample A from a Bernoulli distribution with probability o oixx, Epa(A) (A)(A) (A) 3 + G), where (-) denotes the sigmoid function. The proportion of training data and test data are set to 0.8 and 0.2, respectively.
Performance Comparison. Table I shows the results of baselines and our approach. First, Full and Unaware perform better on RMSE, while Fair, FairRelax, e-IFair, and our approach have a significant advantage on unfairness. Note that our approach outperforms Fair, FairRelax, and c-IFair in all scenarios on both RMSE and unfairness metrics, because the proposed method makes predictions with all attributes and controls unfairness by the adjustment sets, whereas Fair and FairRelax can hardly find the true descendants of the sensitive attribute and e-IFair can hardly find the true causal effects when the learned CPDAG is not accurate in practice. In addition, Figure [ shows the change in RMSE and unfairness as C and increase. When C is increasing, RMSE is decreasing significantly, while unfairness is increasing. Because the larger C' is, the looser the control of causal effects, which is beneficial for prediction performance but hurts fairness. Similar arguments hold for A, where larger À will increase the cost of fairness violations in the optimization problem, thus benefiting fairness but hurting prediction accuracy.
MPDAG with Background Knowledge. Given the CPDAG, we randomly select a certain percentage of the directed edges in the true DAG as background knowledge and impose it on the already learned CPDAG. For example, if A + B is selected from the true DAG, we add this directed edge to the learned CPDAG regardless of the original relationship between A and B in the CPDAG to obtain an MPDAG and then adjust the MPDAG according to the Meek's rule. Figure 2 shows the effect of background knowledge on performance. As the background knowledge increases, the RMSE of Fair and FairRelax increases and the unfairness decreases significantly because more background knowledge forces Fair and FairRelax to have fewer nodes to make predictions. For our approach and F-ir, both prediction and unfairness performance become better as the background knowledge ratio increases, which is attributed to the more accurate identification of the possible adjustment sets and the causal effect. Note that approach stably outperforms e-IFair under varying background knowledge ratios. In addition, Trepot/s the change of precision and recall for finding adjustment sets with increasing background knowledge ratio.
Effect of Different Number of Parent Node. We evaluate the RMSE and unfairness performance with varying numbers p of parent nodes of the sensitive attribute. The results are shown in Table 3
8

---

Full
Ours
c-IFair
Fair
Full
Ours
c-IFair
Fair
1.1
0.20
1.0
0.10
Unfairness
0.08
RMSE
0.9
0.06
0.04
0.7
0.02
0.00
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
Background knowledge ratio
Background knowledge ratio
(a) RMSE
(h) Unfairness
Figure 2: RMSE and unfairness performance under varying background knowledge ratio.
Table Average precision and recall for finding the adjustment sets in MPDAG.
Background Knowledge
0%
10%
20%
30%
40%
50%
Precision f
0.438 = 0.489
0.438 = 0.489
0.466 0.492
0.580
0.486
0.642 = 0.471
0.742 = 0.437
Recall f
0.265 + 0.353
0.265 + 0.353
0.274 + 0.350
0.329 0.355 0.353 + 0.347 0.400 + 0.346
Table 3: Average RMSE and unfairness for synthetic datasets the held-out test set with different numbers p of parent nodes of the sensitive attribute.
Unaware
Fair
FairRelax
e-IFair
Ours
RMSE (p = 0)
0.601 1 0.235
0.642 1 0.241
0.635 1: 0.240
0.633 1 0.237
0.635 1: 0.233
Unfairness (p 0)
0.063 + 0.082
0.041 = 0.051
0.045 + 0.050
0.041 + 0.021
0.035 = 0.016
RMSE (p 2)
0.528 + 0.298
0.601 + 0.291
0.597 + 0.291
0.590 + 0.206
0.581 + 0.201
Unfairness 2)
0.111 + 0.090
0.099 = 0.075
0.100 1:00+:000 0.075
0.100 0.088
0.096 + 0.107
RMSE (p- 4)
0.718 + 0.281
0.860 0.261
0.834 0.253
0.818 + 0.237
0.792 0.228
Unfairness (p- 4)
0.129 + 0.073
0.113 = 0.058
0.120 = 0.050
0.110
0.103 = 0.086
As increases, due to the presence of more backdoor paths, the RMSE performance and unfairness performance of methods decreases, however, our method still outperforms the baseline methods.
The Performance on the Classification Problem. We conduct more experiments to examine the proposed method can be applied classification problems. Specifically, we modify the data generation process (DGP) to clip the outcome variable Y to 1 if Y O, and to 0 if Y < 0, and the rest DGP remains the same. this scenario, we adopt AUC the evaluation metric instead RMSE. The experiment results are shown in T We find that both Full and Unaware perform better on AUC, while Fair, FairRelax, e-iir and our approach perform better on unfairness. Note that our approach outperforms Fair, FairRelax, and e- e-ir in all scenarios for both AUC and unfairness.
Case Study. The sensitive attributes contained in many widely used datasets for fair machine learning have no parent nodes, such as sex in the Adult datase (Kohavi 1996] and race in the COMPAS datase [Angwin al. 2022). Because sex, race, and age cannot be affected by other collected features, we further consider the Open University Learning Analytics Dataset (OÚLAD) datase Kuzilek et al. 2017). in which disability is treated the sensitive attribute and final grade is treated as the outcome. The COMPAS dataset contains 6,172 units with 9 attributes such as gender and number of priors, the Adult dataset contains 48,88 842 units with 14 attributes such as age, education, and race, and the OULAD dataset contains 32,593 units with 11 attributes including demographic information such as gender, age, education level, etc. For this case study, we first learn CPDAG from the raw data using the PC algorithm in the causal-learn package and obtain an MPDAG with the background knowledge such as sex can not caused by other attributes. Second,
"htpps//archive.ics.uu htt.:s..cc..cuu/daasss///aaaaaa
hhtps://wwwkkagett comdlantseevdaanoffrrooopss
archive www.hiiiw.s.......dddaaaaeeeeeea wtwpa//wwww....oooooooaoaeeeeeee

---

Table 4: Average AUC and unfairesss for synthetic datasets the test set on classification problem.
Nuise 2,5
NODE
10, EDGE
NODE
20, EDGE
NODE
30, EDGE
60
NODE
40, EDGE 80
Method
A
Unfairness
A
Unfairness
AUC1
Unfaimess
AUC
Unfairness J
Oracle
10815 0.0%4
0.000 + 0.000
0.805 + 0.148
0,000 0.000
0.819 + 0.149
0.000 0.000
[0818 + 0.087
0,000 + 0,000
0.845 0.071
0,038 0.051
0.115
0.855
0.087
00842
0.106
Unaware
0.833 0.020
0.017 +: 0.021
0.886 + 0.080
0.105 + 0.155
0.853 +: 0.114
0.076
0.082
0.837 0.090
0.113 + 0.121
FairRelax
0,255 + 0.057
0.017 + 0.021
0.857 : 0.130
0.084
+ 0.148
0.845
+ 0.117
0.074
+ 0,106
0.824 = 0.082
0,112 + 0.119
Fair
0.060
0,015 0.021
0.779 0.200
0.082
0.143
0.844
0.116
0.074
0.106
0,822
0.994 0.122
e-IFair
0.43 + 0.049
0.018
:: 0.017
0.883
= 0.081
0.081
+ 0.087
0.843
4: 0.115
0.068
: 0.088
0.833 = 0.085
0.098 + 0.105
Ours
0.844 0.051
0.015 0.016 0.886 1: 0.077 0,080 0.10 0.855 1: 0.109
0.069 0.094 0.835 0.088
0.089 +: 0.119
Table 5: Real-world experiment results on the COMPAS, Adult, and OULAD datasets. For the COMPAS dataset, the sensitive attribute is race, and for the Adult dataset, the sensitive attribute is sex. Both of the sensitive attributes have no parent nodes. For the OULAD dataset, the sensitive attribute is disability, which can have parent nodes.
COMPAS
Full
Unaware
FairRelax
Fair
c-lFair
Ours
RMSE
0,256 + 0.022
0.261 + 0.023
0.261 + 0.023
0.263 + 0.022
0.235 + 0.091
0.219 + 0.020
Unfaimess
0.273 = 0.048
0.269 + 0.052
0.260 + 0.045
0.238 + 0.045
0.190 = 0.120
0.179 = 0.011
Adult
Full
Unaware
FairRelax
Fair
c-lFair
Ours
RMSE
0.433 0.024
0,436 + 0.024
0.607 + 0.131
0.611 0.128
0.413 0.010
0.375 = 0.019
Unfaimess
0.506 + 0.021
0,409 = 0.029
0.209 = 0.205
0.187 0.205
0.155 + 0.009
0.171 = 0.021
OULAD
Full
Unaware
FairRelax
Fair
e-IFair
Ours
RMSE
0.502 0.041
0.502 + 0.042
0.503 + 0.041
0.503
0.041
0.499
0.041
0.491 = 0.040
Unfaimess
0.088 1 0.024
0.031 1 0.058
0.029 1 0.023
0.029 0.023
0.027 1 0.020
0.024 1 0.018
2.5
0.3
0.0
0.5
25
01 9A
(F-0),, RMSE 0.502
(ffrrt, 0,503
Iffiinflelax, RNSE 0.503
Ptylfair), AMSE 6.499
P0Duss) RMSE 0.491
(a) Full
(b) Fair
(c) FairRelax
(d) e-
(e) Ours
Figure 3: Density plot of the predicted |do(AA = a) and Ý Ido(AA a') on OULAD data.
we randomly generate DAG as the ground-truth from the learned MPDAG. After obtaining the DAG, we then divide the data into 100 random batches, and for each batch, we learn an MPDAG from the observed data and background knowledge. The path coefficients linear regression and regard the residual of the regression as noise. The subsequent steps are the same as in the synthetic study. The experiment results are shown in Table with density plots in Figure First, both e- e-lr and our method demonstrate the superiority of our approach in both prediction performance and fairness compared to other baselines. In addition, our method stably outperforms e- e-IFair, further validating the effectiveness of the proposed min-max joint learning approach.
determined based on
5 Conclusion
This paper aims achieve interventional fairness from observational data when the causal graph is unknown or partially known. Interestingly, we show it is actually sufficient to enumerate all possible parental sets of the sensitive attributes via local approach, instead of enumerating all DAGs at high computational cost. We then propose a general min-max optimization framework to achieve interventional fairness that is easy applicable to CPDAGs and maximally oriented PDAGs (MPDAGs) with the added background knowledge. One limitation of our approach is due to the proposed approach relying on a CPDAG given by the causal discovery algorithm and estimations of the propensities, which may lead to mild violations of interventional faimess when the CPDAG or estimates are inaccurate. Another possible limitation, which also serves as a future research direction, is to achieve interventional faimess in the presence of hidden variables with partially known DAGs.
10

---

Acknowledgments and Disclosure of Funding
The authors thank the anonymous reviewers for their valuable comments. This work was supported in part by National Natural Science Foundation of China (623B2002, 12201629). Yue Liu was supported by the MOE Project of Key Research Institute of Humanities and Social Sciences (22IID110001).
References
Alekh Agarwal, Alina Beygelzimer, Miroslav Dudík, John Langford, and Hanna Wallach. reductions approach to fair classification. In ICML, 2018.
Steen A Andersson, David Madigan, and Michael D) Perlman. A characterization of markov equiva- lence classes for acyelic digraphs. The Annals of Statistics, 25(2):505-54) 1997.
Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner. Machine bias. Ethics of data and analytics, pages 254-264. Auerbach Publications, 2022.
Tim Brennan, William Dieterich, and Beate Ehret. Evaluating the predictive validity of the compas risk and needs assessment system. Criminal Justice and behavior, 36( 36(1):21-40, 2009.
Irene Chen, Fredrik D Johansson, and David Sontag. Why is my classifier discriminatory? In NeurIPS, 2018.
Silvia Chiappa. Path-specific counterfactual fairness. In AAAI, 2019.
David Maxwell Chickering. Learning equivalence classes of Bayesian-netwof structures. Journal of Machine Learning Research, 2:445-498, 2002a.
David Maxwell Chickering. Optimal structure identification with greedy search. Journal of Machine Learning Research, 3:507-554, 2002b.
Yoichi Chikahara, Shinsaku Sakaue, Akinori Fujino, and Hisashi Kashima. Learning individually fair classifier with path-specific causal-effect constraint. In AISTATS, 2021.
Alexandra Chouldechova. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. Big Data, 5(2):153-163, 2017.
Alexandra Chouldechova, Diana Benavides-Prado Oleksandr Fialko, and Rhema Vaithianathan. A case study of algorithm-assised decision making child maltreatment hotline screening decisions. In ACM FAccT, 2018.
Diego Colombo, Marloes H Maathuis, et al. Order--ndeeenddet constraint based causal structure learning. Journal of Machine Learning Research, 15(1):374137822, 2014.
William Dicterich, Christina Mendoza, and Tim Brennan. Compas risk scales: Demonstrating accuracy equity and predictive parity. Northpointe Inc, 7(7.4):1, 2016.
Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. Fairness through awareness. In ITCS, 2012.
Marco F Figenmann, Preetam Nandy, and Marloes H Maathuis. Structure learning of linear gaussian structural equation models with weak edges. In UAI, 2017.
Zhuangyan Fang and Yangbo He. IDA with background knowledge. In UAI, 2020.
Sainyam Galhotra, Karthikeyan Shanmugam, Prasanna Sattigeri, and Kush R Varshney. Causal feature selection for algorithmic faimess. In SIGMOD, 2022.
Moritz Hardt, Eric Price, Eric Price, and Nati Srebro. Equality of opportunity in supervised learning. In NewrIPS, 2016.
Alain Hauser and Peter Bühlmann. Characterization and greedy learning of interventional markov equivalence classes directed acyelic graphs. The Journal of Machine Learning Research, 13(1): 2409-2464, 2012.

---

Yangbo He, Jinzhu Jia, and Bin Yu. Counting and exploring sizes of Markov equivalence classes of directed acyelic graphs. Journal of Machine Learning Research, 16:2589-2609 2015.
Mitchell Hoffman, Lisa Kahn, and Danielle Li. Discretion in hiring. The Quarterly Journal of Economics, 133(2):765 800, 2018.
Patrik Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Bernhard Schölkopf. Nonlinear causal discovery with additive noise models. In NeurlPS, 2008.
Jinqiu Jin, Haoxuan Li, and Fuli Feng. On the maximal local disparity of aaire classifiers. In ICML., 2024a.
Jinqiu Jin, Haoxuan Li, Fuli Feng, Sihao Ding, Peng Wu, and Xiangnan He. Fairly recommending with social attributes: flexible and controllable optimization approach. In NeurlPS, 2024b.
Aria Khademi, Sanghack Lee, David Foley, and Vasant Honavar. Fairness in algorithmic decision making: An excursion through the lens of causality. In WWW, 2019.
Niki Kilbertus, Mateo Rojas Carulla, Giambattista Parascandolo, Moritz Hardt, Dominik Janzing, and Bernhard Schölkopf. Avoiding discrimination through causal reasoning. NeurIPS, 2017.
Ron Kohavi. Scaling up the accuracy of naive-bayes classifiers: decision-tree hybrid. In SIGKDD, 1996.
Matt Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva. Counterfactual faimess. In NeurIPS, 2017.
Jakub Kuzilek, Martin Hlosta, and Zdenek Zdrahal. Open university learning analytics dataset. Scientific Data, 4(1):188, 2017.
Haoxuan Li, Chunyuan Zheng, Yixiao Cao, Zhi Geng, Yue Liu, and Peng Wu. Trustworthy policy learning under the counterfactual no-harm criterion. In JCML, 2023.
M. H. Maathuis, M. Kalischand, and P. Bühlmann. Estimating high-dimensional intervention effects from observational data. The Annals of Statistics, 37(6A)33133-3164 2009.
Christopher Meek. Causal inference and causal explanation with background knowledge. In UAI, 1995.
Razich Nabi and Ilya Shpitser. Fair inference on outcomes. In AAAI, 2018.
Judea Pearl. Probabilistic reasoning in intelligent systems: nelworks of plausible inference. Morgan Kaufmann San Matco, CA, 1988.
Judea Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669-688, 12 1995.
Judea Pearl. Causality. Cambridge University Press, 2009.
Emilija Perkovic. Identifying causal effects in maximally oriented partially directed acyclic graphs. In UAI, 2020.
Emilija Perković, Johannes Textor, Markus Kalisch, and Marloes Maathuis. A complete generalized adjustment criterion. In UAI, 2015.
Emilija Perković, Markus Kalisch, and Maloes H Maathuis. Interpreting and using epdags with background knowledge. In UAI, 2017.
Jonas Peters, Joris M Mooij, Dominik Janzing, and Bernhard Schölkopf. Causal discovery with continuous additive noise models. Journal of Machine Learning Research, 2014.
Dominik Rothenhäusler, Jan Ernest, and Peter Bühlmann. Causal inference in partially linear structural equation models. The Annals of Statistics, 46(6A))2904-2938 2018
Babak Salimi, Luke Rodriguez, Bill Howe, and Dan Suciu. Interventional faimess: Causal database repair for algorithmic fairness. In SIGMOD, 2019.
12

---

Shohei Shimizu, Patrik O Hoyer, Aapo Hyvärinen, Antti Kerminen, and Michael Jordan. A linear non-gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7(10), 2006.
Peter Spirtes and Clark Glymour. An algorithm for fast recovery of sparse causal graphs. Social Science Computer Review, 91):622-7, 1991.
Peter Spirtes, Clark N Glymour, Richard Scheines, and David Heckerman. Causation, prediction, and search. MIT press, 2000.
Caroline Uhler, Garvesh Raskutti, Peter Bühlmann, and Bin Yu. Geometry of the faithfulness assumption in causal inference. The Annals of Statistics, pages 436-463, 2013.
Yuhao Wang, Liam Solus, Karren Yang, and Caroline Uhler. Permutation- based causal inference algorithms with interventions. In NeurIPS, 2017.
Yongkai Wu, Lu Zhang, and Xintao Wu. Counterfactual fairness: Unidentification, bound and algorithm. In IJCAI, 2019a.
Yongkai Wu, Lu Zhang, Xintao Wu, and Hanghang Tong. PC-Fairness: A unified framework for measuring causality-based faimess. In NeurIPS, 2019b.
Mikhail Yurochkin, Amanda Bower, and Yuckai Sun. Training individually fair ml models with sensitive subspace robustness. In ICLR, 2019.
Junzhe Zhang and Elias Bareinboim. Fairness in decision- making the causal explanation formula. In AAAI, 2018.
Kun Zhang and Aapo Hyvärinen. On the identifiability of the post-nonlinear causal model. In UAI, 2009.
Lu Zhang, Yongkai Wu, and Xintao Wu. Achieving non-discrimination data release. In KDD, 2017a
Lu Zhang, Yongkai Wu, and Xintao Wu. A causal framework for discovering and removing direct and indirect discrimination. In IJCAI, 2017b.
Lu Zhang. Yongkai Wu, and Xintao Wu. Achieving condiisrrminntionn in prediction. In IJCAI, 2018a.
Lu Zhang, Yongkai Wu, and Xintao Wu. Causal modeling- discrimination discovery and removal: criteria, bounds, and algorithms. IEEE Transactions on Knowledge and Data Engineering, 31(11):2035-200 2050, 2018b.
Aoqi Zuo, Susan Wei, Tongliang Liu, Bo Han, Kun Zhang, and Mingming Gong. Counterfactual fairness with partially known causal graph. In NeurlPS, 2022.
Aoqi Zuo. Yiqing Li, Susan Wei, and Mingming Gong, Interventional fairness on partially known causal graphs: A constrained optimization approach. In ICLR, 2024.
13

---

Broader Impacts
This paper proposes general min-max optimization framework that can effectively achieve interven- tional fairesss when the true causal graph is unknown or partially known. In contrast to statistical fairness, interventional fairness considers possible counterfactual decision-makings. not just based on the observed data. The implications of achieving interventional fairness in algorithms where the true causal graph is unknown or partially known are mainly the following aspects. First, reducing bias: machine learning models may learn and reflect bias in the training data and unconsciously apply this bias to individuals in their predictions. interventional fairesss helps reduce this possible bias. Second, improve fairness: if we achieve interventional fairness partially known causal graphs, our models will be prevented from treating unfairly because of the sensitive attributes. Third, enhancing trust:
able process data in fairer way, people's trust in those algorithms increases. This is critical in many areas, such as healthcare, finance, and justice. Fourth, promote policy making: understanding and addressing interventional fairness issues in algorithms can help policy makers better understand and regulate these technologies to ensure their fairness and transparency in practice. In nutshell, studying how effectively achieve interventional fairness in scenarios such as unknown causal graphs or the presence of hidden variables is both challenging and socially significant, and deserves more effort.
algorithms
A More Discussion on the Previous Work
To tackle the above problem, recent work [uoo al. 2022 proposes observed data to first classify variables into three categories: definite non-descendants, possible descendants, and definite descendants of the sensitive attributes. Next, by noting that prediction model would be counterfactually fair the prediction model is a function of the non-descendants of sensitive attributes [usner et al. 2017], shown in Table FAIR method proposed to use only the definite non-descendants, and FAIRRELAX method further incorporates possible descendants.
Table 6: Comparison of methods to achieve interventional fairness from PDAGs. Both FAIR and FAIRRELAX employ a two-stage approach: they first learn a CPDAG from observed data, and then make prediction with the definite non-descendants (and possible descendants) of the sensitive attribute. Our method alternatively updates the predictions using all variables and possible counterfactual treatment effects via min-max optimization.
Variables (example in Figure 4(b)
FAIR
FAIRRELAX
OURS
Definite non-descendants (0)
Possible descendants (X1, X2)
Definite descendants (0)
However, both FAIR and FAIRRELAX forbid the use of definite descendants for model prediction, which greatly compromises the prediction accuracy. In particular, the sensitive attribute usually an inherent nature of data, making most of the attributes are its descendants (Wu et al. 2019a
Undirected edges
X1
X1
A
Y
A
Y
X
X
(a) A sample true DAG
(b) CPDAG of (a)
Figure 4: A toy example for illustration: FAIR has no available variables for prediction; FAIRRELAX
uses Xx) without further fairness constraint; OURS uses X1, X2) with min-max constraint bounding all possible counterfactual treatment effect.
proceed with a toy example for illustration: Figure 4 shows a sampled DAG as the ground-truth, and given the observed data, FAIR and FAIRRELAX algorithms first learn Markov equivalence

---

Algorithm 3: A local algorithm for finding possible adjustment sets and estimating corresponding propensity model parameters of the sensitive attribute A.
Input:
Sensitive attribute A, CPDDGG
Set SA and m 1;
2 for each ((m) C sib( (*) such that orienting (m)) + A and A sib(A, does not introduce uny v-structure collided on A do
for number of steps for training the possible propensity model on ((m) do
Sample batch of units {amm Amglstin )) k=
Update 36m) by descending along the gradient (lm) ((B(m)); S(m))
end
SA < SAUSE (m and m m
end
Output: set SA of possible adjustment sets (m)) and propensity model parameters 3(m))
Rule
Rule 2
Rule 3
Rule 4
Figure 5: The visualization of four rules of Meek's criteria. If the graph on the left-hand side of a rule is induced subgraph of PDAG G. then orient the undirected edge such that the resulting subgraph the one on the right-hand side of the rule.
class of DAGs that encode the same set of conditional independencies from the data, also known as completely partially directed acyclic graph (CPDAG), as shown in Tigureeib One on hand, sensitive attribute A is a empty set, thus FAIR is unable to return
the definite non-descendants valid prediction model. On the other hand, the possible descendants of the sensitive attribute A are X2), thus FAIRRELAX uses both X1 and X2 to predict Y by minimizing the empirical risk without imposing any further fairness constraint. However, such relaxation would lead to serious violation of interventional fairness, due to the nodes used (X, and X₂ in this example) for outcome regression might be descendants of the sensitive attribute A in the true DAG, in Figure 4 a).
When the true causal graph is unknown, the best of our knowledge, Zuo al. [2022) performed the first work to obtain interventional fairness predictor an MPDAG, which focuses utilizing the properties of the causal graph (Level in Kusner et al. (017 - to make predictions with the definite non-descendants (and possible descendants) of the sensitive attribute, as shown in Table
6 However, further incorporating the descendants of sensitive attribute into the predictor may also achieve interventional fairness by "cancelling out" the counterfactual treatment effects, which utilizes the observed variables more sufficiently and further improves the accuracy the prediction.
We provide an intuition for the rationality and the advantages of using all variables by adopting the toy example in Figure Suppose the structural equations in Figure 4a) are: A = U. .y, A + U2. X₂ A U2 and = 2XX, + X₂ + Uy which satisfies the faithfulness assumption [Uhler ct al. 2013). In such a case, as discussed before, the FAIR algorithm proposed in Zuo et al. [022 prevents all variables from predicting Y. while the FAIRRELAX algorithm uses both X1 and X₂ to predict Y without imposing any fairness constraints, and therefore cannot achieve causal fairness, since X1 and X2 are descendants of A. To achieve more accurate predictions with interventional
15

---

Algorithm 4: Constructing the MPDAG H. from CPDAG G and By Using Meek's 's Rules Input: A CPDAG G*, set of directed edges Ba. Output: An MPDAG H or FAIL..
Set H = =
2 while Ba * 0 do
Choose an edge - from Ba;
B² = Ba {u + v}:
u - u or u
H. then
Orient - v in Mt;
Orient the other edges under the Meek's rules in Figure
else end
return FAIL;
end
12 return MPDAG H;;
fairness guarantees, one may notice that a function of X1 X2 can be used predict Y. On the one hand, this is strictly counterfactually fair due to the fact that X - X2 = U1 U2, which is independent of the sensitive attribute A. On the other hand, this is informative for predicting due to Cov(X1 - X2, Y) 2 Var(U1) - Var(U>) + O.
However, the true DAG and the corresponding structural equations unknown in many real-world scenarios, which poses a great challenge to estimate the possible counterfactual treatment effects. To address this problem, intuitive approach is first find Markov equivalence class over all vertices, which can be achieved using standard causal discovery methods, e.g., PC (Spirtes et al. 2000 and GES [Chickering] 20022), and then globally enumerate all the possible DAGs in the equivalence class and estimate their causal effects for each. However, as discussed in Section of Zuo et al. 2022), this intuitive way enumerate all DAGs is computationally expensive and unrealistic.
16

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: In the abstract and the introduction, accurately describe the paper's contri- butions from four aspects.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification:
We present two limitations in the conclusion.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g.. independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged judgment and recognize that individual actions in favor of transparency play impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
the paper. The authors should use their best
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
17

---

Justification:
We present detailed proofs in the Appendix.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: The codes are provided the supplemental material, and the results can be reproduced by running the provided code.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, No answer this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how reproduce that algorithm. (b) If the contribution is primarily
model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome deseribe the particular way they provide for reproducibility, In the case of closed-source models, it may be that access to the model is limited in some way (e.g., registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
18

---

Answer: [Yes]
Justification:
The codes are provided in the supplemental material and the used dataset is public available.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips.cem cc/
Polyyyy for more details.
While encourage the release of code and data, understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https
7/nips. //nips.cc/ppl http/ubllcgguuueeee
ltt.ssabmisiiooooioooooooooo onPolicy) for more details.
The authors should provide instructions data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental Setting/Details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification:
The experimental details are presented in the experiment section.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment Statistical Significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer:
[Yes]
Justification:
The experimental results are reported with error bars.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the enror bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
19

---

It should be clear whether the error bar is the standard deviation or the standard error of the mean.
is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative enror rates).
If error bars reported in tables plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
Justification: There no requirement for the experiments, and we only use the laptop CPU all experiments.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (c.g., preliminary or failed experiments that didn't 't make into the paper).
9. Code Of Ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics
Answer: [Yes]
Justification: There is no ethical problem with our code.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., there a special consid- eration due to laws or regulations in their jurisdiction).
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: present broader impact Appendix.
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
20

---

The conference expects that many papers will be foundational research and not tied particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology being used intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms monitor how system learns from feedback over time, improving the efficiency and accessibility of MI..
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: There is no such risks about our paper.
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Lieenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: We have included CC-BY 4.0 license.
Guidelines:
The answer NA means that the paper does existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (c.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (c.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, papprswiiiiiie thcode com datastts has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
21

---

If this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
We do not release assets.
Guidelines:
The answer NA means that the paper does not release new assets,
Researchers should communicate the details of the dataset/code/moodl as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
We do not involve crowdsourcing nor research with human subjects
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According the NeurIPS Code of Ethies, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification:
We do not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does involve crowdsourcing research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurlPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
22


Reviews:
[{"review_id": "86qtZL8aJU", "reviewer": "Reviewer_t9F5", "summary": "This paper aims for interventional fairness with sufficient prediction accuracy by employing a min-max optimization framework. The proposed approach aims for partially directed acyclic graphs (PDAGs) and extends itself to maximally oriented PDAGs (MPDAGs). Finally, the approach is evaluated on synthetic and real-world datasets.", "questions": "Below I provide my questions:\n* Line 178: why can it not be $X_i \\leftarrow A \\rightarrow X_j$?\n* What is the computational complexity of the proposed approach?\n* How can the proposed approach be connected with the min-max approach proposed by Xia et al [1] and the invariant prediction problem proposed by Subbaswamy et al [2]\n\n\n[1] Xia, Kevin, et al. \"The causal-neural connection: Expressiveness, learnability, and inference.\" Advances in Neural Information Processing Systems 34 (2021): 10823-10836.\\\n[2] Subbaswamy, Adarsh, Peter Schulam, and Suchi Saria. \"Preventing failures due to dataset shift: Learning predictive models that transport.\" The 22nd International Conference on Artificial Intelligence and Statistics. PMLR, 2019.", "limitations": "The authors properly discussed their limitations.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The authors addressed a nice and interesting fairness problem in this paper. The proposed approach seemed to be theoretically robust. They also provided detailed empirical analysis.", "weaknesses": "Below I share some weaknesses.\n\n* Above line 188: It is unclear what $S^{(i)}(A); 1<=i<=M$ refers to. The authors should explain the superscripts.\n* Equations should be numbered. The Equation after line 192 should be explained in more detail.\n* Line 201: The authors should provide some intuition/explanation about the backdoor adjustment formula.\n* It seems that there are no baselines based on prior works. Is there no work that solves the fairness problem with causal effect estimation?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s great efforts and insightful comments to improve our manuscript. In below, we address these concerns point by point and try our best to update the manuscript accordingly.\n\n## Weaknesses\n\n> **Above line 188 : It is unclear what $S^{(i)}(A) ; 1<=i<=M$ refers to. The authors should explain the superscripts.**\n\n- We know that causal discovery algorithms can only return CPDAGs with undirected edges, not DAGs where all edges are directed.\n\n- Let $sib(A)$ be a set containing all nodes that have an undirected edge with $A$, and our algorithm needs to find the set of all possible parent sets of the sensitive attribute $A$.\n\n- $S^{(i)}(A) ; 1<=i<=M$ is the subsets of $sib(A)$, such that $\\mathcal{S}_A=\\\\{p a(A)\\cup \\mathbf{S}^{(1)}(A), \\ldots, p a(A)\\cup \\mathbf{S}^{(M)}(A)\\\\}$ consists of all possible parent sets of the sensitive attribute $A$ obtained from our algorithm.\n\n> **Equations should be numbered. The Equation after line 192 should be explained in more detail.**\n\n- The propensity $P(A \\mid p a(A)\\cup \\mathbf{S}^{(m)}(A))$ is defined as the probability of the (binary) sensitive attribute $A$ given its (possible) parent set $pa(A)\\cup \\mathbf{S}^{(m)}(A)$.\n\n- The Equation after line 192 is the binary classification cross-entropy loss for training the propensity model $g(X; \\hat \\phi^{(m)})$ for estimating the true propensity $P(A \\mid p a(A)\\cup \\mathbf{S}^{(m)}(A))$.\n\n- The input is covariates $X$ restricted on the possible parent set $p a(A, \\mathcal{H})\\cup \\mathbf{S}^{(m)}(A)$, denoted as $X|_{p a(A, \\mathcal{H})\\cup \\mathbf{S}^{(m)}(A)}$.\n\n> **Line 201: The authors should provide some intuition/explanation about the backdoor adjustment formula.**\n\n- Whenever we undertake to evaluate the effect of one factor $A$ on another $Y$, the question arises as to whether we should adjust our measurements for possible variations in some other factors $X$, otherwise known as \"confounders\".\n\n- The backdoor adjustment formula aims to partitioning the population into groups that are homogeneous relative to $X$, assessing the effect of $A$ on $Y$ in each homogeneous group, and then averaging the results.\n\n- Formally, the backdoor adjustment formula is $P(Y=y \\mid do(A=a))=\\sum_x P(Y=y \\mid A=a, X=x) P(X=x)$, which computes the association between $A$ and $Y$ for each value $x$ of $X$, then averages over those values, also referred as \"adjusting for $X$\".\n\n> **It seems that there are no baselines based on prior works. Is there no work that solves the fairness problem with causal effect estimation?**\n\n- For compared baselines, we would like to clarify that FairRelax [R1], Fair [R1], and $\\epsilon$-IFair [R2] are all prior works solving the same problem (causal fairness with partially known causal graph). We will explicitly add citations in our revised \"Baselines\" part.\n\n- To the best of our knowledge, we have compared the most comprehensive and SOTA methods, e.g., [R2] is published in ICLR 24.\n\n- The important reason for the limited baselines is because most previous causal fairness work requires known causal graphs (DAGs), whereas we only need tabular data (from which we can only obtain CPDAGs).\n\n## Questions\n\n> **Line 178: why can it not be $X_i \\leftarrow A \\to X_j$?**\n\n- From Definition 3.1 in Line 157, the v-structure is defined as $X_i \\to A \\leftarrow X_j$, not $X_i \\leftarrow A \\to X_j$.\n\n- From Lemma 3.2 in Line 170, a set $\\mathbf{S}(A) \\subset sib(A)$ is a possible parent set of the sensitive attribute $A$, if and only if there is no more v-structures.\n\n- Thus, as stated in Line 178, if $X_i$ and $X_j$ are not adjacent, they cannot be both in the parent set of $A$. Otherwise, a new v-structure $X_i \\to A \\leftarrow X_j$ is formed.\n\n> **What is the computational complexity of the proposed approach?**\n\n- First, our proposed approach mainly includes the following 3 steps: constructing the CPDAG from the tabular data, estimating all possible propensities (Alg. 1), and learning fair classifier via minimax approach (Alg. 2).\n\n- From [R1], the complexity of constructing the CPDAG in the worst case is $\\mathcal{O}(|sib(S, \\mathcal{G})+ch(S, \\mathcal{G})| *|E(\\mathcal{G})|)$, where $|E(\\mathcal{G})|$ is the number of edges in $\\mathcal{G}$.\n\n- For Alg. 1 and Alg. 2, the proposed local method needs to identify the v-structures for every node pairs in $sib(S, \\mathcal{G})$, thus the computational complexity is $\\mathcal{O}(|sib(S, \\mathcal{G})|^2)$.\n\n- Thus, the overall computational complexity is $\\mathcal{O}(|sib(S, \\mathcal{G})+ch(S, \\mathcal{G})| *|E(\\mathcal{G})|+|sib(S, \\mathcal{G})|^2)$, which also illustrates the advantage of our local method instead of enumerating on all possible DAGs.\n\n> **How can the proposed approach be connected with the min-max approach proposed by Xia et al [1] and the invariant prediction problem proposed by Subbaswamy et al [2]?**\n\n- For the min-max approach proposed by Xia et al [1], one similar point is that the input is also the observational data, rather than requiring the known DAG. \n\n- They found that a neural net is unable to predict the effects of interventions given observational data alone (Thm. 1), which motivates to introduce **a special type of SCM called a neural causal model (NCM)** for performing causal inferences.\n\n- However, our paper still aims to conduct the causal inferences **in the context of general SCM**, with minimax approach for bounding the uncertainties of the true DAGs.\n\n- For the invariant prediction problem proposed by Subbaswamy et al [2], they aims to solve the **dataset shift problem with different training and target distributions.**\n\n- However, our study aims to achieve causal fairness without the explicit knowledge of causal DAGs, with **same training and target distributions.**\n\n\n****\n\n**References**\n\n[R1] Counterfactual fairness with partially known causal graph. NeurIPS, 2022.\n\n[R2] Interventional fairness on partially known causal graphs: A constrained optimization approach. ICLR, 2024.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QHpcyqEyzr", "reviewer": "Reviewer_3Hwm", "summary": "This work investigates interventional fairness given partially known causal graphs. Compared to existing methods, it employs all variables and does not need to rely on additional strong assumptions for identification. Specifically, it offers a min-max optimization framework which produces counterfactual fairness in maximally oriented PDAGs. Experiments on synthetic and real-world datasets showcase the efficacy of the proposed method.", "questions": "See the weakness above.", "limitations": "The limitation has been well discussed in the conclusion part.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The manuscript is commendably clear in its presentation, particularly in articulating the research gap. This clarity facilitates a quick comprehension of the central theme and the contributions of the study.\n\nThe toy examples provided are meticulously crafted, serving effectively to elucidate several underlying concepts integral to the research.\n\nThe application of a min-max optimization algorithm is a highlight of this work, elegantly addressing the research problems identified.", "weaknesses": "Overall, the paper is well-composed, substantiated by robust experimental outcomes and comprehensive theoretical analysis. However, some sections could benefit from further clarification to enhance understandability. Specific points of ambiguity are addressed below:\n\n•\tThe statement “One may notice that a function of X1 − X2 can be used to predict Y” appears somewhat abrupt and disrupts the flow of reading. A detailed explanation or a graphical illustration might clarify the independence between U1 − U2 and A.\n\n•\tAt line 165, the rationale behind transforming certain undirected edges into directed edges in the opposite direction remains unclear. Elaboration on this would be beneficial.\n\n•\tThe term ‘direct causal information set’ warrants a clear, explicit definition within the main text to avoid potential misunderstandings.\n\n\n---\n\nThank you very much for providing thorough answers to my queries, which have satisfactorily addressed my concerns. I am pleased to adjust my score.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s great efforts and insightful comments to improve our manuscript. In below, we address these concerns point by point and try our best to update the manuscript accordingly.\n\n> **The statement “One may notice that a function of X1 − X2 can be used to predict Y” appears somewhat abrupt and disrupts the flow of reading. A detailed explanation or a graphical illustration might clarify the independence between U1 − U2 and A.**\n\n**Response:** We thank the reviewer for the helpful comments. We will add the following toy example to illustrate such claims. \n\n- Let $A = U_A, X_1 = A + U_1, X_2 = A + U_2$, and $Y = 2X_1 + X_2 + U_Y$.\n- Because $Cov(X_1 – X_2, Y) \\neq 0$, thus $X_1 – X_2$ is useful in predicting $Y$. \n- Meanwhile, $X_1 – X_2 = U_1 – U_2$ is a constant for any value of $A$, so $U_1 – U_2$ is independent of $A$. \n- Thus, using $X_1 – X_2$ as the variable for the prediction can still achieve counterfactual fairness.\n\n> **At line 165, the rationale behind transforming certain undirected edges into directed edges in the opposite direction remains unclear. Elaboration on this would be beneficial.**\n\n**Response:** We would like to clarify the need for ‘transforming certain undirected edges into directed edges’ as follows.\n\n- We know that causal discovery algorithms such as the PC algorithm can only return CPDAGs with undirected edges, not DAGs where all edges are directed.\n\n- By definition, $sib(A)$ is a set containing all nodes that have an undirected edge with $A$, and our algorithm needs to find the set of all possible parent sets of the sensitive attribute $A$.\n\n- Thus, we need to enumerate all possible cases (i.e., for some $Z \\in sib(A), Z \\rightarrow A$, and for other $Z \\in sib(A), A \\rightarrow Z$), and determine whether a new v-structure is generated.\n\n- Formally, let $\\mathbf{S}(A)$  be a subset of $sib(A)$, we can obtain a DAG from CPDAG by changing all undirected edges $\\\\{Z-A, \\forall Z\\in \\mathbf{S}(A)\\\\}$ into the directed edges $\\\\{Z\\to A, \\forall Z\\in \\mathbf{S}(A)\\\\}$ as parents, and all of other undirected edges $\\\\{Z- A, \\forall Z \\not\\in \\mathbf{S}(A)\\\\}$ into the directed edges with opposite direction $\\\\{Z\\leftarrow A, \\forall Z \\not\\in \\mathbf{S}(A)\\\\}$ as children.\n\n> **The term ‘direct causal information set’ warrants a clear, explicit definition within the main text to avoid potential misunderstandings.**\n\n**Response:** The ‘direct causal information set’ is assumed to be the direct causal information in the form $X \\to Y$, indicating that $X$ is a direct cause of $Y$. Such direct causal information are usually determined by the prior knowledge. Meanwhile, it is worth noting that other forms of background knowledge, such as tier orderings, specific model restrictions, or data obtained from previous experiments, can also induce MPDAGs [1-4]. As suggested by the reviewer, we will add the above explicit definition within the main text to avoid potential misunderstandings.\n\n***\n\n**References**\n\n[1] Alain Hauser and Peter Buhlmann. Characterization and greedy learning of interventional Markov equivalence classes of directed acyclic graphs. JMLR, 2012.\n\n[2] Marco F Eigenmann, Preetam Nandy, and Marloes H Maathuis. Structure learning of linear gaussian structural equation models with weak edges. ArXiv:1707.07560, 2017.\n\n[3] Yuhao Wang, Liam Solus, Karren Yang, and Caroline Uhler. Permutation-based causal inference algorithms with interventions. NeurIPS, 2017.\n\n[4] Dominik Rothenhausler, Jan Ernest, and Peter Buhlmann. Causal inference in partially linear structural equation models. The Annals of Statistics, 2018.\n\n***\n\n**We hope the above discussion will fully address your concerns about our work.** We look forward to your insightful and constructive responses to further help us improve the quality of our work. Thank you!", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "z4HZcpwd2l", "reviewer": "Reviewer_BRmE", "summary": "In this paper, the author(s) talk about a better way to estimate interventional fairness in the case where we have partially known Directed Acyclic Graphs (DAG). It lists out how most previous works have only done this for fully known DAG's and recent attempts to address the case of unknown graphs have utilized strong assumptions and or reduced prediction accuracy. \nThe authors introduce  a min-max optimization framework to achieve interventional fairness and boost accuracy and test it on both generated data and real-world data.", "questions": "CPDAG was used on line 57 but was but the full meaning was not given until line 96.", "limitations": "The author brings it up clearly in their conclusion that their method relies on Completely Partially Directed Acyclic Graph (CPDAG) and on the propensity scores returned and therefore any errors in these can affect their method.\nAlso, the case where we have latent variables or confounders was not addressed.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper does a good assessment of former work on the topic and builds directly on it. \nThe terms and definitions and clearly outlined and explained adequately for easy understanding. The authors put their theory into practice and using both synthetic and real-world data.", "weaknesses": "The paper's approach relies heavily on the accuracy of the CPDAG and on the generated propensity scores. The paper could address measures to address interventional fairness in the case of inaccuracies coming from the CPDAG", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s great efforts and insightful comments to improve our manuscript. We are also encouraged by the very positive comments on our paper.\n\n> The paper's approach relies heavily on the accuracy of the CPDAG and on the generated propensity scores. The paper could address measures to address interventional fairness in the case of inaccuracies coming from the CPDAG.\n\nAs suggested by the reviewer, we added more simulation experiments to explore the effect of the degree of inaccuracy of CPDAG on the performance of our method. The experimental results are shown in Fig. 1 in the Supplementary PDF.\n\n> CPDAG was used on line 57 but was but the full meaning was not given until line 96.\n\nWe thank the reviewer for pointing out the typo. In our revised manuscript, we will provide the full name of CPDAG and more explanation at Line 57.\n\n> Flag For Ethics Review: Ethics review needed: Data privacy, copyright, and consent\n\nWe're guessing you misclicked the flag for ethics review. We would like to clarify that all used datasets in our original manuscript and rebuttal are public and widely adopted datasets.\n\n- The OULAD dataset: https://www.archive.ics.uci.edu/dataset/349/open+university+learning+analytics+dataset\n\n- The Adult dataset: https://archive.ics.uci.edu/dataset/2/adult\n\n- The COMPAS dataset: https://www.kaggle.com/datasets/danofer/compass\n\n***\n\nWe hope the above discussion will fully address your concerns about our work. We look forward to your insightful and constructive responses to further help us improve the quality of our work. Thank you!", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nHcjWk6Ib3", "reviewer": "Reviewer_GJNz", "summary": "This paper develops a method for satisfying interventional fairness when the causal graph is partially known. The paper first develops a method for checking possible parental sets as well as a method for estimating propensities given a possible parental set. Then, the interventional distribution is computed using the parental set as a back-door adjustment set. Finally, a min-max joint learning approach is proposed to learn a fair model based on the worst-case violation of interventional fairness.", "questions": "In the experiments, the accuracy is measured by RMSE. Can the proposed method be applied to classification problems?\n\n===============post-rebuttal comments============\n\nThe authors partially addressed my concerns, and I value the contributions made by the paper. I have raised my score.", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Originality\nThe paper extends the previous methods to MPDAGs and develops a novel method for learning fair models based on the worst-case violation of interventional fairness.\n\nQuality\nThe proposed method is theoretically sound. It builds upon established theoretical results from previous works while also providing rigorous theoretical analysis of the newly proposed approaches.\n\nClarity\nThe main results are summarized as lemmas. However, a lemma is usually used to help prove a larger theorem. The authors may change lemmas to propositions.\n\nSignificance\nThe experiment results show that the proposed method can achieve good fairness and accuracy performance without providing exact causal graphs.", "weaknesses": "This paper presents a solid work. The main reason I am inclined towards rejection is the applicability of this work. The proposed method relies on the existence of parent nodes of the sensitive attribute. However, as also mentioned in the paper, in most datasets the sensitive attributes have no parent nodes. To be more specific, most sensitive attributes have no parents in nature. For example, common sensitive attributes like sex/gender, race, nationality, etc., are determined at birth. As a result, the proposed method is applicable to very limited sensitive attributes like disability, which significantly limits the importance of this work. I recommend that the authors may apply the proposed method to more general causal inference scenarios where the treatment node may generally have parent nodes.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s great efforts and insightful comments to improve our manuscript. We are also encouraged for the reviewer to recognize the **Originality**, **Quality**, **Clarity**, and **Significance** of our work. In below, we would like to address the reviewer's concerns regarding the **Applicability** of our proposed method from both theoretical and experimental aspects.\n\n***\n\n### **Main Result 1: The proposed method can also be applied in scenarios where sensitive attributes are identified as root nodes (W1).**\n\n- We **agree** with the reviewer that most sensitive attributes have no parents in nature. However, we respectively **disagree** that the proposed method relies on the existence of parent nodes of the sensitive attribute.\n\n### **Theoretical Analysis**\n\n- For our studied interventional fairness, if the sensitive attribute $A$ is as a root node in the causal DAG, then there will be **no backdoor path** from the sensitive attribute $A$ to the outcome variable $Y$. \n\n- At this point, **the causal estimands in the definition of interventional fairness and our method degenerate to the correlation estimands.** \n\n- Compared to previous Fair, FairRelax, and $\\epsilon$-IFair methods, **our method doesn't rely on variable selection and strong causal assumptions**, thus still should be considered as a more plausible method.\n\n### **More Real-World Experiments**\n\n- We **add experiments on two widely used real-world datasets**: the **Adult** and the **COMPAS** datasets, where **the sensitive attributes are \"sex\" and \"race\" as the root nodes**, respectively.\n\n|COMPAS|Full|Unaware|FairRelax|Fair|ε-IFair|Ours|\n| :--: | :--: | :--: | :--: | :--: | :--: | :--: |\n|RMSE|0.256 ± 0.022|0.261 ± 0.023|0.261 ± 0.023|0.263 ± 0.022|0.235 ± 0.091|0.219 ± 0.020 |\n|Unfairness|0.273 ± 0.048|0.269 ± 0.052|0.260 ± 0.045|0.238 ± 0.045|0.190 ± 0.120|0.179 ± 0.011|\n||\n\n|Adult|Full|Unaware|FairRelax|Fair|ε-IFair|Ours|\n| :--: | :--: | :--: | :--: | :--: | :--: | :--: |\n|RMSE|0.433 ± 0.024|0.436 ± 0.024|0.607 ± 0.131|0.611 ± 0.128|0.413 ± 0.010|0.375 ± 0.019|\n|Unfairness|0.506 ± 0.021|0.409 ± 0.029|0.209 ± 0.205|0.187 ± 0.205|0.155 ± 0.009|0.171 ± 0.021|\n||\n\n- From the above experimental results, we find that **our method still outperforms previous methods when the sensitive attribute is used as the root node**.\n\n\n### **More Synthetic Experiments with Varying Number of Parent Nodes**\n\n- **To further explore the impact of the number of parent nodes of sensitive attributes on the accuracy and fairness performance, we also added more synthetic experiments with varying number of parent nodes.**\n\n|Metrics|Unaware|Fair|FairRelax|$\\epsilon$-IFair|Ours|\n|:--:|:--:|:--:|:--:|:--:|:--:|\n|RMSE (p = 0)|0.601 ± 0.235|0.642 ± 0.241|0.635 ± 0.240|0.633 ± 0.237|0.635 ± 0.233|\n|Unfairness (p = 0)|0.063 ± 0.082|0.041 ± 0.051|0.045 ± 0.050|0.041 ± 0.021|0.035 ± 0.016|\n|RMSE (p = 2)|0.528 ± 0.298|0.601 ± 0.291|0.597 ± 0.294|0.590 ± 0.206|0.584 ± 0.201|\n|Unfairness (p = 2)|0.111 ± 0.090|0.099 ± 0.075|0.100 ± 0.075|0.100 ± 0.088|0.096 ± 0.107|\n|RMSE (p = 4)|0.718 ± 0.281|0.860 ± 0.261|0.834 ± 0.253|0.818 ± 0.237|0.792 ± 0.228|\n|Unfairness (p = 4)|0.129 ± 0.073|0.113 ± 0.058|0.120 ± 0.050|0.110 ± 0.072|0.103 ± 0.086|\n||\n\n- **As the number of parent nodes increases, the advantage of our method over **Fair**, **FairRelax**, and **$\\epsilon$-IFair** gradually increases.** This is because when there are more parent nodes, there are more backdoor paths, thus our method can find these backdoor paths to better control the causal effect of the sensitive attribute on the outcome variable.\n\n***\n\n### **Main Result 2: The proposed method can also be applied to the Classification Problems (Q1).**\n\n### **Real-World Experiments**\n\n- For real-world experiments, we would like to kindly remind the reviewer that **the outcome variable is binary in the OULAD dataset** adopted in our original manuscript. **The outcome variables in our added experiments on the Adult and COMPAS datasets are also binary.**\n\n### **Synthetic Experiments**\n\n- For synthetic experiments, as suggested by the reviewer, **we conduct more experiments using AUC as the evaluation metric instead of RMSE**. Specifically, we modify the data generation process (DGP) to clip the outcome variable $Y$ to 1 if $Y > 0$, and to 0 if $Y < 0$, and the rest DGP remains the same. The experiment results are shown below. \n\n|Noise = 2.5|Node = 10, Edge = 20 (AUC) |Node = 10, Edge = 20 (Unfairness) |Node = 40, Edge = 80 (AUC)|Node = 40, Edge = 80 (Unfairness)|\n|:--:|:--:|:--:|:--:|:--:|\n|Oracle|0.815 ± 0.094|0.000 ± 0.000|0.828 ± 0.087|0.000 ± 0.000|\n|Full|0.845 ± 0.071|0.038 ± 0.051|0.842 ± 0.086|0.143 ± 0.106|\n|Unaware|0.843 ± 0.070|0.017 ± 0.021|0.837 ± 0.090|0.113 ± 0.121|\n|FairRelax|0.825 ± 0.057|0.017 ± 0.021|0.824 ± 0.082|0.112 ± 0.119|\n|Fair|0.819 ± 0.060|0.015 ± 0.021|0.822 ± 0.128|0.094 ± 0.122|\n|$\\epsilon$-IFair|0.843 ± 0.049|0.018 ± 0.017|0.833 ± 0.085|0.098 ± 0.105|\n|Ours|0.844 ± 0.051|0.015±0.016|0.835 ± 0.088|0.089 ± 0.119|\n||\n\n- From the above results, **Full** and **Unaware** perform better on AUC, while **Fair**, **FairRelax**, **$\\epsilon$-IFair**, and our approach perform better on unfairness. Note that our approach outperforms **Fair**, **FairRelax**, and **$\\epsilon$-IFair** in all scenarios on both AUC and unfairness.\n\n***\n\n### **(Minor) Clarity Issues (S3)**\n\n> Clarity: The main results are summarized as lemmas. However, a lemma is usually used to help prove a larger theorem. The authors may change lemmas to propositions.\n\n- We thank the reviewer for raising such useful suggestions. In our revised version, we have changed all *Lemma*s to *Proposition*s.\n\n***\n\n**We hope the above discussion will fully address your concerns about the applicability of our work, and we would really appreciate it if you could consider to raise your score.** We look forward to your insightful and constructive responses to further help us improve the quality of our work. Thank you!", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear all reviewers and AC,

Please kindly find the attachment as our added one page experimental results.

Thanks,

Authors from Submission #18019

Meta-review: The authors proposed a method for achieving interventional fairness when the causal graph is unknown.
The proposed method enumerates only those equivalent PDAGs that affect interventional fairness among the equivalent causal DAGs, and then trains a model to minimize the worst-case fairness within these equivalent PDAGs.
Through discussions with the reviewers, the authors clarified the effectiveness of the proposed method and its broad applicability.
Although there were comments from the reviewers regarding the differences in assumptions between the proposed method and existing methods, as well as their performance comparison, the authors responded to these points appropriately. Incorporating the content of these discussions into the manuscript would further improve the quality of the paper.
Decision: Accept (poster)