Towards counterfactual fairness through auxiliary variables
Authors: Bowei Tian
Ziyao Wang
Shwai He
Wanghao Ye
Guoheng Sun
Yucong Dai
Yongkai Wu
Ang Li
Venue: ICLR 2025 Poster (2025)
Keywords: Counterfactual
Fairness
Auxiliary variables
Primary area: alignment, fairness, safety, privacy, and societal considerations
TL;DR: We introduce EXOC, a causal model framework that achieves counterfactual fairness without sacrificing accuracy.

Abstract: The challenge of balancing fairness and predictive accuracy in machine learning models, especially when sensitive attributes such as race, gender, or age are considered, has motivated substantial research in recent years. Counterfactual fairness ensures that predictions remain consistent across counterfactual variations of sensitive attributes, which is a crucial concept in addressing societal biases. 
However, existing counterfactual fairness approaches usually overlook intrinsic information about sensitive features, limiting their ability to achieve fairness while simultaneously maintaining performance. To tackle this challenge, we introduce EXOgenous Causal reasoning (EXOC), a novel causal reasoning framework motivated by exogenous variables. It leverages auxiliary variables to uncover intrinsic properties that give rise to sensitive attributes. Our framework explicitly defines an auxiliary node and a control node that contribute to counterfactual fairness and control the information flow within the model. Our evaluation, conducted on synthetic and real-world datasets, validates EXOC's superiority, showing that it outperforms state-of-the-art approaches in achieving counterfactual fairness without sacrificing accuracy. Our code is available at https://github.com/CASE-Lab-UMD/counterfactual_fairness_2025.

Published as a conference paper at ICLR 2025
TOWARDS COUNTERFACTUAL FAIRNESS AUXILIARY VARIABLES
THROUGH
Bowei Tian', Ziyao Wang', Shwai He', Wanghao Ye! Guoheng Sun', Yucong Dai Yongkai Wu Ang Li 'University of Maryland, College Park, Clemmso University (btianl, ziyaow, shwaihe, wy891, ghsun, angliece) @und edu yucongdeg, clemson edu, wttp:k/wwwwleemmm.eeed.
ABSTRACT
The challenge of balancing fairness and predictive accuracy in machine learning models, especially when sensitive attributes such race, gender, age con- sidered, has motivated substantial research in recent years. Counterfactual fair- ness ensures that predictions remain consistent across counterfactual variations of sensitive attributes, which is crucial concept in addressing societal biases. However, existing counterfactual fairness approaches usually overlook intrinsic information about sensitive features, limiting their ability to achieve fairness while simultaneously maintaining performance. To tackle this challenge, we introduce EXOgenous Causal reasoning (EXOC), novel causal reasoning framework mo- tivated by exogenous variables. It leverages auxiliary variables to uncover in- trinsic properties that give rise sensitive attributes. Our framework explicitly defines an auxiliary node and a control node that contribute to counterfactual fair- ness and control the information flow within the model. Our evaluation, con- ducted on synthetic and real-world datasets, validates EXOC's superiority, show- ing that it outperforms state-of-the-ee approaches in achieving counterfactual fair- ness. Our code is available at https: /tt:://gitit thub. com/ CASE- Lab- UMD counterfactual fairness 2025
INTRODUCTION
Machine learning has been widely adopted in prediction tasks (Brennan et al. 2009 Corbett-Davies al. 2023) such as personalized recommendation (Mehrotra et al. [2018) Wu al. 2021) and image classification (Bhojanapalli et al.) 2021 Chen et al. 2021). Recent literature shows that predictions based on traditional machine learning methods often exhibit bias against certain demographic sub- groups, which are described by sensitive attributes such as race, gender, age, and sexual orientation. Therefore, developing a fairer predictor has attracted considerable attention (Bellamy et al.) 2019 Bird et al. 2019 Caton & Haas 2024). Among them, counterfactual fairness applies causal mech- anisms to model how discrimination occurs and measure societal bias at an individual level, using Pearl's causal structural models (Pearl 2009). The idea behind counterfactual faimess is to ensure that predictions from the same individual remain consistent even if their sensitive attribute would have changed. Kusner al. (2017) introduce the framework for counterfactual fairness the indi- vidual level using causal models. After that, several works focus on counterfactual fairness. [Russll] al. (2017) propose new counterfactual fairness framework by integrating multiple counterfactual assumptions, aiming to address inconsistencies in fairness across different causal models. The paper uses Bayesian approach to unify different counterfactual assumptions into a probabilistic model, thereby better handling complex fairness issues. Wu et al. (2019) presents unified definition that covers most the previous causality-based fairness notions, namely the path-specific counterfac- tual fairness (PC fairness), and proposes estimation approach for unidentified causal quantities.
Ma et al. (2023) propose a method to achieve counterfactual faimness without requiring a predefined causal graph by learning directly from observational data. The approach involves creating terfactually fair dataset through augmentation and using a carefully designed loss function to ensure
fairness during model training.

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We observe that the majority of existing methods for counterfactual fairness focus analyzing causal inference and their counterfactual framework (Kusner et al., 2017 Russell et al. (017) or creating counterfactual-fai augmentation dataset that is agnostic 2023), which can be inferred directly using the augmentation dataset and crafted loss design. How- ever, to the best of knowledge, most existing works assume sensitive attributes should not be causally influenced by any other variables (Kusner et al., 2017; Berk et al. [021]] Ma et al.) (202)) This assumption overlooks the essentials of sensitive features, i.c., which part of the sensitive fea- ture is intrinsic or essential for the inference and which part should be neglected. Also, existing methods usually into specific scenario where the causal relationship from the sensitive attribute to the target attribute is fixed. For example, race should generally not influence decision-making at all, making it hard to extend and distribute in real-world scenarios. For example, in a demography experiment, race distribution can be deduced from the geographic distribution of the population, which can not be causally neglected.
casual graph (Ma et al.
To tackle these challenges, propose a novel framework, EXOC, which introduces intuitive mod- ifications to the causal model. This framework utilizes the auxiliary variables in causal inference, extracting essential information from sensitive attributes and effectively controlling the flow of in- formation from the sensitive attribute to the target attribute. We summarize our contributions as follows:
We develop framework that utilizes auxiliary variables in causal inference, extracting essential information from sensitive attributes and enhancing fairness without sacrificing much accuracy.
We formalize method to regulate the flow of information from the sensitive attribute to the target attribute, effectively controlling the balance between accuracy and faimesss
We provide theoretical analysis and conduct extensive baseline and ablation experiments to validate the effectiveness of our approach.
PRELIMINARIES
2.1 COUNTERFACTUAL FAIRNESS
Counterfactual fairness (Kusner et al. 2017) is an individual-level faimnsss notion based on the causal model. constructed on the Pearl's causal framework (Pearl 2009), which is defined as a triple FU) so that:
U is set of latent background variables, which are exogenous and not caused by any variable in the set V:
V is set of observed variables, which are endogenous and determined by UUV;
F set functions (m) on for each V; E V, so that V; = -i(pp; Upa.), where pa; C V (V.) and Up, C U are variables that directly determine V,-
A causal model is associated with causal graph, which is directed acyclic graph (DAG). Each node in the causal graph represents a variable in the causal model, and each directed edge corre- facilitated
sponds a causal relationship. In the causal model, the counterfactual estimands by interventions through do-calculus, which simulates the physical interventions that force some variables take certain values. For example, for observed variables A and the value of the counterfactual "what would A have been if B had been b is denoted by Ag-b-
Counterfactual fairness Kusner et al. 2017: Wu al. 2019): Given factual condition 0 the predictor Y = =()) is counterfactually fair if under any context O, PYYs., y o) P y / vs # where O {S, Sis the sensitive attribute and X is observed non-sensitive attributes.
Approximate counterfactual fairness (Russell et al. 2017): A predictor Y = (() satisfies (8,0)-
approximate counterfactual fairness if, given the factual condition O o, have:
[[(Ys-: Ys-v) o] < 8, Vs
This approximate metric measures counterfactual fairness in practical manners. Unless otherwise specified, refer to this approximate metric as counterfactual fairness in theoretical analysis.
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X
S
Y
K
S
S'
Y
K
U
U
S"
(a) Fair-K
(b) EXOC
Figure 1: The causal models of Fair-K and EXOC. S is the sensitive attribute, X is observed non- sensitive attributes, is the target attribute, K the latent domain knowledge, and 5' and 5 are latent auxiliary nodes, U is the exogenous variable. The solid lines represent designed causal relationships, and dashed lines mean our focused existing relationships in implementation. illustrated in 3.2.2 (note that U have existing causal relationships with every node (Pearl 2009)).
2.2 COUNTERFACTUALLY FAIR LEARNING APPROACHES
In the counterfactually fair machine learning literature, Fair-K K (Kusner et al. 2017 Ma et al. 2023 widely adopted framework. As illustrated Fig. [a] it is designed on the Law school dataset (Krueger et al.) 202T) and assumes a node K representing domain knowledge that can act as non- deterministic causes of X. Then, trains a predictor using K to predict Y, e.g., using logistic regression and achieves counterfactual fairness improvement. However, the causal effect trans- mitting from exogenous variables U is not fully utilized the deployments of counterfactually fair predictors, potentially leading to significant performance decreases.
3 DESIGN
3.1 CAUSAL MODEL OVERVIEW
To address the above limitations, we propose EXOC, a novel framework that introduces the auxil- iary node and the control node, which instantiates the exogenous variable U. Specifically, the model leverages controllable auxiliary nodes to simultaneously capture intrinsic, latent information from X, Y, and S. The model enhances the overall performance and fairness balance by incorporating this additional auxiliary compared to Fair-K in Fig. Ta Motivated by the controllable nature of 5", we further introduce S with the aim that S" support S' in controlling the balance between fair- ness and accuracy. Strengthening the relationship between S' and 5" indicates stronger alignment with fairness, whereas weakening allows 5 tap into more intrinsic information from S and em- phasize performance. Through this flexible mechanism, we can prioritize faimess or accuracy in various real-world scenarios. We explain the components of the EXOC framework in the following subsections.
3.2 S:: THE AUXILIARY NODE
3.2.1 ILLUSTRATING AUXILIARY NODE S' IN A SIMPLIFIED CASE
We observe that previous works fail to dive into the essential of sensitive features and consider U as unknown background variables in causal model. Rethinking the role of exogenous variable U. we devise an idea to instantiate U into an auxiliary node S' in the model. To demonstrate how the auxiliary node S' achieves counterfactual faimess, first take simple example: consider the ideal linear model (the model can ideally fit the determination of the real world, where is eliminated) with normal distributions. In Fig. Ta for each individual, the causal relationship can be written
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as:
Y = as+ BBK,
(3)
where and B are path coefficients (Pearl 2009). Therefore, in causal inference,
Ys Ys+ = a (Sst o)
+ B. aas+ + Bk, k~N(HR ok).
(4)
(Ys-s* Ys++ 0),, - asss" -s B(1 - -k),,
(5)
where the same alphabet with different subscript numbers is sampled from the same corresponding distribution, asss-S is fixed, and (kk - Ko) (0,
Similarly in Fig. lb we have:
Y aS' + BKK
(6)
where the alphabet with the tilde operator has similar meanings. Since the model structure differs between Fig. la and Fig. 1b we distinguish their values using tilde. Therefore,
Ys-s Ō (Sses o) B- = ās' +Bk, ~~~ ( k a~
(7)
(Yst> Ys-s 0)b = acs} + ko);
apply Three Sigma Rule (Pukelshei 1994 these results:
(Ys-s' Yst 1) (a<(s' - 5) + [k) ko) =30
a(ss s) = 33 2- OK
(10)
(Ys- Ys-ss (0)=
1:3 = =a(ss - s6) + [(1 ko)) 130
(11)
= 3 oZ + +
(12)
where these bounds are not exceeded in 99.7% cases, so the exceptions are negligible. Therefore, estimate the upper bound of [Ys+=' Ys-s o]|, i.e., approximate counterfactual fairness bound in these equations as:
(S--'' Ys S - =3 =333 [x
= =la<(s = 1B|ox 8..
(13)
[ Yss 0/2l S $3/2 20100 ( +B
Sp.
(14)
where the value corresponds to 8 in the approximate counterfactual faimess definition. As a(s -) the counterfactual parity that plays more important role than the standard deviation, we showcase that Sz Sp. so the scenario in EXOC is tighter than Fair-K in the constraint of counterfactual fairness. Therefore, 5' theoretically helps improve the counterfactual fairness in this case.
3.2.2 EXTENDING AUXILIARY NODE S' INTO A GENERAL CASE
To mitigate the strong assumptions that the model ideal and the distribution of nodes is normal, we use arbitrary functions for the causal model. We have Y = f(. K, U) in Fair-K K and Y = =f' "S'KS"KUU" K.U) EXOC. When we calculate the counterfactual fairness, we will similarly operate counterfactual parity between different sensitive attributes as in Eq. 5 and 8 i.e., (J(S, in Fair-K and (f(S', K,U)s~> - f(S', K, K,U)KK ())+r)))))))))))))))))) in EXOC. As S' is a non-descendant of S, there should also be an climination of S parity when calculating the counterfactual faimess in EXOC. Therefore, we expect a promotion of counterfactual fairness.
f(S, K, UU)+)))
Since the analysis based on counterfactuals from Pearl's SCM framework PPaal 2009) conflates the predictor with the outcome Y (Kusner et al. 2017). and it does not explicitly incorporate probabilistic deployment of causal models. So, when reconsider causality from the information perspective, we discover that S' not only achieves counterfactual fairness but also has the nature of controlling information flows from S to Y and from K Y.

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Specifically, in the deployment of EXOC, we first perform inference the model using observed training set to estimate the posterior distribution of P(K 0). Subsequently, we train the logistic re- gression predictor Ý = (() to model the relationship between O and Y. This predictor 5(-) focuses on the correlation without the craft of causality and this ignorance of causality can be depicted as the impact from the exogenous variable U, focusing on two causal relations of U - K and U - S. They yield backdoor path nhe (K U + S). Without loss of generality, conduct a path analysis of nt:
K| n (U)), 5 )z())
(15)
where Φ1 and Φ are arbitrary causal functions, acting as an extension to the path coefficient in (Pearl 2009). These functions measure the intensity of the causal relationship from input output. Then we have the impact from K to S:
S ne = (Φ2 o o (K),
(16)
where 0i is the inverse function of 61, whose intensity has negative correlation with the intensity of Φ1. due to the unidirectional deduction in causal inference. (02 e o shows the non-negligible correlation between K and S. Consequently, in Fig. la there is the frontdoor path Tu = {S - }. so
even if we exclude S as a factor in logistic regression, the correlation (02 o 1) will cause impact from S to Y:
Y n = (S).
(17)
where (02 o 0,-1) is included in S. so this equation expressed when inferring from K is:
Y n x π. (Ø) c 02 0, (K),
(18)
where O3 merely depends on the distribution of S, which is fixed. However, the causal graph in Fig. lb bypasses this frontdoor path by creating Tu = {S < S +.. Similar as Th, we have in n₁;
S πΙ (S'), Y πhp (S'),
(19)
Y T& = (0s - 04 )S))
(20)
Y Frg x nup (Øs o 0, o Φ2 o o1 (X),
(21)
where (Øs o 1) shows the correlation between S and Y. Notably, (Øs o 1) can be controlled by the auxiliary node S", because du and Øs both take S rather than S as inputs. This framework provides flexibility for users to balance fairness and accuracy by controlling whether 5' should more likely result in or Y. Specifically, if we strengthen the intensity of d4. S' then have a stronger causal effect to S. According to Eq. 20 and 21 both the correlation from S to Y and from K to Y are minimized, where the former contributes to counterfactual fairness, with a tradeoff of accuracy. Intuitively, we can consider minimizing correlations as controlling two information flows: one flowing from to Y and the other flowing from K to Y. Minimizing the correlations weakens the information flows, thus promoting counterfactual fairness and decreasing performance. So, we can conclude that introducing
improves counterfactual fairness and naturally made to control information flows. Since we face the challenge of realizing this control process, develop control node S" to tackle it.
3.3 S": THE CONTROL NODE
Now, introduce S", where we design custom loss that connects S and S acting the key factor supporting information flow control. It minimizes the distance between S and 5":
1 Σ S/ D
Ss(
(22)
where D is the training dataset length, and ||11/2 is the Euclidean norm (L2 norm). Here, we aim to deduce s" as the descendant of Y for calculating C.(S' (s In Pearl's SCM theory, deduced variables often represent hidden factors that cannot be directly observed. Estimating the posterior distribution of these latent variables is challenging (Kingmal 2013) Blei et al., 2017). Therefore, the implementation of the causal graph resorts to the ELBO technique (Jordan et al.) 1999). which defines a guide model equipped with an assumed posterior distribution to fit the rules of the causal graph. In our case, we realize the Evidence Lower Bound (ELBO) loss as:
LELBO (0) (L || p(Z(O)).
(23)
5

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P(S)
P(Y)
P(Y)
P(S)
P(S)
P(S)
P(S)
P()
P(P)
P(S")
P(S")
P(S")
(a) High y-
(b) Low y.
Figure 2: The distribution mapping regarding "S" which can be seen probability infer- ence perspective the partial causal graph in Fig. Tb where blue arrows mean the distribution parity tightened, red arrows mean loosened, the dashed line circle is the true distribution, the full line circle is the inferred distribution. Blue circles are close to true distributions, and orange circles far from true distributions. Note that the y positively related to the effect of (S "S". so the constraint of (5 ) in Fig 2a is tighter, in Fig 2b is looser.
where Z - (K, are the latent variables, p(O) is prior, (Z(O) is the approximate posterior distribution that tends fit the true posterior distribution p(Z KL|- || the Kullback-Leibler (KL) divergence. The formula expresses that LELBO is the negative marginal log likeliiodd log (O) and the KL divergence between gZ 0) and p(Z O). Minimizing LELBO effectively minimizes the KI. divergence, which in turn provides better approximation of the true posterior p(Z]O)).
Here the overall loss is defined as: CELBO + y RSC.(S (")), where y is the that balances the two losses, R is a normalization scale ensuring LELBO and (S .S") are initialized with the same order of magnitude.
To understand why S can control the information flows, we need to rethink the ELBO technique from the implementation perspective. Specifically, what we do the training process is to use ELBB to construct approximate posterior distribution (Z(O) that fits the true posterior distri- bution OZ)O). After obtaining (Z)O), during the inference, we can predict ý and based on the approximate distribution. Note that S is different from observable variable S, where S is inferred from (O))
Next, to distinguish the behavior of probability inference from causal inference, notate the distri- butions of the nodes by P(-), and the distributions will simplify (. (.1o) expression. Fig. 2 shows the distribution mapping under different y. Different from causal inference, probability inference is method that focuses on correlation rather than causal relations. For example, although S a descen- dent of S' in the causal graph, P(S) will impact the inference result of P(S') because the correlation between S and 5' is mutual.
In the scenario of probability inference, guarantee counterfactual fairness, we should ensure the causal relationship between 5 and S is 5 + S according to Fig. 1b Applying it to probability inference, when we infer from P(S') to P(S). P(S) should approximate P(S). Therefore, KL(P(S)
P())) is a valuable property to estimate counterfactual faireess. For accuracy, we can estimate it as KL(P(Y) || P())).
The effect of Ls("", in distribution manner is minimizing KL(P(S") || P(S'). Now we can view the loss from two distinct perspectives:
Fairness: Since both and are descendants S is challenging for S' to simultaneously infer both $ and Y that closely match their true distributions. The decreased accuracy in fitting PYY with P(f) creates opportunity for S' to better infer PS)). thereby minimizing

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(L(((S) || P(S)). Consequently, this reduction in KL divergence enhances counterfactual fairness.
Accuracy: since we formulate deduction from Y to S" causal graph (Pearl 1995). the posterior distribution of S" are constrained by Y. i..., KL(P()) P((S)) <1, where is positive. Because of the triangle inequality for KL divergence, we have:
(L(P(P) P(S)) SI+ KL((("))) P(S"))
(24)
which demonstrates the upper bound of KL(((?) I| P(S)) is constrained by KL(P(S")
P(S")), therefore minimizing KL.(P(5") || P(S")) also expect to minimize KI.(P(P) P(S)))
This minimization will lead to p()) aligning more closely with P(S"). which may compro- mise its alignment with P(Y), resulting a trade-off in accuracy.
Connection between KL divergence and causal functions: We observe that the greater the inten- sity of causal functions, the fewer distribution parities between inferred and true variables. For ex- ample, more intense 44 with less (I.(P(S) || P(S)), and more intense with less (I.(P(P) P(Y)). Therefore, minimizing (S' ,5") can be viewed as a maximized causal intensity in 4s and mini- mized causal intensity in @5. According to Eq. 20 and 21 within path ny X Tg, the causal intensity of (Ø3 ) is minimized. This minimization indicates less correspondence between S and Y, contributing to counterfactual fairness, and less correspondence between K and Y, compromising predicted accuracy.
The benefit of 5" compared to P: The purpose of using S in the custom loss rather than ÿ is to provide additional flexibility in controlling the influence of S' on both Y and By minimizing the distance between S' and 5", the model can dynamically adjust the extent to which S' influences both and during the optimization process. Compared to directly minimizing the distance between S and P, this approach allows S' to influence the prediction more subtly through the intermediate node S" This gives the model greater freedom to prioritize fairness while maintaining performance.
EXPERIMENTS
4.1 EXPERIMENT SETTINGS
Baselines: To investigate the effectiveness of our framework in learning counterfactually fair predic- tors, we compare the proposed framework with multiple tfate-oftthe-as methods. First, we briefly introduce all the compared baseline methods and their settings:
produces constant output. We obtain it by finding a constant mini- mizing the mean squared error (MSE) loss on the training data.
Constant Predictor:
Full Predictor: It takes X and S as input for prediction.
Unaware Predictor: It takes X as input for prediction to achieve fairness through unaware-
ness (Dwork et al. 20122)
Counterfactual Fairness Predictors: Fair-K (Kusner et al. 2017) reaches counterfactual fairness using the latent variables and non-descendants of the sensitive attribute in the pre- diction model. CLAIRE (Ma et al., (023) involves creating a counterfactually fair dataset through augmentation and using a carefully designed loss function ensure fairness during model training.
For baselines Full, Unaware, and Counterfactual Fairness Predictors, we use linear regression for re- gression and logistic regression for classification. Details about implementations, including datasets, environments, and
are in Appendix
Evaluation Metrics: Generally speaking, the evaluation metrics consider two different aspects: prediction performance and counterfactual fairness. To measure the model prediction performance, employ the widely used metrics- Root Mean Square Error (RMSE) (Chai et al. 2014) and Mean Absolute Error (MAE) Yuan, 2022) for regression tasks and accuracy for classification tasks. To evaluate different methods concerning counterfactual faimess,, we compare the distribution diver- gence of the predictions made on different counterfactuals in synthetic or real-world datasets. De- tailed information about how generate these counterfactuals is in the Appendix C If predictor is

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counterfactually fair, the distributions of the predictions under different groundtruth counterfactuals are expected to be the same. Here, we use two distribution distance metrics (including Wasserstein- 1 distance (Wass) (Chen et al., 2017) and Maximum Mean Discrepancy (MMD) (Long et al. 2015 Shalit et al. 2017 ) to measure the distribution divergence. We compute the divergence of prediction distributions in every pair of counterfactuals (S and S t- s", As + s), then take the average value as the final result. The smaller the average values of MMD and Wass are, the better predictor performs in counterfactual fairness.
4.2 BASELINE STUDY
Baselines synthetic datasets: For better measurement of counterfactual fairness, we generate synthetic dataset for each real-world dataset, where we will italicize the synthetic datasets. Details about generating these synthetic datasets are in Appendix The results are shown in Table where RMSE and MAE are performance metrics, and MMD and Wass are fairness metrics, Compared with Constant, Full, and Unaware baselines that omit the definition of counterfactual fairness, we showcase considerably better faimess. Compared with Fair-K and CLAIRE, demonstrate not only better performance but also surpassing fairness.
Table 1: The comparison on synthetic datasets among Constant, Full, Unaware, Fair-K (Kusner ct al. 2017). ). CLAIRE (Ma et al. 2023) and EXOC (Ours) on Law school (Krueger et al. 2021 and
Adult (Bekker & Kohavi 1996) dataset.
Law school
Adult
Method
RMSE (4)
MAE (L)
MMD (L)
Wass ())
Accuracy (1)
MMD (.)
Wass (1)
Constant
0.938-+0.004
0.759 20.000
0.000-00000
0.000
0.737 20.000
0.000.a..m
0.000
Full
0.862
00.09
278.918-25.814
69.248 136
+0.005
$2.515
6.11660.637
Unaware
0,900
0.72610.007
40.25613.187
10,256
0.80410.000
19.732.22480
2,004
Fair-K
0.894
0.718+p.ook
4.313_0.393
3.733
0.745
3.597+4.256
1.533_0.173
CLAIRE
0.897 002
0.719
6.717_0442
4.073 0.199
0.748 10.005
4.760.10.275
1.584_0.203
EXOC
0.87440.003
0.702+0.008
3.824_0.553
3.590-00259
0.760
2.958
1.428-
Baselines on real-world datasets: the result is shown in Table Although compared with synthetic dataset results, we observe that the majority of the baselines demonstrate degradation in fairness and accuracy, we are still able to surpass the counterfactual-avar and fairness and are fairer than counterfactual-une unawwae models. This observation demonstrates the robustness of our method in real-world scenarios.
models in both performance
Table 2: The comparison on real-world datasets among Constant, Full, Unaware, Fair-K (Kuuege al. (02T), CLAIRE (Ma et al. 2023) and EXOC (Ours) Law school (Krueger et al. 2021 and
Adult (eekkrr & Kohavi 1996)
dataset.
Law school
Adult
Method
RMSE (4)
MAE (L)
MMD (L)
Wass ())
Accuracy (1)
MMD (1)
Wass (L)
Constant
0.940+0.005
0.762++p.004
0.000
0.000
0.724
0.000
0.000
Full
0.883 10.004
0.701
574.013
82.74648.20
0.791 +0.007
78.392_5.723
6.989_1.738
Unaware
0.917+0.005
0.73140.007
48.738443.8
12.38441.54
0.800
21.729-2.77
2.425
Fair-K
0.904
0.72310.005
5.341
3.980 10.275
0.727
4.381 10.214
1.619
CLAIRE
0.910
0.735
7,8914a.50
4.095 +0.146
2..377..004
5. 1...0
1.671 1.671 20.
EXOC
0.902-10.000 10.05
0.720
4.73920.553
3.87910.236
0.748
3.891 3.891:0.055 20,095
20.05
4.3
ABLATION STUDY
4.3.1
ABLATION ON Y
We perform an ablation study y. shown in Tab. B This experiment evaluates the effect of controlling fairesss-accuraa balance, running on synthetic datasets. The results show that as y increases from I to observe the performance gradually decreases, but the counterfactual
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0.92
0.78
0.90
5.0
0.76
0.88
Accuney
RMSE
6.0
MMD
0.86
0.74
4.0
0.82
STRMSE
MDMMD
0.72
5 S'cccumcy
5DMMD
y: RMSE
9:
2.0
9: Accuracy
Y: MMD
2.0
0.80
1.2
12
188
Y
(a) Law school
(b) Adult
Figure 3: The ablation study on S and y
fairness gradually increases. Also, observe a better fairness- tradeoff, i.e., an increased fairness without sacrificing much accuracy. We attribute this to the introduction the auxiliary node S which serves intrinsic information capable of deducing S. The result aligns with our theoretical analysis in Section 3.3 where node and the custom loss (S can control the fairness- accuracy tradeoff. We observe that when 1.2, there is generally excellent balance between accuracy and fairness. Therefore, we set y= 1.2 in our experiments.
Table 3: The ablation study on
Law school
Adult
RMSE (.)
MAF (4)
MMD (L)
Wass (1)
Accuracy (t)
MMD())
Wass (4)
0.875 20.006
0.608
4.489
3.905
0.765
3.532+0:283
1.539+00245
1.2
0.867 0.867 10.003
0.706
3.83220.623
3.580+0.256
0.760+0.005
2.961 10.124
1.426+0.095
1.4
0.886
0.724440.005
3.377
3.352
0.755
2.628 10.107
1.352
1.6
0.900
0.728
3.08944.421
3..03
0.757+4.005
2.458
1.297 +0.098
1.8
0.903
0.73 0.731 +0.005
2.034.0.322
2.89010.241
0.75 40.006
2.068 40.085
1.204
2
20.003
0.735
1.34220.121
2.82410.204
0.74620.006
1.792.0.074
1.18420.069
4.3.2 ABLATION ON S" AND ?
We perform an ablation study on whether S" or 8 should be used the custom loss, i.e., (S' or (S, .P), where the experiment runs on synthetic datasets and the results are shown in Fig. The results show that when apply S" in the custom loss, decrease on the Law school dataset and an around 0.02 Accuracy increase on the Adult dataset, indicating better performance. This observation aligns with analysis in Section 3.3 Moreover, the fairness metrics are slightly better when we apply s", Therefore, we find it necessary to use S in the custom loss S""
observe around 0.02 RMSE
CONCLUSION
This paper introduces EXOC, a novel framework aimed at achieving counterfactual fairness while addressing the limitations of existing approaches. The key innovation lies in the revelation of in- trinsic properties that are overlooked in previous works, through the introduction of auxiliary node S' and control node S" We demonstrate that they increase counterfactual faimness and also provide more refined control over the flow intrinsic information beneath the concept of faimess and ac- curacy. Moreover, detailed analysis and extensive experimental evaluations both synthetic and

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Published as a conference paper at ICLR 2025
real-world datasets demonstrate the framework's effectivvnesss, showing that EXOC outperforms state-of-the-art models in improving counterfactual fairness without sacrificing much accuracy.
Future work could explore scaling the framework to more complex datasets and simplifying its implementation for broader use. Theoretically, the connections between causal inference and its probability implementations niques for balancing the trade-off between fairness and accuracy, especially in high-dimensional data, could improve scalability and performance.
also of great interest. Developing more efficient optimization tech-
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11


Reviews:
[{"review_id": "p03BRGEZmz", "reviewer": "Reviewer_5RPc", "summary": "This work aims to address the issue that previous methods ignored the intrinsic information in the exogenous variable U that does not lead to discrimination in predicting labels, leading to degraded performance. They propose a new causal graph with latent variables S' and S''.", "questions": "1. How is validation/model selection performed for the experiments?\n2. Why do the authors only consider S' and S''? Is there any other latent variable that can be considered in that are not in the causal graph of Fig. 1 (b)?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed framework allows a trade-off between utility and fairness by adjusting the strength of the correlation between S' and S''.\n- In linear cases, they provide bounds on the counterfactual fairness error to show the benefit of S'. They also provide an analysis with unknown functions in the causal graph.\n- Strong empirical performance across different datasets and settings.\n- Ablation study on \\gamma and S'' supported the analysis in Section 3.", "weaknesses": "- The proposed method heavily rely on the assumed causal graph in Fig. 1 (b). I wonder how general this causal graph can be.\n- The experiments are only conducted on two small tabular datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OAj2JaMqLS", "reviewer": "Reviewer_LPsA", "summary": "This paper introduces the EXOC framework, a causal model leveraging auxiliary and control nodes to enhance counterfactual fairness while maintaining predictive accuracy. It shows improved fairness compared to existing methods, validated on both synthetic and real-world datasets.", "questions": "1) Could you provide more details on the approach to tuning hyperparameters, especially γ, within the causal structure? How sensitive are your fairness and accuracy results to changes in this parameter, and what guidelines would you recommend for tuning it on different datasets?\n\n2) Could you offer a practical, real-world scenario where the auxiliary node S′ and control node S′′ would interact causally in the manner illustrated in Figure 1? Examples or applications with concrete causal relationships would help clarify the intended structure and interpretability of the model.\n\n3) Why did you choose a VAE for synthetic data generation over more conventional methods in fairness literature? Additionally, could you provide performance metrics or examples demonstrating the validity of the VAE-generated counterfactuals?", "limitations": "", "rating": 5, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1) EXOC introduces a new approach by utilizing auxiliary variables to capture latent information, improving counterfactual fairness and accuracy, which is new to my knowledge\n2) Demonstrates competitive accuracy and fairness on benchmark datasets, outperforming prior models. Though in many cases the improvement is only with accuracy or fairness, not as impressive as claim in the intro.", "weaknesses": "1) The paper does not adequately explain the process for tuning hyperparameters, particularly within the causal structure. Specifically, the role of hyperparameter 𝛾 in balancing fairness and accuracy is crucial, yet the approach to fine-tuning this balance remains unclear. And I am wondering if a efficient tuning strategy exist as it involves a tradeoff.\n2) The causal diagram in Figure 1, illustrating the roles of auxiliary nodes S' and control nodes S'' , lacks practical grounding. The authors could strengthen their framework by providing realistic examples or settings in which these nodes causally impact the system in a way that aligns with the proposed structure. This would improve interpretability and offer a clearer justification for the chosen causal relationships.\n3) Apart from above two, a major weakness I would point out is the experiment section:\n\nAlthough synthetic data is a valid approach, as we could assume a groundtruth, Generating synthetic data with a Variational Auto-Encoder (VAE) introduces additional assumptions and potential challenges, such as ensuring the VAE accurately represents counterfactual distributions. The authors do not report VAE performance metrics or provide convincing examples of generated counterfactuals, which makes it difficult to verify if the synthetic data truly captures the desired counterfactual fairness properties. Without this validation, the reliability of the synthetic evaluation is unclear.\n\nBoth the use of real-world and synthetic datasets is very limited. For example, other commonly used fairness benchmarks include insurance dataset, Crime dataset. More datasets would strengthen the validation and help demonstrate EXOC’s assumption robustness across diverse real-world settings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jo5hsCEHPH", "reviewer": "Reviewer_HYEE", "summary": "This paper aims to achieve counterfactual fairness through the introduction of auxiliary variables $S’$ and control node $S’’$.", "questions": "Please refer to the weaknesses above.", "limitations": "", "rating": 3, "confidence": 4, "soundness": 1, "presentation": 1, "contribution": 1, "strengths": "The introduction of auxiliary variables $S’$ and control node $S’’$ is interesting.", "weaknesses": "- The authors assert in the introduction part that their framework can \"enhance fairness without compromising accuracy.\" However, the evidence supporting this claim is not readily discernible in the paper. A clearer demonstration or justification of this statement is needed.\n\n- The methodology section lacks clarity in several areas. For instance, in lines 199-201, the statement \"As $\\alpha(s*−s)$ is the counterfactual parity that plays a more important role than the standard deviation, we showcase that δa>δb\" is not intuitively clear. Additionally, there is no distinction between probability inference and causal inference, leaving the reader confused about the framework. Furthermore, the rationale behind choosing $S''$ over $\\hat{Y}$ is unclear. Lines 345-350 do not provide enough context or reasoning.  \n\n- The experimental results raise several questions. Specifically, why does the EXOC model outperform the Full model in accuracy when $\\gamma=1$ on the Law School dataset? This outcome appears coincidental, and further analysis or explanation is necessary.\n\n- The paper does not include comparisons with other models aimed at achieving counterfactual fairness, such as mCEVAE (Pfohl et al., 2019), DCEVAE (Kim et al., 2021), ADVAE (Grari et al., 2023), or CFGAN (Xu et al., 2019). A comparison with these established methods is necessary for evaluating the effectiveness and innovation of the proposed framework.\n\nStephen R. Pfohl, Tony Duan, Daisy Yi Ding, and Nigam H Shah. Counterfactual reasoning for fair clinical risk prediction. In Machine Learning for Healthcare Conference, 2019.\n\nHyemi Kim, Seungjae Shin, JoonHo Jang, Kyungwoo Song, Weonyoung Joo, Wanmo Kang, and Il.Chul Moon. Counterfactual fairness with disentangled causal effect variational autoencoder. In AAAI, 2021.\n\nVincent Grari, Sylvain Lamprier, and Marcin Detyniecki. Adversarial learning for counterfactual fairness. Machine Learning, 2023.\n\nDepeng Xu, Yongkai Wu, Shuhan Yuan, Lu Zhang, and Xintao Wu. Achieving causal fairness through generative adversarial networks. In IJCAI, 2019.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "M6UVn9gpdD", "reviewer": "Reviewer_ZZ7B", "summary": "The paper presents a novel framework called EXOC designed to achieve counterfactual fairness using auxiliary variables. By introducing an auxiliary node and a control node, the framework uncovers information from intrinsic properties that previous works have overlooked. EXOC demonstrates improved counterfactual fairness and offers finer control over the flow of intrinsic information, enhancing both fairness and accuracy. The framework is evaluated on synthetic and real-world datasets, showing superior performance compared to state-of-the-art models in terms of both counterfactual fairness and accuracy.", "questions": "1. The introduction of causal functions involves the presence of $U$, but its specific contribution to the analysis is unclear. I also have concerns about the path analysis in your framework, as it appears to condition on specific paths (e.g., $\\pi_{\\vartheta}$￼ and $\\pi_{b}$). Could other unconsidered paths also affect counterfactual fairness? If so, could these alternative pathways introduce unintended changes to fairness outcomes? How does your framework address or mitigate these potential effects?\n\n2. Deterministic models like MLP and ResNet-50 are widely used due to their simplicity and effectiveness in optimization. Could you clarify the rationale for opting for probabilistic models and probabilistic inference in this context? Additionally, please elaborate on the relationship between probabilistic inference and causal inference in your framework. How do these two forms of inference interact, and what benefits does this interaction offer for achieving fairness and accuracy?\n\n3. The trade-off between fairness and accuracy is well-documented in fairness research, often implying that gains in accuracy come at the cost of fairness. Could you elaborate on how controlling information flow differs from managing this trade-off, and provide justification for the distinction?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The theoretical analysis is compelling. The authors first outline an ideal model scenario, clearly demonstrating improved counterfactual fairness. They then extend their analysis beyond these assumptions to broader contexts, showing consistent performance improvements. The connection to information flow is creative, highlighting how the presence of $U$￼ leads to information leakage from $K$￼to $S$￼, while introducing $S'$￼ helps regulate the flow from ￼$S$ to $Y$￼.\n\n2. The visual representations in Figure 2 effectively illustrate the impact of the control node. The clear differentiation between ￼$P(\\hat{S})$ and $P(S)$￼ offers a fresh perspective on fairness issues, making the framework’s approach to counterfactual fairness more intuitive.\n\n3. The paper is well-written, with clear and coherent presentation of research concepts. The experiments are thorough and well-designed, providing strong evidence to support the effectiveness of the proposed framework.", "weaknesses": "The introduction could be more comprehensive by including additional related works. While the authors discuss counterfactual fairness within the ML domain, there is a growing body of research that could complement the current submission, such as:\n\n[1] Garg, Sahaj, et al. “Counterfactual fairness in text classification through robustness.” Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, 2019.\n\n[2] Madras, David, et al. “Fairness through causal awareness: Learning causal latent-variable models for biased data.” Proceedings of the Conference on Fairness, Accountability, and Transparency, 2019.\n\n[3] Zuo, Zhiqun, Mahdi Khalili, and Xueru Zhang. “Counterfactually fair representation.” Advances in Neural Information Processing Systems 36, 2023.\n\nIncluding these references would strengthen the discussion of related works.\n\nIn Section 3.2.1, the authors provide an example to illustrate improvements in counterfactual fairness, further extended in Section 3.2.2. However, some aspects of the explanation could be clearer—please see the questions below for more details.\n\nIt is not clear why the authors chose a probabilistic model to implement the causal model, given that deterministic models (e.g., MLPs with activation functions) are often effective for inference tasks.\n\nSome experimental settings and hyper-parameters could be better reported to ensure clarity and reproducibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)