Conformalized Fairness via Quantile Regression
Anonymous Author(s)
AffiliationAddress email
Abstract
Algorithmic fairness has received increased attention in socially sensitive domains.
While rich literature on mean fairness has been established, research on quantilefairness remains sparse but vital.
To fulfill great needs and advocate the significanceof quantile fairness, we propose a novel framework to learn a real-valued quantilefunction under the fairness requirement of Demographic Parity with respect to sensitive attributes, such as race or gender, and thereby derive a reliable fair predictioninterval.
Using optimal transport and functional synchronization techniques, weestablish theoretical guarantees of distribution-free coverage and exact fairness forthe induced prediction interval constructed by fair quantiles.
A hands-on pipelineis provided to incorporate flexible quantile regressions with an efficient fairnessadjustment post-processing algorithm.
We demonstrate the superior empiricalperformance of this approach on several benchmark datasets.
Our results showthe model’s ability to uncover the mechanism underlying the fairness-accuracytrade-off in a wide range of societal and medical applications.
Introduction
We are increasingly leaning on machine learning systems to tackle human problems.
A primaryobjective is to develop intelligent algorithms that can automatically produce accurate decisions whichalso enjoy equitable properties, as unintended social bias has been identified as a major concern[13, 17].
As a means of providing quantitative measures of fairness, a number of metrics have been proposed. These metrics can be categorized into three broad categories: group fairness [3], individual fairness[22], and causality-based fairness [27].
In contrast to causality-based fairness that requires domainknowledge to develop a fair causal structure and individual fairness that seeks equality only betweensimilar individuals, group fairness does not require any prior knowledge and seeks equality for groupsas a whole [6]. Among the metrics defined for group fairness such as equalized odds [9, 26] andpredictive rate parity [10], demographic parity (DP) is generic since it does not allow predictionresults in aggregate to depend on sensitive attributes [1, 19, 12, 34]. In particular, an algorithm is saidto satisfy DP if its prediction is independent of any given sensitive attribute. There have been a number of studies on algorithmic fairness concerning DP [1, 11, 12, 19, 28, 34].
In the context regression analysis, much attention have been paid on conditional mean inferences[1, 11, 12, 28], few are concerned with conditional quantiles [32, 34]. As real-world data oftenexhibit heterogeneity, contain extreme outliers, or do not meet satisfactory distributional assumptions,like Gaussianity, a fairness discussion on conditional quantiles may be more rational and essentialsince they are able to provide a more complete understanding of the dependence structure betweenresponse and explanatory variables [34], as well as better accommodate asymmetry and extreme tail behavior [33].
It should also be noted that bias or unfairness that arises in mean regression mayalso be propagated through quantile regression, therefore it must be properly dealt with separately: agraphic demonstration can be found in Figure 1. More intuitively, we may take an example from a Spanish labor market study [16, 17].
The study found that in Spain, also in line with other countries,the mean wage gap between men and women is quite substantial: on average, women earn around 70percent of what men earn. While wage gaps are not uniform across all pay scales, they are greaterat higher quantiles than at lower quantiles.
As biases and disparities at different quantiles tend tobe overshadowed by the mean behavior of the entire population, we propose a novel frameworkfor searching for fair predictions at different quantiles. It uses optimal transport techniques [2, 12]by transforming bias-affected distributions into an only-fair Wasserstein-2 barycenter through akernel-based functional synchronization method [8, 38], in order to provide fair quantile estimators.
Since quantile fairness poses a number of theoretical challenges, no previous literature has beenable to provide any inference results such as prediction intervals concerning quantile fairness. Itis imperative to keep in mind that fairness is only one of two legs of the primary goal of modernmachine learning algorithms, the other being accuracy.
Building a reliable prediction with validconfidence is a significant challenge that is encountered by many machine learning algorithms [37]. Towards this end, we propose the conformalized fair quantile prediction (CFQP) inspired by theworks of Romano et al.
Our analysis demonstrates, both mathematically and experimentally,that CFQR provides finite sample, distribution-free validity, DP fairness for different quantiles, andprecise control of the miscoverage rate, regardless of the underlying quantile algorithm.
Contributions and Outlines.
In this paper, we propose a new quantile based method with validinference that enhances both accuracy and fairness while maintaining a balance between the two. It isa novel framework that allows an exact control of quantile prediction miscoverages while ensuringquantile fairness simultaneously. The main contributions are summarized as follows:
i. We successfully transform the problem of searching quantiles under DP fairness to the construction of multi-marginal Wasserstein-2 barycenters via the optimal transport theory [2, 12, 18].We incorporate a novel kernel smoothing step into the preceding method, which is particularlyadvantageous for subgroups whose sample sizes are too small to obtain reliable quantile functionestimations.ii.
In Section 4, we propose a conformalized fair quantile regression prediction interval (CFQP) inspired by the works of Romano et al.
It is mathematically proved to achieve a finitesample, distribution-free validity, demographic parity on different quantiles, and an exact controlof miscoverage rates, regardless of the quantile algorithm used.
The theoretical validity ofprediction interval constructed by CFQP and exact DP of the fair quantile estimators are given inSection 5 and the supplement.iii.
The experimental results presented in Section 6 include a numerical comparison of the proposedCFQP and fair quantile estimation with both state-of-the-art conformal and fairness-oriented
methods. By reducing the discriminatory bias dramatically, our method outperforms the state-ofthe-art methods while maintaining reasonable short interval lengths.
Related works.
Existing approaches for building a fair mean regression broadly fall into three classes:pre-processing, in-processing and post-processing.
In particular, preprocessing methods focus ontransforming the data to remove any unwanted bias [5, 27, 36]; in-processing methods aim to buildin fairness constraints into the training step [1, 4, 22, 26]; post-processing methods target to modifythe trained predictor [11, 12, 25].
As few previous works have focused on the quantile fairness ofand fair prediction interval, the most related are Yang et al.
[34], where a different fairness measurewas used. While Agarwal et al. [1] mentioned that their reduction-based approach can be adaptedinto quantile regression, Williamson and Menon [32] brought forward a novel conditional varianceat risk fairness measure aiming to control the largest subgroup risk. For interval fairness measure,the approach by Romano et al.
[29] achieved equalized coverage among groups without fairnesson interval endpoints.
Methodologically, integrating algorithmic fairness with Wasserstein distancebased barycenter problem has been studied in [2, 11, 12, 18, 20].
Both in-processing [1, 20] andpost-processing [11, 12] methods were proposed to solve classification and mean regression problems.
As a post-processing method, our work is distinct from above-mentioned methods by constructingthe DP-fairness for each population quantile, and generating a fair prediction interval accordingly.
Notations.
We denote by [ K ] the set { 1 , . . .
, K } for arbitrary integer K .
|S| represents the cardinalityfor a finite set S .
E and P represent the expectation and probability and 1 {·} is the indicator function.
Let { Z n } ∞ n =1 be a sequence of random variables, and { k n } ∞ n =1 be a sequence of positive numbers,we say that Z n =
O p ( k n ) , if lim T →∞ lim sup n →∞ P ( | Z n | > Tk n ) = 0 , then Z n /k n =
O p (1) .
Todenote the equality in distribution of two random variables A and B , we write A d = B .
Problem statement
Consider the regression problem where a “sensitive characteristic” S is available, which by the
U.S. law [18, 29] can be enumerated as sex, race, age, disability, etc.
One observes the triplets( X 1 , S 1 , Y 1 ) , . . .
, ( X n , S n , Y n ) , where we denote ( X i , S i , Y i ) by Z i , i = 1 , . . . , n and Z i is arandom variable in R p ×
[ K ] × R , and aim to predict the unknown value of Y n +1 at a test point X n +1 , S n +1 .
Let P be the joint distribution of Z , we assume that all the samples { Z i } n +1 i =1 aredrawn exchangeable,
where i.i.d. is a special case.
Our goal is to construct a marginal distribution-free prediction band C ( X n +1 , S n +1 ) ⊆
R that islikely to cover the unknown response Y n +1 with finite-sample (nonasymptotic) validity.
Formally,given a desired miscoverage rate α , the predicted interval satisfies
[Equation]
for any joint distribution P and any sample size n , while the left and right endpoint of C ( X n +1 , S n +1 )satisfies the fairness constraint of Demographic Parity concerning the sensitive variable S .
Demographic Parity. We introduce the quantitative definition of DP on fair regression and connectthe DP-fairness with a quantile regressor q α . We invoke the result that q α can be projected tofairness using optimal transport. In particular, given a fixed quantile level α (it may refer to q α lo orq
α hi indicating the left and right endpoint, which will then given by the upper and lower quantileestimates for the prediction band C ( X n +1 , S n +1 ) ), let q α : R p ×
[ K ] → R represent an arbitraryconditional quantile predictor.
Denote by ν q α | s the distribution of ( q α ( X, S ) | S = s ) , the Cumulative
Distribution Function (CDF) of ν q α | s is given by
[Equation]
The quantile function Q ν qα | s = F − 1 ν qα | s : [0 , 1] → R ,namely, the generalized inverse of F ν qα | s , canthus be defined as for all levels t ∈ (0 , 1] ,
[Equation]
Q
[Equation]
qα | s
[Equation]
[Equation]
Q
[Equation]
qα | s
[Equation]
[Equation]
To simplify the notations, we will write F q α | s and Q q α | s instead of F ν qα | s and Q ν qα | s respectively,for any prediction rule q α .
In the following, we introduce the definition of Demographic Parity (DP), which is most commonlyused in the context of fairness research [1, 11, 12, 19, 26].
Definition 1 (Demographic Parity) .
An arbitrary prediction g : R d × [ K ] → R satisfies demographic parity under a distribution P over ( X, S, Y ) if g ( X, S ) is statistically independent of the sensitive attribute S , namely, for every s, s ′ ∈
[ K ] ,
[Equation]
Demographic Parity (DP) requires the predictions to be independent of the sensitive attribute, and itdemands the Kolmogorov-Smirnov distance [23] (the difference between CDFs measured in the l ∞norm) between ν g | s and ν g | s ′ to vanish for all categories s, s ′ .
Quantile Regression and Conformal Prediction
In this section, we recall the CQR approach for finite sample, distribution-free prediction intervalinference.
Quantile regression was proposed by Koenker and Bassett [21] to estimate the α -th quantileof the conditional distribution of Y given ˜ X :=
( X, S ) for some quantile level α ∈ (0 , 1) , sincethen it has become more pervasive with various applications, such as providing prediction intervals,detecting outliers, or perceiving the entire distribution [24].
Denote the conditional cumulativedistribution of Y given ˜ X by F ( y | ˜ X = ˜ x ) :
= P { Y ≤ y | ˜ X = ˜ x } .
The α -th conditional quantileprediction is defined as q α (˜ x ) :=
inf { y ∈ R : F ( y | ˜ X = ˜ x ) ≥ α } .
Quantile regression can becast as an optimization problem[24], by minimizing the expected check loss function E ( ρ α ) =
E [ ρ α ( y, q ) | ˜ X = ˜ x ] , where
[Equation]
Quantile regression offers a principled way of judging the reliability of predictions by buildinga prediction interval for the new observation ( ˜ X n +1 , Y n +1 ) .
In contrast to asymptopia, Romanoet al.
[29, 30] brought forward the conformalized quantile regression (CQR) by combining themerits of robust quantile regression with conformal prediction; thus finite sample validity in (1)is guaranteed.
Inspired by the split conformal method, a split CQR likewise starts with splittingthe data into a proper training set and a calibration set, indexed by I 1 , I 2 respectively.
Given anyquantile regression algorithm Q , we then fit two conditional quantile functions ˆ q α lo and ˆ q α hi onthe proper training set: { ˆ q α lo , ˆ q α hi } ← Q (cid:16)(cid:110)(cid:16)
˜ X i , Y i (cid:17) : i ∈ I 1 (cid:111)(cid:17) .
Then the conformity scores arecalculated to quantify the error made by the plug-in prediction interval ˆ C (˜ x ) =
[ˆ q α lo (˜ x ) , ˆ q α hi
(˜ x )] .
We evaluate the scores on the calibration set as E k := max (cid:110)
ˆ q α lo
( ˜ X k ) − Y k , Y k − ˆ q α hi
( ˜ X k ) (cid:111)for each k ∈ I 2 , where both undercoverage and overcoverage of the interval are taken intoconsideration [30].
Given a new input data ˜ X n +1 , we construct the prediction interval for Y n +1 as C (cid:16) ˜ X n +1 (cid:17) = (cid:104)
ˆ q α lo (cid:16) ˜ X n +1 (cid:17) − Q 1 − α ( E, I 2 ) , ˆ q α hi (cid:16) ˜ X n +1 (cid:17) + Q 1 − α
( E, I 2 ) (cid:105) , where
Q 1 − α ( E, I 2 ) :
= (1 − α )(1 + 1 / |I 2 | )
-th empirical quantile of { E k : k ∈ I 2 } conformalizes theplug-in prediction interval.
Note that the constructed interval C ( ˜ X n +1 ) could be highly influencedby the sensitive variable S .
Conformal fair quantile prediction (CFQP)
We formally describe our proposed conformal fair prediction (CFQP) framework for constructing
DP fairness constrained prediction intervals in this section.
A kernel smoothing quantile functionis introduced during the functional synchronization, which can improve the estimation when somesubgroups are too small to give reliable sample quantile function estimations.
Definition 2 (Wasserstein-2 distance) .
Let µ and ν be two univariate probability measures with finite second moments.
The squared Wasserstein-2 distance between µ and ν is defined as
[Equation]
where Γ µ,ν is the set of probability measures (couplings) on R × R having µ and ν as marginals.
Proposition 1 (Fair optimal prediction [12]) .
Assume, for each s ∈
[ K ] , that the univariate measureν q α | s has a density and let p s =
P ( S = s ) .
Then,
[Equation]
Moreover, if g α and ν solve the l.h.s.
and the r.h.s.
problems respectively, then ν =
ν g α andspecifically,
[Equation]
Proposition 1 implies that the optimal fair quantile predictor for an input ( x, s ) is obtained by anonlinear transformation of the vector [ q α ( x, s )]
Ks =1 linking to a Wasserstein barycenter problem[2,12].
The explicit closed form solution comes from [2, 12, 15], which relies on the classical characterizationof optimal coupling in one dimension of the Wasserstein-2 distance.
A rigorous proof is given in[12, 20].
It shows that a minimizer g α of the L 2 -risk can be used to construct ν and vice-versa, givenν , there is a explicit expression Eq.
(6) for the multi-marginal Wasserstein barycenter [2].
We start with splitting the whole training data into a proper training set I 1 and a calibration set I 2 , thenfit an arbitrary quantile regression algorithm Q on I 1 , { ˆ q α lo , ˆ q α hi } ← Q (cid:16)(cid:110)(cid:16)
˜
X i , Y i (cid:17) : i ∈ I 1 (cid:111)(cid:17) . We apply the fitted quantile algorithm Q on the calibration set I 2 to obtain the predicted{ ˆ q α lo ( ˜ X i ) , ˆ q α hi ( ˜ X i ) } i ∈I 2 . Since the quantile estimates for I 2 will be used for conformalization,it is essential to transform them into fair ones, i.e. ˆ g α,i , ∀ i ∈
I 2 (Eq.
(9)), through Algorithm 2. Finally, for a test point ˜ X n +1 , we will predict two quantile estimates ˆ q α ( x, s ) affected by the sensitivevariable S by Q , then apply the functional synchronization (details in Algorithm 2) and calibration(Algorithm 1) steps in turn to generate the fair constraint prediction interval C ( ˜ X n +1 ) for Y n +1 . Next, we explicate in detail how to remove the effect of the sensitive variable for the predictedquantile estimates.
By Proposition 1, the optimal fair quantiles take the form of Eq.
Therefore,we propose an empirical optimal fair quantile estimator ˆ g α that relies on the plug-in principle.
Inparticular, Eq.(6) indicates that for each quantile level α and each category s ∈
[ K ] , we only needestimators for the regression function ˆ q α , the proportion ˆ p s , the cumulative distribution function F ˆ q α | s and the quantile function Q ˆ q α | s .
Note that we can empirically estimate the CDF and quantile function for each sensitive group in the calibration set I 2 separately.
Hence for each quantile level α , let N s := |I
s 2 | , and the quantile estimators (ˆ q s 1 , ˆ q s 2 , . .
, ˆ q sN s ) 1 are calculated through the fitted quantile regression Q with training set
I 1 .
We define the augmented random variable for each data point in I 2 ,
[Equation]
where U si are i.i.d.
random variables, uniformly distributed on [ − σ, σ ] for some small positive σ ,and independent from all the previously introduced random variables.
it serves as a smoothingrandom variable, for the random variables ˜ q si := ˆ q si + U si are i.i.d.
continuous for any P ( Y | ˜ X ) and
Q .
Otherwise, the original ˆ q si might have atoms resulting in a non-zero probability to observe tiesin the group { q si } for s = 1 , . . .
, K .
This trick, also called jittering [7, 12] is often used for data visualization for tie-breaking.
Using the above quantities, we build the CDF and quantile functionestimators for each subgroup s ′ ∈
[ K ] as follows,
[Equation]
The smoothed kernel estimator Eq.(8) was firstly proposed by Cheng and Parzen [8], where K h ( · ) =
K ( · /h ) /h is a kernel function chosen as a probability density function that is symmetric around zerowith bandwidth parameter h > 0 .
If the quantile functions Q 2 ,q α | s ′ is differentiable, the derivative Q ′ s ′ ( t ) :
= Q ′ 2 ,q α | s ′ ( t ) for t ∈ (0 , 1)is the quantile density function [8, 38].
We hereby give an estimation bound for Eq.
(8) using kernelsmoothing.
For this purpose, we invoke the conditions (A1) - (A3) that are needed for deducingthe following proposition.
They can also be found from [38] and are included in the supplementarymaterial.
Proposition 2.
Under conditions (A1), (A2), and (A3), we have
[Equation]
The motivation for including a smoothing step is twofold: First, smoothing the quantile functioneliminates the troublesomeness in defining arbitrary quantiles from the empirical one when thesample sizes of subgroups are small.
Second, the proposed kernel smoothing improves second-orderefficiency by alleviating the so-called relative deficiency [14, 38].
Remark 1.
One can utilize various kernels such as the Gaussian or Epanechnikov kernel withadaptive bandwidth for better practical performance.
Furthermore, other smoothing methods such assplines or local linear fitting can be applied with equal effectiveness.
Consequently, for each quantile level α , the functional synchronized quantile estimator is
[Equation]
The proposed estimator can be deemed as the empirical counterpart with additional randomization ofthe explicit fair optimal formula Eq.(6).
To conformalize the adjusted fair quantiles Eq (9), we need to compute the conformity scores
E i for each i ∈ I 2 that quantify the error made by the plug-in fair prediction interval
ˆ C g
˜ x ) =[
ˆ g α lo (˜ x ) , ˆ g α hi (˜ x )] .
The scores are evaluated on the calibration set as
[Equation]
At the last stage, for a new data point ˜ X n +1 =
( x, s ) , and α ∈ { α lo , α hi } , by defining ˜ q s 1 ,
i =
ˆ q si + U si ([ − σ, σ ]) ∀ i ∈
I sand ˜
q α ( x, s )
= ˆ q α ( x, s )
+ U ([ − σ, σ ]) .
We use the empirical CDF of training set {+
+}
[Equation]
to estimate the location ˆ F 1 ,q α | s ◦ ˜
q α ( x, s ) .
Thus the fair quantile estimator is built as follows
[Equation]
The fair prediction interval for Y n +1 is constructed as
[Equation] [Equation]
where Q 1 − α ( E, I 2 ) :
= (1 − α )(1 + 1 / |I 2 | )
-th empirical quantile of { E i : i ∈ I 2 } will adjust theplug-in fair prediction interval.
We present the pseudo-codes of CFQP as well as the construction ofˆ g α for Eq. 9 in Algorithm 1, 2 respectively.
= ( x, s ) .
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Theoretical results
In this section, we provide a statistical analysis of the proposed algorithm with coverage and DPfairness guarantees.
Theorem 1 (Prediction coverage guarantee) .
If ( ˜ X i , Y i ) , i = 1 , . . . , n + 1 are exchangeable, then the prediction interval C ( ˜ X n +1 ) constructed by the split CFQP algorithm satisfies
[Equation]
Moreover, if the conformity scores E i are almost surely distinct, the prediction interval is nearly exactly calibrated,
[Equation]
Remark 2.
We give an extension for the conformalization step which allows coverage errors to bespread arbitrarily over the left and right tails as Corollary 1 in the supplement.
Controlling the leftand right tails independently yields a stronger coverage guarantee.
Theorem 2 (Demographic parity guarantee) .
For any joint distribution P of ( X, S, Y ) , any σ > 0 , as well as the base quantile estimator
ˆ q α : R p ×
[ K ] → R constructed on labeled data, the estimator ˆ g α defined in Eq.
(12) satisfies
[Equation]
To prove Theorem 2, we leverage distribution-free properties on rank and order statistics presented in
Lemma 3 in the supplementary material.
Theorem 2 and the estimator
ˆ g improve upon the estimatorof Chzhen et al.
[12], Chzhen and Schreuder [11], for which only approximate fairness is established.
Experiments
To evaluate our proposed method 3 , we report the performance of post-processing fairness adjustmenton quantiles through four benchmark datasets: Law School (LAW), Community&Crime (CRIME),
MEPS 2016 (MEPS), Government Salary (GOV).
A detailed description of these datasets can befound in the supplementary material.
We measure the violation of DP-fairness of the quantiles required by Definition 1 through theempirical Kolmogorov-Smirnov (KS) distance.
The value represents the disparity between groups
Z s = { ( X, S, Y ) ∈
Z : S = s } for all s ∈ [ K ] ,
[Equation]
Experiment results.
In Table 1, we report the average performance of the proposed CFQP over200 randomly training-test splits as well as the baseline model CQR by the coverage rate, length ofprediction interval, and the KS distance of the interval endpoint.
80% of the examples are used fortraining, and the remaining 20% to test.
We split the training part into proper training and calibrationsets of equal sizes.
Throughout the experiments, the nominal miscoverage rate is fixed to α = 0 .
Among preexisting quantile algorithms, we select three leading variants: linear model [21], randomforests [24] and neural networks [31].
Overall, our CFQP likewise CQR constructs prediction bandsattaining desirable coverage around 90%, as claimed in Theorem 1.
Random forest based approachestend to be slightly more conservative than the other two w.r [Table] t the coverage rate [Table]
[Table] regression neural network In the KS column concerning the DP-fairness of interval endpoints, our CFQP method greatly reducesthe discriminatory bias (quantified by KS) by 70% up to 90% compared to that of CQR.
In addition,the lengths of the prediction intervals mostly remain the same except for the Crime dataset, which isdue to the reason of its inherent high discriminatory bias among sensitive groups.
Figure 2 presents the comparison of our post-processing fairness adjustment procedure on quantilesˆ g α on the test set Z test =
{ ( X i , S i , Y i ) } n test i =1 with some state-of-the-art fairness algorithms.
{+Since
+}
we utilize the local linear fitting smoothing method in the experiments.
most of the algorithms are targeted at mean prediction, there is no direct comparison with our quantilefairness method; we accordingly modified the existing methods into quantile versions for comparison.
A detailed description can be found in the supplementary material.
The points in Figure 2 represents the mean of 200 repeated experiments with x -axis as KS distanceand y -axis as Mean Absolute Error(MAE), defined as MAE( g α ) = 1 / n test (cid:80) Z test | Y − g α
( X, S ) |to measure the prediction error of quantiles, the bars is the standard error on both axis.
The optimalpoint should be located at the bottom left corner of the graph, where the smaller KS distance, aswell as smaller MAE, are achieved.
In each subplot, our method consistently performs better withthe smallest KS distance while keeping the MAE equal or slightly higher than the others or unfairversion.
Overall, CFQP is more robust according to the standard errors over experiment repetitions.
Note that due to the highly right skewness of real datasets, the MAE of the upper quantile estimationis larger than that of the lower quantile for all approaches as well as datasets.
Conclusion and future work
Conformal fair quantile regression is a novel approach for creating fair prediction intervals thatattain valid coverage and reach independence between sensitive attributes while making minimalmodifications to the quantile endpoints simultaneously.
It becomes superior within heteroskedasticand/or asymmetric datasets and robust to outliers.
CFQP is supported by rigorous distribution-freecoverage and exact DP-fairness guarantees, as proved in theoretical parts.
We conducted severalreal data examples demonstrating the effectiveness of our method in achieving exact coverage whileimposing DP-fairness in practice.
The method outperforms several state-of-the-art approaches bycomparison.
A limitation in our numerical experiments is that we simply utilize the local linear smoothing methodin defining quantile functions of the subgroups; we believe incorporating flexible kernel smoothingapproaches [35, 38] would improve the experimental performances.
As potential future works, itwould be valuable to introduce a DP relaxation framework based on an unfairness measure in a similarmanner as [11, 32], allowing controlling the level of unfairness in quantile estimates.
We also expectto extend the scope to other potential fairness metrics which is dependent on the underlying responselike equalizing quantile loss across groups by incorporating a fairness penalty term in training, or thefairness metric defined for conditional variance-at-risk.
Checklist
For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
[Yes](b) Did you describe the limitations of your work?
[Yes] See Section 7(c)
Did you discuss any potential negative societal impacts of your work?
[No] We currently don’tforesee the potential negative societal impacts of our work, however, we anticipate this methodto be used in the future by practitioners to real-life scenarios involving job markets, medicalscience, etc., and to potentially help making decisions that would protect the vulnerable andpropel social harmony. (d) Have you
read the
ethics review guidelines and ensured that your paper conforms to them
Yes
If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
[Yes] (b) Did you include complete proofs of all theoretical results?
[Yes]
If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
[Yes]
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