Fair Bayes-Optimal Classifiers Under Predictive Parity
Anonymous Author(s)
Abstract
Increasing concerns about disparate effects of AI have motivated a great deal ofwork on fair machine learning.
Existing works mainly focus on independenceand separation-based measures (e.g., demographic parity, equality of opportunity,equalized odds), while sufficiency-based measures such as predictive parity aremuch less studied.
This paper considers predictive parity, which requires thesame probability of success given a positive prediction, among different protectedgroups.
We prove that, if the overall performances of different groups vary onlymoderately, all fair Bayes-optimal classifiers under predictive parity are group-wisethresholding rules.
Perhaps surprisingly, this may not hold if group performancelevels vary widely; in which case we find that predictive parity among protectedgroups may lead to within-group unfairness.
We then propose an algorithm we call
FairBayes-DPP, aiming to ensure predictive parity when our condition is satisfied.
FairBayes-DPP is an adaptive thresholding algorithm that aims to achieve predictiveparity, while also seeking to maximize test accuracy.
We provide supportingexperiments conducted on synthetic and empirical data.
Introduction
Due to an increasing ability to handle massive data with extraordinary model accuracy, machinelearning (ML) algorithms have achieved remarkable success in many applications, such as computervision [43, 45, 23, 46] and natural language processing [44, 48, 12, 54].
However, empirical studieshave also revealed that ML algorithms may incorporate bias from the training data into modelpredictions.
Due to historical biases, vulnerable groups are often under-represented in available data[26, 59, 47].
As a consequence, without fairness considerations, ML algorithms can be systematicallybiased against certain groups defined by protected attributes such as race and gender.
As algorithmic decision-making systems are now widely integrated in high-stakes decision- makingprocesses such as in healthcare [21] and criminal prediction [26], fair machine learning has grownrapidly over the last few years into a key area of trustworthy AI.
A main task in fair machine learningis to design efficient algorithms satisfying fairness constraints with a small sacrifice in model accuracy.
This field has made substantial progress in recent years, as many effective approaches have beenproposed to mitigate algorithmic bias [56, 32, 3, 51, 4, 9, 34, 10, 24, 5, 57].
An important fundamental benchmark for fair classification is provided by fair Bayes-optimalclassifiers, which maximize accuracy subject to fairness [35, 57].
A key class of classifiers isgroup-wise thresholding rules (GWTRs) over the feature-conditional probabilities of the target label,for each protected group (e.g., probability of repaying a loan given income).
Intuitively, beinga GWTR is a minimal requirement for within-group fairness: the most qualified individuals areselected in every group.
[8, 35, 7, 1, 42, 57] have studied fair Bayes-optimal classifiers under variousfairness constraints and proved that, for many fairness metrics, the optimal fair classifiers are GWTRs.
Moreover, the associated thresholds can be learned efficiently [35, 57].
Current literature on Bayes-optimality focuses mainly on the independence- and separation-basedfairness measures (e.g., demographic parity, equality of opportunity, equalized odds; see Section2.1 for definitions and a review).
However, sufficiency-based measures such as predictive parity areless commonly considered, possibly due to the complexity of their constraints.
Sufficiency-basedmeasures are often applied to assess recidivism prediction instruments [19, 13, 6].
[30] show thata particular sufficiency-based measure, group calibration, is implicitly favored by unconstrainedoptimization: calibration error is bounded by the excess risk over the unconstrained Bayes-optimalclassifier.
For selective classification, [29] find that sufficiency-based representation learning leads tofairness.
Despite these findings, little is known about (1) what are the optimal fair classifiers undersufficiency-based measures and (2) how to learn them.
In this paper, we aim to answer these twoquestions. We consider predictive parity, which requires that the positive predictive value (probabilityof a successful outcome given a positive prediction) be similar among protected groups.
In creditlending, for example, predictive parity requires that, for individuals who receive the loans, therepayment rates in different protected groups are the same.
We study fair Bayes-optimal classifiers under predictive parity.
Perhaps surprisingly, our theoreticalresults reveal that the optimal fair classifiers may or may not be a GWTR, depending on the datadistribution.
We identify a sufficient condition under which all fair Bayes-optimal classifier are GWTRs.
Without this condition, we show that fair Bayes-optimal classifiers may not be a GWTRwhen the minority group is more qualified than the majority group.
In these cases, predictive paritymay have limitations as a fairness measure, as it can lead to within-group unfairness for the minoritygroup.
Our findings are a reminder that the improper use of fairness measures may result in severeunintended consequences.
Careful analysis before applying fairness measures is necessary.
We then develop an algorithm, FairBayes-DPP, aiming for predictive parity.
Our method is a twostage plug-in method.
In the first step, we use standard learning algorithms to estimate group-wiseconditional probabilities of the labels.
In the second step, we first check our sufficient condition. Ifthe sufficient condition holds, we apply a plug-in method for estimating the optimal thresholds underfairness for each protected group.
We summarize our contributions as follows.
• We show that Bayes-optimal classifiers satisfying predictive parity may or may not begroup-wise thresholding rules (GWTRs), depending on the data distribution.
• We identify a sufficient condition under which all fair Bayes-optimal classifiers are GWTRs.
However, when the sufficient condition is not satisfied, the fair Bayes-optimal classifier may lead to within-group unfairness for the minority group.
• We propose the FairBayes-DPP algorithm for binary fair classification.
The proposedFairBayes-DPP is computationally efficient, showing a solid performance in our experiments.
Related Literature2.1 Fairness Measures
Various fairness metrics have been proposed to measure aspects of disparity in ML.
Group fairness[2, 15, 22] targets statistical parity across protected groups, while individual fairness [25, 28, 40]aims to provide nondiscriminatory predictions for similar individuals.
In general, group fairnessmeasures can be categorized into three categories.
The first group consists of independence-based measures, which require independence betweenpredictions and protected attributes; this includes demographic parity [27, 56] and conditionalstatistical parity [8, 1].
In credit lending, independence means that the proportion of approvedcandidates is the same across different protected groups.
However, as discussed in [22], independencebased measures have limitations; and applying them often leads to a substantial loss of accuracy.
The second group consists of separation-based measures, which require conditional independencebetween predictions and protected attributes, given label information.
Typical examples in thisgroup are equality of opportunity [22, 58] and equalized odds [22, 55].
In credit lending, separationbased measures require, that the individuals who will pay back (or default on) their loan have anequal probability of getting the loan, despite their race or gender.
Compared to independencebased measures, separation-based measures take label information into account, allowing for perfect predictions that equal the label.
However, these measures are hard to validate in certain applicationsas the label information is often unknown for some groups.
For example, the repayment status ismissing for individuals whose loan application is declined.
As a result, measuring predictive bias is more widely applicable.
This leads to the third class,sufficiency-based measures [38, 6, 30], where the label is required to be conditionally independent ofthe protected attributes, given the prediction.
In credit lending, this requires that among the approvedapplications, the proportion of individuals who pay back the loan is equal across different groups.
Unlike independence- and separation-based measures that are well studied with solid theoreticalbenchmarks and efficient algorithms, sufficiency-based measures are less commonly investigated.
Apossible reason is that conditioning on the prediction leads to a complex constraint, which is thuschallenging to study and enforce algorithmically.
2.2 Algorithms Aimed at Fairness
Literature on algorithms for fairness has grown explosively over the past decade.
Existing algorithmsfor fairness can be categorized broadly into three categories.
The first category is pre-processingalgorithms aiming to remove biases from the training data.
Examples include transformations[17, 33, 3, 24], fair representation learning [56, 32, 34, 10] and fair data generation [52, 41, 53,39].
The second group is in-processing algorithms, which handle fairness constraints during thetraining process.
Two common strategies are penalized optimization [20, 37, 9, 5] and adversarialtraining [58, 49, 51, 4].
The former incorporates fairness measures as a regularization term into theoptimization objective and the later tries to minimize the predictive ability of the model with respectto the protected attribute.
The third group is post-processing algorithms, aiming to remove disparities from the model output.
The most common post-processing algorithm is the thresholding method [18, 35, 1, 42, 57], adjustingthresholds for every protected group to achieve fairness.
In this paper, we propose a post-processingalgorithm, FairBayes-DPP, to estimate the fair Bayes-optimal classifier under predictive parity.
Problem Formulation and Notations
In this paper, we consider classification problems where two types of feature are observed: the usualfeature X ∈ X , and the protected feature A ∈ A .
For example, in loan applications, X may refer tocommon features such as education level and income, and A may correspond to the race or gender ofa candidate.
As multiclass protected attributes are often encountered in practice, we allow A to haveany number |A| ≥ 1 of classes, and let A = { 1 , 2 , ..., |A|} .
We denote by Y the ground truth label.
In credit lending, Y may correspond to the status of repayment or defaulting on a loan.
The output ˆ Yof the classifier aims to predict Y based on observed features.
We consider randomized classifiersdefined as follows:
Definition 3.1 (Randomized classifier) .
A randomized classifier is a measurable function 1 f :
X ×
A → [0 , 1] , indicating the probability of predicting (cid:98) Y = 1 when observing X = x and A = a .
We denote by ˆ Y f
= ˆ Y f ( x, a ) the prediction induced by the classifier f .
Group-wise thresholding rules (GWT rules/classifiers or GWTRs) over conditional probabilities areof special importance.
Consider an appropriate dominating sigma-finite measure µ on X (such asthe Lebesgue measure for measurable subsets of R d , d ≥ 1 , or the uniform measure for finite sets),and suppose that for all a ∈ A and y ∈ Y , the features X have a conditional distribution P X | a,ygiven
A = a, Y = y with a density dP X | a,y with respect to µ .
For all 2 x ∈ X and a ∈ A , letη a ( x ) =
P ( Y = 1 | X = x, A = a ) .
Definition 3.2 (GWT classifier) .
A classifier f is a GWTR if there are constants t a , a ∈ A , andfunctions τ a : X → [0 , 1] , a ∈ A , such that for all x ∈ X and a ∈ A
[Equation]
where I ( · ) is the indicator function.
Clearly, GWTRs choose individuals with the highest conditional probability in each group.
Thisproperty is a minimal requirement for within-group fairness.
For example, a GWT recruitment toolensures that the most qualified candidates are approved in every protected group.
We consider predictive parity, which aims to ensure the same positive predictive value amongprotected groups:
Definition 3.3 (Predictive Parity) .
A classifier f satisfies predictive parity if for all a ∈ A ,
P
[Equation]
Y [Equation] Y f
[Equation]
[Equation] (cid:98)
[Equation]
f = [Equation]
We follow [5, 57] to use the difference between positive predictive values to measure the degree of unfairness, defining the Difference in Predictive Parities (DPP) of a classifier f as
[Equation]
Fair Bayes-optimal Classifiers under Predictive Parity
Since predictive parity is commonly considered under the scenarios where false positives are particularly harmful [29], we study cost-sensitive classification.
For a cost parameter c ∈ [0 , 1] 3 , the cost-sensitive 0-1 risk of the classifier f is defined as
[Equation]
An unconstrained Bayes-optimal classifier for the cost-sensitive risk is any minimizer f ⋆ ∈argmin
f R c ( f ) .
A classical result is that all Bayes-optimal classifiers have the form f ⋆ ( x, a ) =
I ( η a ( x ) > c ) + τI ( η a ( x ) = c ) , where τ ∈ [0 , 1] is arbitrary [16, 35].
Taking predictive parityinto account, a fair Bayes-optimal classifier is any minimizer of the cost-sensitive risk among fairclassifiers:
[Equation]
4.1 GWT Fair Bayes-Optimal Classifiers under Predictive Parity
We first identify a sufficient condition under which all fair Bayes-optimal classifier under predictiveparity are GWTRs.
Condition 4.1 (Sufficient condition for Bayes-optimal classifiers to be GWTRs) .
[Equation]
The sufficient condition 4.1 requires that the minimal group-wise positive predictive value P ( Y =1 | η a ( X ) ≥ c, A = a ) of the unconstrained Bayes-optimal classifier is lower bounded by the maximalproportion of positive labels P
( Y = 1 |
A = a ) among groups.
In other words, the performancesof different groups vary only moderately: the average performance of the most qualified class ofeach group—the points with x such that η a ( x ) ≥ c —should be better than the overall performance
P ( Y = 1 | A = a ) of any of the other groups.
Condition 4.1 holds if P ( Y = 1 | A = a ) ≤ c for alla ∈ A , because P ( Y = 1 | η a ( X ) ≥ c, A = a ) ≥ c .
These conditions are applicable in settings where c is large, such as in credit lending where falsepositives are more harmful than false negatives, or if p Y | a , a ∈ A are small, such as in job recruitmentor school admissions where the number of slots is much smaller than the number of applications.
Under this condition, we present our main result.
Theorem 4.2 (Main result) .
Consider the cost-sensitive 0 - 1 risk with cost parameter c .
If Condition4.1 holds, then all fair Bayes-optimal classifiers under predictive parity are GWTRs.
Thus, for allf ⋆PPV from (2) , there are ( t a ) |A| a =1 ∈ [0 , 1] |A| and functions τ a ( x ) : X → [0 , 1] such that (1) holds.
Unlike for demographic parity or for equality of opportunity, where the fairness constraint is linearwith respect to the probability predictions of the classifier f [35], the DPP constraint is non-linearwith respect to f .
As a consequence, previously used theoretical tools such as the Neyman-Pearsonargument from hypothesis testing [57] are no longer valid in this case.
Instead, we prove the resultusing a novel constructive argument.
When Condition 4.1 is satisfied, for any classifier satisfyingpredictive parity, which is not a GWTR, we construct a GWTR that satisfies predictive parity andachieves a smaller classification error.
As a result, under Condition 4.1, all fair Bayes-optimalclassifiers are GWTRs.
Overall, the proof of Theorem 4.2 is quite involved, and requires a lot ofcareful casework and analysis.
4.2 Fair Bayes-optimal Classifiers under Predictive Parity do not Need to be ThresholdingRules
Next, we consider the case when the sufficient condition 4.1 does not hold.
For simplicity, weconsider a binary protected attribute a ∈ { 0 , 1 } with
[Equation]
Our result shows that, under condition (3), there exist class probabilities p a , a ∈ A , such that
no
Bayes-optimal classifier under predictive parity is a GWTR.
Theorem 4.3.
Suppose that condition (3) holds.
Denote t 1 = inf { t : P ( Y = 1 | η 1 ( X ) ≥ t, A =1) >
P ( Y = 1 | A = 0) } .
Suppose there exist δ 1 , δ 2 > 0 such that P ( c + δ 1 < η A ( X ) < t 1 |
A
=1) =
δ 2 > 0 .
Then, for all p 1 > 2 2+ δ 1 δ 2 , no fair Bayes-optimal classifier under predictive parity is a
GWTR.
The condition involving the constants δ 1 , δ 2 > 0 ensures that η 1 ( X ) has positive probability to bestrictly larger than c , which is a technical condition needed in the proof.
Theorem 4.3 shows thatpredictive parity may lead to within-group unfairness, whereby the most qualified individuals arepredicted to be unqualified, for a better overall accuracy.
By definition, predictive parity requires thatthe qualifications of selected individuals are similar across the protected groups.
Suppose there existsa highly qualified minority group in which most individuals are qualified.
Selecting the most qualifiedindividuals in this group leads to a very high standard.
As a result, many qualified individuals inother majority groups may be predicted to be unqualified using this standard, leading to accuracy loss.
Conversely, if we select less qualified individuals in the highly qualified group, the lower standardallows more qualified individuals from the other groups to be selected, and increases accuracy.
FairBayes-DPP: Adaptive Thresholding for Fair Bayes-optimality
In this section, we propose the FairBayes-DPP algorithm (Algorithm 1) for fair Bayes-optimalclassification under predictive parity.
As mentioned, the DPP constraint is non-linear with respectto the classifier f , and is also highly non-convex with respect to the model parameters, even if boththe classifier f and the risk function are convex with respect to these parameters.
In such cases,incorporating fairness constraints as a penalty in the training objective may be hard due to potentiallocal minima.
Therefore, we consider a different approach, developing a new two-step plug-in methodbased on Theorem 4.2.
Suppose we observe data points ( x i , a i , y i ) ni
=1 drawn independently andidentically from a distribution D over the domain X ×
A × Y .
Step 1.
In the first step, we apply standard machine learning algorithms to learn the feature- andgroup-conditional label probabilities η based on the whole dataset.
Consider a loss function L ( · , · )and the function class F = { f θ : θ ∈ Θ } parametrized by θ .
The estimator of η is obtained byminimizing the empirical risk, ˆ η a ( x ) :
= f ˆ θ ( x, a ) , where
[Equation]
Here we use the cross-entropy loss, as minimizing the empirical 0-1 risk is generally not tractable.
Atthe population level, the minimizers of the risks induced by the 0-1 and cross-entropy losses are boththe true conditional probability function [36]. Algorithm
Define, for all t ,
[Equation]
if min a (cid:100) PPV a ( c ) < max a ˆ P ( Y = 1 |
A = a ) then
We recommend considering other fairness measures.
else
Let t min = min { t : (cid:100) PPV 1 ( t ) ≥ max a ˆ P ( Y = 1 |
A = a ) } .
for t ∈ T
=
[ t min , max j ˆ η 1 ( x (1) j )] do for a ∈ A \ { 1 }
do
Find ˆ t a ( t ) such that (cid:100) PPV a (ˆ t a ( t ))
≈ (cid:100) PPV 1 ( t ) .
end for
Let ˆ f ( x, a, t ) =
˜ f
(cid:0) x, a ; ˆ t 1 ( t ) , ˆ t 2 ( t ) , ..., ˆ t |A| ( t ) (cid:1) =
I (cid:0)(cid:98)
η a ( x ) ≥ ˆ t a ( t ) (cid:1) .
Let R c ( t ) = 1 n (cid:80) n i =1 c (1 − y i ) (1 − c )
y i I
( y i ̸
= ˆ f ( x i , a i , t )) .
end for
Find ˆ t = argmin t ∈T n R c
( t ) .
Output: (cid:98) f
[Equation]
end if
Step 2.
In the second step, we first check the empirical version of Condition 4.1 for the classifier derived in the first step.
To be more specific, we divide the data into |A| parts, according to the value of A : for a ∈ A , S a
= { x ( a ) j , y ( a ) j } n a j =1 , where a ( a ) j = a .
Let, for all t for which it is defined,
[Equation]
[Equation]
and ˆ P (
[Equation]
n a
[Equation]
We only divide by nonzero quantities here and below.
To ensure that the quantities we divide byare nonzero, we restrict to t a ∈ [0 , max j (ˆ η a ( x ( a ) j ))] when evaluating (cid:100) PPV a ( t a ) .
We check whethermin a (cid:100) PPV a ( c ) ≥ max a ˆ P ( Y = 1 | A = a ) .
If this is not satisfied, we recommend considering otherfairness measures, as predictive parity may not be appropriate in this case, see the discussion after Theorem 4.3.
If it is satisfied, we then adjust the thresholds of the classifier aiming for predictiveparity.
Based on Theorem 4.2, we consider the following deterministic classifiers:
[Equation]
5) where ˆ η is the estimate of η from the first step, and t a , a ∈ A , are parameters to learn.
We use the following strategy to estimate t a , a ∈ A : First, we fix the threshold for the group witha
= 1 , say t .
The positive predictive value for this group can then be estimated by (cid:100) PPV 1 ( t ) .
Toachieve predictive parity, we need to find thresholds for the other groups such that the positivegroup-wise predictive values are the same 5 , i.e., find t a , a = 2 , 3 , . .
. , |A| , such that
[Equation]
As stated in Lemma A.1, the positive predictive value for each group in the population is always non-decreasing with the thresholds t a increases.
As a consequence, we can search over t a , a = perform a Since mean n ni iid even values t a , t they are as possible.
[Table]
, |A| , efficiently via, for instance, the bisection method.
Correspondingly, we consider the following range of t : T =
[ t min , max j η 1 ( x (1) j )] with
[Equation]
We denote by ˆ t a ( t ) , a = 2 , 3 , . . .
, |A| , the estimated thresholds given by (6), writing ˆ t 1 ( t ) =
t forconvenience.
We consider the classifier (5) with these thresholds:
[Equation]
Lastly, we find t that minimizes the cost-sensitive risk on the training data by searching over a grid
T n within T :
[Equation]
Our final estimator of the fair Bayes-optimal classifier is ˆ f PP =
ˆ f ˆ t .
The FairBayes-DPP algorithmis related to the algorithms proposed for other fairness measures in [57], where a binary protectedattribute is considered and closed-form optimal thresholds are derived.
In contrast, FairBayes-DPPcan handle multi-class protected attributes and does not rely on closed-form thresholds.
Similar to[57], our algorithm enforces fairness only in the fast second step, where no gradient-based techniqueis applied.
Thus, it is computationally efficient and the non-convexity of fairness constraint is nolonger problematic.
Our experimental results demonstrate that our method removes disparities andpreserves accuracy.
Experiments
6.1 Synthetic Data
We first study a synthetic dataset to compare our method with the true Bayes-optimal fair classifierderived analytically using the true data distribution.
Statistical model.
Let X =
( X 1 , X 2 ) ∈ R 2 be a generic feature, A ∈ { 0 , 1 } be the protectedattribute and Y ∈ { 0 , 1 } be the label.
We generate A and Y according to the probabilities P ( A = 1) ,
P ( Y = 1 | A = 1) and P ( Y = 1 |
A = 0) , specified below.
Conditional on A = a and Y =
y , Xis generated from a bivariate Gaussian distribution N ((2 a − 1 , 2 y − 1) ⊤ , 2 2 I 2 ) , where I p is thep -dimensional identity covariance matrix.
In this model, η a ( x ) has a closed form, and we use it tofind the true fair Bayes-optimal classifier numerically under the Condition 4.1.
More details aboutthis synthetic model can be found in Section C of Appendix.
Experimental setting.
We randomly sample 50 , 000 training data points and 5 , 000 test data points.
In the Gaussian case, the Bayes-optimal classifier is linear in x and thus we employ logistic regressionto learn η 1 ( · ) and η 0 ( · ) .
We then search over a grid with spacings equal to 0 .
001 over the range weidentified in Section 5 for the empirically optimal thresholds under fairness.
We denote (cid:98) f and (cid:98) f
PPVthe estimators of the unconstrained and fair Bayes-optimal classifiers, respectively.
We first evaluate the FairBayes-DPP algorithm under the Condition 4.1.
We set the cost parameterc = 0 .
5 , while P ( A = 1) = 0 .
3 and P ( Y = 1 | A = 0)
= 0 .
It can be calculated that P ( Y = 1 | η 0 ( X ) > 0 .
5 , A = 0)
≈ 0 .
613 , using (23) in the Appendix.
To consider settings with variedlevels of fairness in the population, we vary p
= P ( Y = 1 |
A = 1) from 0 .
2 to 0 .
6 , with the DPP ofunconstrained Bayes-optimal classifier grows from 0 to 0 .
Table 1 presents the classification accuracy and DPP of the true fair Bayes-optimal classifier and
FairBayes-DPP trained via logistic regression over 100 simulations 7 .
Our first observation is that,under predictive parity, the accuracy of true unconstrained and fair Bayes-optimal classifiers is almostidentical, indicating that predictive parity under Condition 4.1 requires a very small loss of accuracy.
This finding is consistent with the results in [30] that sufficiency-based measures are favored byunconstrained learning.
Second, our FairBayes-DPP method closely tracks the behavior of the fair Bayes-optimal classifier, controlling the accuracy metric ACC and unfairness metric DPP on the test data effectively.
When | P ( Y = 1 | A = 1) − P ( Y = 1 |
A = 0) | is small, FairBayes-DPP performs similarlyto the unconstrained classifier.
However, when the data is biased against protected groups and| P ( Y = 1 | A = 1) − P ( Y = 1 | A = 0) | is large, FairBayes-DPP mitigates the disparity of the unconstrained classifier effectively, while preserving model accuracy.
We further conduct extensivesimulations to evaluate the FairBayes-DPP algorithm with different model and training setups, asshown in the Appendix.
In particular, we also consider the multi-class protected attribute case.
6.2 Empirical Data Analysis
Dataset.
We test FairBayes-DPP on two benchmark datasets for fair classification: “Adult” [14] and“COMPAS” [26].
For each dataset, we randomly sample (with replacement) 70%, 50% and 30% asthe training, validation and test set, respectively.
To further test the performance of our algorithm ona large-scale dataset, we conduct experiments on the CelebFaces Attributes (CelebA) Dataset [31].
• Adult:
The target variable Y is whether the income of an individual is more than $50,000.Age, marriage status, education level and other related variables are included in X , and theprotected attribute A refers to gender.• COMPAS:
In the COMPAS dataset, the target is to predict recidivism.
Here Y indicateswhether or not a criminal will reoffend, while X includes prior criminal records, ageand an indicator of misdemeanor.
The protected attribute A is the race of an individual,“white-vs-non-white”.• CelebA: CelebA dataset is a large-scale dataset with more than 200,000 face images, eachwith 40 attributes (including protected attribute “gender” and other 39 different attributesfor prediction tasks).
Our goal is to predict the face attributes Y based on the images X andremove bias with respect to gender A from the output.
Experimental setting.
As algorithms for predictive parity are rarely considered in the literature,we use unconstrained learning as a baseline for our experiments.
For the “Adult” and “COMPAS”datasets, we adopt the same training setting as in [5, 57].
The conditional probabilities are learned
[Table]
via a three-layer fully connected neural network architecture with 32 hidden neurons per layer.
For“CelebA”
[Table]
we apply the training setting from [50].
We learn the conditional probabilities by traininga ResNet50 model; [23], pretrained on ImageNet [11].
For all the datasets, Over the course oftraining the model on the training set, we select the one with best performance on the validationset.
In addition, we learn the optimal thresholds over the validation set to avoid overfitting.
Allexperiments use PyTorch.
We refer readers to the Appendix for more training details, includingoptimizer, learning rates, batch sizes and training epochs.
We repeat the experiment 100 times for the Adult and COMPAS datasets and 10 times for the CelebA dataset.
Figure 1 presents the average performances of FairBayes-DPP and unconstrained learning on the
Adult and COMPAS datasets.
Our method achieves almost the same accuracy as the unconstrained classifier, and has a smaller disparity.
To better compare our fair classifier with the unconstrained one, we use the paired t -test to compare the DPP of the proposed algorithm (DPP Fair ) and of unconstrained learning (DPP Base ).
We consider the following one sided test:
[Equation]
The p -values of the tests are 3 .
90 × 10 − 4 for the Adult dataset and 3 .
× 10 − 8 for the COMPASdataset.
In both cases, these results provide evidence that our FairBayes-DPP achieves a smallerdisparity than unconstrained learning.
Finally, we test FairBayes-DPP on the CelebA dataset, Here, we only consider 27 attributes 9 with0 .
01 ≤ P ( Y = 1 | M ) , P ( Y = 1 | F ) ≤ 0 . 99 in the training, validation, and test sets
to ensure that thetraining, validation and test sample sizes are large enough for each subgroup.
We further identifyone attribute, “Young”, that violates Condition 4.1.
We calculate the per-attribute accuracies and
DPPs on the test set.
Table 2 presents the results of the first six attributes; the remaining results are inthe Appendix.
As we can see, even for the large-scale CelebA dataset with high dimensional imagefeatures, our algorithm mitigates the gender bias effectively, with almost no loss of accuracy.
Summary and Discussion
In this paper, we investigate fair Bayes-optimal classifiers under predictive parity.
We prove that whenthe overall performances of different protected groups vary only moderately, all fair Bayes-optimalclassifiers under predictive parity are GWTRs.
We further propose a post-processing algorithmto estimate the optimal GWTR.
The derived post-processing algorithm removes the disparity inunconstrained classifiers effectively, while preserving a similar test accuracy.
However, when our sufficient condition is not satisfied, the fair Bayes-optimal classifier underpredictive parity may lead to within-group unfairness for the minority group.
In the current literature,man algorithms directly apply penalized/constrained optimization to impose fairness.
Our negativefinding, however, is an important reminder that careful analysis is required before employing afairness measure.
The improper use of a measure may result in severe unintended consequences.