Pushing the limits of fairness impossibility: Who’s the fairest of them all?
Anonymous Author(s)
AfﬁliationAddress email
Abstract
The impossibility theorem of fairness is a foundational result in the algorithmicfairness literature.
It states that outside of special cases, one cannot exactly andsimultaneously satisfy all three common and intuitive deﬁnitions of fairnessdemographic parity, equalized odds and predictive rate parity.
This result hasdriven most works to focus on solutions for one or two of the metrics.
Ratherthan follow suit, in this paper we present a framework that pushes the limits of theimpossibility theorem in order to satisfy all three metrics to the best extent possible.
We develop an integer-programming based approach that can yield a certiﬁablyoptimal post-processing method for simultaneously satisfying multiple fairnesscriteria under small violations.
We show experiments demonstrating that ourpost-processor can improve fairness across the different deﬁnitions simultaneouslywith minimal model performance reduction.
We also discuss applications of ourframework for model selection and fairness explainability, thereby attempting toanswer the question: who’s the fairest of them all?
Introduction
While fairness in machine learning has received signiﬁcant attention in recent years, most existingworks focus on one of the many fairness criteria [3].
Consequently, practitioners are perhaps leftwith no choice but to use their best judgment to apply a single fairness criterion.
We suspect that theconﬂicting nature of existing mathematical deﬁnitions of fairness might have led to this undesirablepractice of narrowing down fairness-related measurement and mitigation to one chosen deﬁnition.
This is somewhat analogous to the trade-off between precision and recall while evaluating modelperformance [7].
Instead of choosing one of precision or recall to evaluate the performance of aclassiﬁcation model, practitioners often evaluate the trade-off and choose models that can maintaina certain level of precision while optimizing recall or vice-versa [19].
In this paper, we providea framework for explicitly addressing such trade-offs among multiple fairness criteria and modelperformance toward optimal model selection.
One of the most proliﬁc examples of fairness in machine learning arose from the ProPublica recidivismstudy ([2]), in which a risk assessment tool called COMPAS was found to be biased against blackdefendants.
But beyond the immediate implications in criminal justice, the study also promptedmore general studies in algorithmic fairness and, in particular, led to a key result highlighted in [13],[21], and [25] which some colloquially refer to as the "impossibility theorem" in fairness [30, 24].
This theorem essentially states that three common deﬁnitions of algorithmic fairness - demographicparity [16], equalized odds [21], and predictive parity [3], cannot be simultaneously satisﬁed outsideof pathological situations.
Several works following these initial results have therefore focused onsatisfying one metric ([21, 32]) or proposing adaptable methods for different metrics ([11, 37]).
Yet,while we do not deny the conclusions of the impossibility theorem, we also believe there have not been sufﬁcient efforts to reconcile the conﬂicting fairness deﬁnitions to the best extent possible
We ﬁll this gap in the literature by translating the trade-offs among multiple fairness criteria and modelperformance into a constrained optimization problem and propose a post-processing methodology forsimultaneously achieving approximate fairness in the conﬂicting deﬁnitions simultaneously.
We believe our framework would alleviate the practitioner from making the hard choice in choosing aparticular metric.
Instead, if they have a partial ordering of importance amongst the metrics (whichmany possess), our framework would explicitly allow them evaluate such trade-offs.
The applicationwhich we highlight in Section 4 discusses these in detail.
Overall, we make three main contributionsin this work:
We design a ﬂexible optimization framework that returns a post-processing score transformation
function that can make scores group-wise ✏ -fair along three deﬁnitions (demographic parity,equalized odds and predictive rate parity) simultaneously.
This framework can be applied toany binary classiﬁer that produces a continuous score, can be conﬁgured for singular or multiplemetrics of fairness, and also can account for fairness vs. performance trade-offs in terms ofROC-AUC.2.
We present a novel reformulation of this non-convex optimization problem as a Mixed IntegerLinear Program (MILP) [41].
This reformulation allows us to ﬁnd provably globally optimalsolutions.
We further show that in practice, we can consistently ﬁnd better solutions through our global optimization method compared to local optimization methods in a reasonable time.
We discuss and extend our framework from a post-processing mechanism to a tool that can aid practitioners in better understanding their data and models’ empirical fairness characteristics andtrade-offs and compare these traits across models.
The rest of the paper is organized as follows.
In Section 2 we mathematically deﬁne fairness metricsand the multiple fairness optimization problem.
We discuss the optimal solution via MILP in Section3.
We try to answer the question of “who’s the fairest of them all?”
through our applications andexperiments in Section 4.
Finally, we conclude with a discussion in Section 5.
We wrap up thissection with a discussion of the related literature.
Related Work: Early works [13] exploring the conﬂicts between fairness deﬁnitions prove that for abinary predictor, predictive rate parity conﬂicts with equalized odds ([21]) unless base rates are equalor the model is perfectly predictive.
Chouldechova [13] also considers trade-offs between the fairnessdeﬁnitions in the binary prediction case.
Kleinberg et al.
[25] generalizes this study by showing thatstatistical (i.e., demographic) parity is also inconsistent with predictive rate parity and equalized odds.
The same paper studies these inconsistencies in a more general bin-wise prediction setting and showsthat approximate fairness deﬁnitions (predictive rate parity, equalized odds) can simultaneously holdbut only under ✏ -approximate equal base rates or ✏ -approximate perfect performance.
This workfurther proves that there is an inherent trade-off between fairness and loss.
Beyond impossibility theorem results, several works have focused on trade-offs between fairness andmodel performance [21, 17, 37, 42].
They develop in-processing solutions aimed at reducing onemetric while maintaining accuracy.
A few authors have analyzed trade-offs or attempted to achievemultiple fairness.
One example is Pleiss et al.
[36], which shows that predictive rate parity and arelaxed form of equalized odds are reconcilable under a randomized prediction scheme.
Anothernotable example is Celis et al.
[11], which develops a ﬂexible in-processing approach to achievemultiple types of fairness (potentially at the same time).
Like Celis et al.
[11], the framework wedevelop also aims for a ﬂexible approach to focus on one or many fairness metrics simultaneously.
However, our method is distinct in that it is a post-processing based solution and also more general asit works for continuous scores (rather than binary classiﬁcation).
Our work is most closely relatedto Nandy et al.
[32] in terms of the underlying score transformation mechanism, and we leveragesome of their methods.
However, whereas [32] only targets equalized odds, we go further and includedemographic parity and predictive rate parity in our framework—this posits computational challenges,which we address by proposing novel methodology based on integer programming.
As noted at the end of [25], some open questions are how to optimally assign scores to satisfymultiple criteria when base rates are equal and additionally, how to satisfy predictive rate parity andeither equal TPR or TNR when one cost outweighs the other.
To our knowledge, no prior workhas attempted to reconcile all three fairness conditions (demographic parity, equalized odds, andpredictive rate parity) simultaneously with model performance through a post-processing framework.
Although the post-processing methods tend to be less ﬂexible for fairness-performance trade-offs than their in-processing counterparts, they can be much more easily added on top of any existingmodel training pipeline.
This makes a post-processing approach modular, and in particular, moreappealing in complex web-scale recommender systems that use (a combination of) certain predictionscores to rank a list of items.
Multiple Fairness Optimization
We consider a binary classiﬁcation problem where the i -th observation is characterized by theirlabel y i 2 { 0 , 1 } , their group membership g i 2 G , and model predicted probability (also known as 99 a risk score [25]) s i 2 [0 , 1] for i
= 1 , . .
N .
The corresponding random variables are denotedas Y , G and S respectively.
To set up the problem, we discretize the scores into nonempty binsb 2 B := { 1 , . .
. , |B|} by using, for example, a quantile transformation (we will denote |B| as B ).
Additionally, let N [ g ] b + denote the number of group g positive class ( y i = 1 ) instances in bin b , N [ g ] bdenote the total number of instances of group g instances in bin b , N [ g ]+ ( N [ g ]  ) be the total numberof group g positive (negative) instances, and N [ g ] be the total number of group g instances.
Lastly,our approach seeks to achieve fairness by moving instances from one bin to another bin: hence, wedeﬁne variable x
[ g ] bb 0 as the probability of moving an instance of group attribute g and score in bin binto a new bin b 0 1 .
In other words, for every group g , the collection { x [ g ] bb 0 } b,b 0 can be represented asa B ⇥ B transition matrix (with additional constraints, as discussed next).
For the optimization framework and the remainder of this paper, we translate a single fairnessdeﬁnition (e.g. equal true positive rate) as a constraint that controls for the worst-case violationsacross all bins.
Below, we discuss different fairness constraints that we consider in our framework,and describe how they can be represented in terms of the optimization variables { x [ g ] bb 0 } .
2.1 Fairness Constraints Under Binning Framework
Demographic Parity (DP): For simplicity, we assume that there are exactly two groups g 2 { 1 , 2 } as we formulate the fairness metrics, though pairwise constraints can be added to enforce fairness for an arbitrary number of groups.
Starting with demographic or statistical parity from [16], this condition states that the model’s predicted score is independent of group membership.
This is equivalent to
[Equation]
Our version of this constraint uses bins B to empirically approximate the probability P ( S = s |
G = g )
and we also relax the equality to an ✏ -approximate equality (for some pre-speciﬁed ✏ > 0 ).
Therefore, after transforming the scores using { x [ g ] bb 0 } b,b 0 the ✏ DP -approximate DP can be expressedas the following as a linear constraint:
[Equation]
where N [ g ] b denote the number of observations from group g in bin b (before transformation), and
N [ g ] = P b 2 B N [ g ] b .
For reproducibility, ✏ DP should be chosen to be larger than the approximationerror O (1 / p N [ g ] ) for replacing P ( S = s | G = g ) with its empirical counterpart.
Equalized Odds (EOdds): The equalized odds condition for binary predictors given in Hardt et al.
This is a balance condition where the groups must have equal true positive and false positive rates.
For continuous scores, it translates to having equal score distributions for each group conditional on their true labels [32]:
[Equation]
Like demographic parity, our empirical score bin version requires that the distribution of positive ornegative instances be ✏ EOdds -approximately equal between groups in the new bins b 0 .
Both equaltrue positive rate and false positive rate can be expressed as linear constraints, respectively:
[Equation]
Predictive Rate Parity (PRP): Lastly, we examine the predictive rate parity condition popularized in [13].
This condition states that the probability of being a positive instance is independent of group membership when we condition on the score.
Formally:
[Equation]
Using the empirical score bin framework, an approximate version of the above implies that theproportion of positive instances in each bin must be ✏ PRP -approximately equal among groups:
[Equation]
[Equation]
Unlike constraints (1) and (2), which can be expressed as a linear function of the optimizationvariables { x [ g ] bb 0 } , condition (3) yields bilinear terms and is in general a non-convex constraint.
Amain technical difﬁculty of our framework arises from this non-convex fairness constraint—Section3 presents an integer programming framework to handle this non-convexity, ensuring we can obtain aglobally optimal solution to the resulting optimization problem.
Remark.
Our deﬁnition of fairness as the worst-case violation across all bins aims to resembleapproximations of the respective probabilistic deﬁnitions but we have not found identical deﬁnitionsin other works.
We comment on the differences and discuss why it does not contradict the traditionalimpossibility theorem of [25] in Appendix E.
2.2 MFOpt: Multiple Fairness Optimization Framework
We use constraints developed in Section 2.1 to state the multiple fairness optimization (MFOpt)problem:minimize
[Equation]
Above, the optimization variables are the group-speciﬁc movement probability x
[ g ] bb 0 terms and allremaining terms are problem data and/or conﬁgurable hyperparameters.
Starting with (4a), we deﬁne¯ s b as the midpoint score in the bin and hence the objective is the product of the movement distance¯ s b  ¯ s b 0 weighed by the fraction of total samples moved N [ g ] b / N and the amount of movement x
[ g ] bb 0 . (4b)
States that the total movement out of bin b , including the movement back to itself, must sum up to 1and along with (4g) ensures that x [ g ] bb 0 represent probabilities in a transition matrix.
(4c) states that the total movement from bin b back to itself must be lower bounded by hyperparameter ⇠ .
This parametercontrols how far we allow the new scores to stray from the original and is necessary to preventzero denominators in 3 and 4f.
Constraint (4d) represents window constraints to restrict extrememovements of scores beyond w bins.
Constraint (4e) are the fairness constraints in Section 2.1.
(4f)ensures that we preserve the rank-ordering of the scores and expected values, which is desirable forcomparing bins against each other.
The predictive rate parity constraint (3) and (4f) both introducenon-convexities into Problem (4).
These two constraints also require us to assume overlap, or thateach bin contains at least one member from each group.
Without overlap, predictive rate parityis undeﬁned since it is not possible to compare expectations across groups for a given bin anddemographic parity is likely violated as it means one group has 0 probability of landing in a givenbin.
We close this section with two major beneﬁts of this framework compared to inprocessing solutionssuch as adding fairness regularization ([42], [17], [37], ) or using an entirely different fairness-basedmodel ([11], [43]).
First, the number of optimization variables scales in the order of O ( |G||B| 2 ) .
159 This is signiﬁcant as it entails that all of our solutions (solving the optimization once, or over agrid of settings for applications in Section 4) can be applied in arbitrarily large settings as long asscore-based binning is possible.
Applying the transformation is equally tractable, as it only requiresbinning observations and making independent draws from a multinomial distribution with B possibleoutcomes.
The second beneﬁt of our framework is that it returns a highly interpretable solution as it returns oneoptimized B ⇥ B transition matrix per group.
Hence given a newly scored instance, several facts canbe easily read from the corresponding row of the matrix such as the likelihood of moving to a speciﬁcbin b , moving into any higher or lower bin, etc.
These probabilities can also be ﬁnely controlledvia constraints in the optimization as we demonstrate with the window constraints (4d) and maxmovement constraints (4c).
This interpretability is not present in model regularization frameworks,where it is hard to gauge the amount of regularization needed to achieve a certain fairness effect andalso difﬁcult to know how an individual’s score might change when switching from the base model toa fair model.
Finding Optimal Solutions via Mixed Integer Programming (MIP)
The primary difﬁculty of the above optimization problem are the predictive rate parity constraint(3) and rank-order constraint (4f) which turn the problem non-convex.
Non-convex constrainedoptimization is generally NP-hard and traditional methods that seek locally optimal solutions includegradient-based interior point optimization ([40]), sequential quadratic programming ([18]), or algorithms speciﬁc to quadratically-constrainted-quadratic-programs (QCQPs) such as operator splittingmethods, semideﬁnite relaxations, etc.
(see [33] for an overview).
Rather than pursuing a locally optimal solution, we propose a novel reformulation of the problem intoa tractable mixed-integer-linear-program (MILP) which can be solved to global optimality ([10], [9]).
Our reformulation grants two beneﬁts over traditional locally optimal solvers.
First, global strategiestheoretically enable us to ﬁnd the best possible solution.
Second, since our problem primarily scaleswith the number of bins, it is computationally tractable and allows us to utilize the power of modern MIP solvers.
3.1 Tractable reformulations for computational efﬁciency
We ﬁrst observe that a direct reformulation of the fractional terms into bilinear terms in constraints (3)and (4f) will lead to bilinear terms in the order of O ( B 3 ) .
We show that we can reduce this to O ( B )bilinear terms through a substitution that exploits the problem structure.
Next, we take advantage ofthe vastly reduced number of bilinear terms to apply the normalized multiparametric disaggregationtechnique (NMDT, [1]) which we explain in Section 3.1.2.
This allows us to approximate products ofcontinuous variables as products of integer variables, which can be easily linearized and handled by
MIP solvers.
Importantly, this transformation of xy terms requires upper and lower bounds for x andy and we propose a method in 3.1.2 for generating and tightening these bounds by solving fractionallinear program subproblems ([12]).
Taken together, the reduction of bilinear terms combined withthe bound-tightening procedure enable us to effectively apply the NMDT methodology and transformthe problem from a non-convex QCQP to an MILP that can be solved to global optimality.
3.1.1 Step 1: Reducing number of bilinear variables
We reduce the number of bilinear variables in our problem by making a substitution for the fractionterm by introducing new optimization variables v [ g ] b  0 (to represent a sum) and t [ g ] b  0 (to 201 represent the fractional quantity) as additional variables (see Appendix C for more details).
We canthen use them to write equivalent constraints with only O ( B ) bilinear terms.
Let,v [ g ] b 0 = [Equation] B x [ g
[Equation]
bb
[Equation]
N [ g ] b and t [ g
[Equation]
Then we have the following:
[Equation]
3.1.2 Step 2:
NMDT and bound tightening through fractional LP subproblems
Linearizing bilinear terms (NMDT): We show how we can model each bilinear term t [ g ] b v [ g ] b by using a binary expansion for the continuous variables t
[ g ] b , v [ g ] b , and by observing that the product of binary variables can be modeled via integer programming (see [27, 39] for reference).
To this end, we make use of the NMDT transformation [1].
We recap this method below as formulated in [1]
Given any bounded optimization variable x 2 [ x L , x U ] , and precision factor p , a negative integer, we can represent this variable exactly as x = ( x U  x L )  + x L where
[Equation]
where 0     2 p is a remainder term and z l 2 { 0 , 1 } are binary optimization variables.
Dropping 206 the remainder term   gives us the approximate form and product forms of xy become dot productsof several integer variables, which can be effectively handled via modern MIP solvers, such as [20].
Although we are solving an approximation (e.g. precision of 1 e  4 ) this is not a practical problemsince it is precise enough for reasonable choices of ✏ and we do not expect the constraints to holdexactly when we apply the post-processor on the testing data anyways.
Bounds on v [ g ] b , t [ g ] b via Fractional LPs: A key requirement to apply NMDT is that all optimizationvariables in the bilinear terms ( t [ g ] b and v [ g ] b in our case) must be bounded and we need to be ableto accurately estimate these bounds (i.e., tighter bounds leads to faster runtimes [1]).
We proposeobtaining these lower and upper by minimizing and maximizing respectively, the sub-problemswhile keeping all fairness constraints except the quadratic constraints.
Firstly, note that bounds onv [ g ] b can be solved as a simple LP (details in Appendix C).
Meanwhile, obtaining a bound on t [ g ] brequires solving a nonlinear problem due to fact that t [ g ] b represents a fractional objective (ratio of twoafﬁne terms of optimization variables).
However, we observe that we can apply the Charnes-Coopertransformation [12] to reformulate the nonlinear problem into a simple LP (details in Appendix C).
3.2 Choice of algorithm: QCQP (heuristic) vs MIP (optimal solution)
In this section we show the results and beneﬁts of our reformulation from a non-convex QCQP toan MILP.
In Table 1, we take each dataset and train a grid-searched random forest model and scorethe training data.
Next, we discretize the scores into bins, parameterize the problem (# bins, ✏ , maxmovement, window size, solve time) and compare our MILP solution solved by Gurobi ([20]) againstthe QCQP problem solved by IPOPT ([28]), which is a generic interior-point log-barrier penaltymethod for nonlinear constrained optimization.
We discuss the datasets and problem parameters forall experiments in the Appendix D. For each metric such as AUC , we use AUC INT and AUC IPto denote the average result of applying the interior-point (INT, for short) or integer programming(IP) method, respectively.
The metrics used are the objective value, optimality gap ( %  ), 3 and AUC 4 .
We also report the statistical signiﬁcance of the improvement based the p -value from the Wilcoxon signed-rank test to determine if %  IP  %  INT is a consistent result.
Bold ﬁguresindicate statistical signiﬁcance w.r.t.
1 standard deviation 5 .
[Table]
As the results show, the MIP reformulation consistently beats the interior point solver applied on theraw optimization problem, even when solving for only 10 minutes.
We also observe that in mostcases, regardless of the method we choose, we can quickly ﬁnd near optimal solutions that are highperforming in the sense of keeping an AUC close to the original model.
This is a signiﬁcant resultas it means that even a locally optimal solution to our optimization problem can yield a practicallyuseful post-processing result.
We conclude this section by reiterating two beneﬁts of the reformulation.
First, solving a MIP methodyields lower bounds that can be used to prove optimality or otherwise gauge the quality of a feasiblesolution.
Second, by framing the problem as a MIP, we can always theoretically continue improvingthe solution to optimality based on the acceptable time limits.
Applications
In this section, we illustrate a few methods of applying our framework and how it can be used tohelp model developers select and understand models from a fairness perspective.
To apply theseprocedures, we ﬁrst require developing an efﬁcient frontier of fairness solutions to understand which✏ conﬁgurations are feasible for a given model type and dataset.
To generate this frontier, we solvethe problem over a grid of parameters ✏ DP , ✏ EOdds , ✏ PRP .
Each feasible solution will yield a points 2 { ( AUC, ✏ DP , ✏ EOdds , ✏ PRP ) } and the collection of non-dominated points from the solution setyields a efﬁcient frontier.
We show the 2-d proﬁle shots of our 4-d fairness surface in Figure 1 as anexample, where the color gradient represents AUC.
In theory, we could obtain a true Pareto-optimalfrontier since we have devised a method of obtaining globally optimal solutions.
However, wegenerate this frontier using IPOPT due to practical limitations as we are solving a 7 ⇥ 7 ⇥ 7 grid of ✏parameters.
4.1 Understanding fairness tradeoffs
After the Pareto frontier is generated, the modeler can pick an operating point s based on thedesired AUC and tolerable fairness violations ✏ .
However, when communicating fairness propertiesto stakeholders and accounting for potential changes in strategy, it can be helpful to additionallyunderstand the cost of further increasing fairness in terms of ✏ DP , ✏ EOdds , ✏ PRP .
This tradeoff can be understood by looking at the characteristics of points on the frontier near theoperating point.
Suppose we are at an operating point s .
If we want to trade AUC for ✏ EOdds , thenwe would ﬁnd a point s 0 with at least as good ✏ DP , ✏ PRP but worse AUC and better ✏ EOdds .
Moregenerally, if we pick trade performance/fairness characteristic c (cost) for characteristic b (beneﬁt), then we hold all other factors constant and ﬁnd a point with better b and worse c .
We illustrate thisin the Table 2.
The ﬁrst row shows a hypothetical operating point while the following rows showother points on the efﬁcient frontier that we could move to when we make a certain trade.
Blank rowsindicate that no such point was found that permits the desired trade-off.
This trade-off perspective canenable developers to better understand and communicate the costs to performance or other fairnessmetrics when trying to close the disparity in one fairness metric.
[Table]
4.2 Performance Comparison
[Table]
Lastly, we compare our framework against two methods and show that we can satisfy fairness constraint(s) just as well if not better, while yielding signiﬁcantly better performance.
When comparingagainst these methods, we ﬁrst create 20 random splits of the dataset, perform some basic preprocessing, ﬁt a grid-searched random forest model, and score the training and testing data to get thebase scores ˆ y 0 .
Next, we run the methodology that we are comparing against (i.e. build a model orapply the postprocessor) to get method scores ˆ y m .
We then bin the outputs of the base model andcompared method and compute the AUC along with the fairness metrics ( ✏ 0 , ✏ m ).
Next, we solve ourconstrained optimization problem where we set the parameters ✏ to 12 min ( ✏ 0 , ✏ m ) .
After optimizing,we can apply the optimal solutions x
[ g ] bb 0 to assign new bins for scored test instances.
One methodof assigning a group g instance with score s 2 b (denote as s [ g ] b ) would be to make a random drawfrom a multinomial distribution parameterized by probabilities ( x [ g ] b 1 , x [ g ] b 2 , . .
, x [ g ] b |B| ) .
This is a coarsemethod of stochastic assignment and we also propose an alternative method in the Appendix A. Finally, we can compute the resulting AUC and fairness metrics on remapped bins for the testing data(and do the same for ˆ y 0 and ˆ y m on the testing data).
These ﬁgures are shown in Table 3.
We onlyshow the results on the test set due to space constraints and have placed the results for the training setin Appendix D. Bold ﬁgures indicate that a metric is statistically signiﬁcant to 1 standard deviation.
First, we compare our framework against the in-processing framework in Rezaei et al.
[37], which is arobust optimization-based logistic regression model for reducing equality of opportunity violation.
Wefound that this method works well compared to standard logistic regression and managed to decreasefairness violations while maintaining the similar performance.
It improves in ✏ Eodds compared to arandom forest as well.
However, its weakness is that the underlying model is still logistic regressionand therefore has limited expressiveness.
In comparison, MFOpt can be applied on top of any modelclass and thus the performance advantages of more ﬂexible models are better maintained. [Table]
Next, we compare our framework against the post-processing framework in Pleiss et al.
In thismethod, the authors use randomization to maintain the model’s calibration while simultaneouslysatisfying a relaxed equalized odds condition (whereby a linear combination of TPR and TNR aresatisﬁed).
Again, we see that this method maintains close performance as the base model, successfullyshrinks equalized odds violations, and even decreases demographic parity violations too.
However, itresults in large violations of predictive rate parity based on our deﬁnition in contrast to our method.
4.3 Who’s the fairest of them all?
In industry, machine learning model selection is guided by many factors including performance,speed, interpretability, among others.
Yet, the fairness dimension is commonly overlooked unlessthe developers speciﬁcally induce it in their model.
Even then, picking a speciﬁc fairness metric tooptimize for can be a nebulous task.
Rather than focus on a single metric, we propose a simple andintuitive method of gauging a model’s efﬁciency in trading between different fairness deﬁnitions.
To do so, we propose constructing the frontier for the two models in question and then ﬁltering allpoints on the efﬁcient frontier with tolerable performance AUC  AUC min .
Next, ﬁnd the point onthe respective frontiers with minimum Euclidean distance to the origin.
The model with the shortestdistance to their frontier can then be declared as the model that has better tradeoff properties.
We donot show an example due to lack of space, but remark that this procedure along with the trade-offanalysis can be useful when a developer is iterating between models that were not designed forfairness, but wants a model that can be ﬂexibly made more fair through the postprocessing frameworkthat we propose.
As fairness requirements may change over time, the model that can yield the besttradeoffs between different deﬁnitions can offer the most overall utility.
Discussion
In our study, we have devised a ﬂexible, tractable, and interpretable post-processing method formaking any binary classiﬁer more fair.
We then apply our methodology to push the limits of theimpossibility theorem and show that while theoretical limitations remain undisputed, there is a pathforward to practically reconciling the conﬂicting fairness deﬁnitions.
Interestingly, our results extendthe ﬁndings of [38], which ﬁnds that the trade-off of fairness and accuracy are negligible in practice.
Our work reinforces this claim but also adds on that trade-offs between fairness deﬁnitions can benegligible as well.
On the technical end, there are other optimization methodologies for solving theproblem to global optimality such as spatial branch and bound. We considered these solutions duringour investigation but ultimately opted for a MIP approach due to the maturity and availability ofsolvers (see Appendix F for details).
Another potential avenue of further research is to improve theconsistency of the PRP violation reduction, as we observed the largest standard error in reducingthis metric.
This could be due to us using random forests for all experiments, which is known tobe an uncalibrated model [6].
One method to address this is therefore ﬁrst calibrating the modelthrough other methodologies such as Platt scaling ([35]) or binning-based calibration ([26]), as
MFOpt appears to have no issue at least maintaining the level of calibration.
We could also considerincorporating uncertainty in the training data through principled approaches such as stochastic orrobust optimization.
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- [Table]
mental results (either in the supplemental material or as a URL)?
[Yes] The code will be uploaded within the week of paper submission as permitted by NeurIPS submission
guidelines.
Instructions and data used in all experiments are clearly outlined in D and
(b) Did you specify all the training details (e.g., data splits, hyperparameters,
how they
were chosen)?
[Yes] Referenced in D
[Table]
(c) Did you report error bars (e g., with respect to the random seed after running experiments multiple times)?
[Yes]
(d) Did you include the total amount of compute and the type of resources used (e g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] Referenced in D
If you are using existing assets (e
g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
[Yes] License and references
can be found in the Appendix Section D
(b) Did you mention the license of the assets?
[Yes] License are mentioned alongside
open-source package references
(c) Did you include any new assets either in the supplemental material or as a URL?
[No]
(d) Did you discuss whether and how consent was obtained from people whose data you’re
using/curating?
[N/A]
(e) Did you discuss whether the data you are using/curating contains personally identiﬁable
information or offensive content?
[Yes] Under the Experimental Data Section D
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable?
[N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount
spent on participant compensation?
[N/A]
[-All experiments were run on a MacBook Pro with a 2.4GHz 8-Core Intel Core i9 processor with 
-]