Fair Without Leveling Down: A New Intersectional Fairness Definition
Authors: Gaurav Maheshwari
Aurélien Bellet
Pascal Denis
Mikaela Keller
Venue: EMNLP 2023 Main (2023)
Keywords: Fairness; Intersectional; Leveling Down
TL;DR: A new intersectional fairness definition which takes into account worst case performance.

Abstract: In this work, we consider the problem of intersectional group fairness in the classification setting, where the objective is to learn discrimination-free models in the presence of several intersecting sensitive groups.
First, we illustrate various shortcomings of existing fairness measures commonly used to capture intersectional fairness. 
Then, we propose a new definition called the $\alpha$-Intersectional Fairness, which combines the absolute and the relative performance across sensitive groups and can be seen as a generalization of the notion of differential fairness. 
We highlight several desirable properties of the proposed definition and analyze its relation to other fairness measures.
Finally, we benchmark multiple popular in-processing fair machine learning approaches using our new fairness definition and show that they do not achieve any improvement over a simple baseline. 
Our results reveal that the increase in fairness measured by previous definitions hides a ``leveling down'' effect, i.e., degrading the best performance over groups rather than improving the worst one.

Fair Without Leveling Down: A New Intersectional Fairness Definition
Gaurav Maheshwari, Aurélien Bellet, Pascal Denis, Mikaela Keller Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France first_name. f last_mame@inria.ff
Abstract
Female African-Americans) For example, Buo- lamwini and Gebru (2018) showed that commer- cially available face recognition tools exhibit sig- nificantly higher error rates for darker-skinned fe- males than for lighter-skinned males. Similar obser- vations have been made by several studies in NLP including contextual word representation (Tan and Celis, 2019), and generative models (Kirk et al., 2021). These findings resonate with the analytical framework of intersectionality (Crenshaw, 1989), which argues that systems of inequality based on various attributes (like gender and race) may "inter- sect" to create unique effects.
In this work, we consider the problem of in- tersectional group fairness in the classifica- tion setting, where the objective is to learn discrimination-free models the presence of several intersecting sensitive groups. First, we illustrate various shortcomings existing fair- ness measures commonly used to capture in- tersectional fairness. Then, we propose a new definition called the a-Intersectional Fairness, which combines the absolute and the relative performance across sensitive groups and can be seen generalization of the notion of differential fairness. We highlight several de- sirable properties the proposed definition and analyze its relation to other fairness sures. Finally, we benchmark multiple popular in-processing fair machine learning approaches using
To capture these effects in the context of machine learning, several intersectional fairness measures have been proposed (Kearns et al., 2018; Hébert- Johnson et al., 2018; 1; Foulds et al., 2020). Amongst them the most commonly used (Lalor al., 2022; Zhao al., 2022; Subramanian al., 2021) is Dif- ferential Fairness (DF) (Foulds et al., 2020), which the log-ratio of the best-performing group to the worst-performing group for a given performance measure (such as the True Positive Rate). While DF has many desirable properties, in this work we emphasize that DF implements a "strictly egal- itarian" view, i.e., it only considers the relative performance between the group and ignores their absolute performance. In particular, trivial way improve fairness as measured by DF is by harming the best-off group without improving the worst-off group. This phenomenon, known as leveling down, does not fit the desired fairesss requirements in many practical use-cases (Mittelstadt et al., 2023; Zietlow et al.. 2022). Yet, we empirically observe that (i) popular fairness- prommoing approaches tend to level down more in intersectional fairesss, and (ii) this often goes unnoticed in the overall perfor- mance of the model due to the large number of groups induced by intersectional fairness.
fairness definition and show that they do not achieve any improvement over a simple baseline. Our results reveal that the in- crease in fairness measured by previous def- initions hides a "leveling down" effect, i.e., degrading the best performance over groups rather than improving the worst one.
1 Introduction
The aim of fair machine learning is to develop models that are devoid of discriminatory behav- ior towards sensitive subgroups of the population. A broad range of approaches have been devel- oped (see e.g., Zafar et al., 2017: Denis et al., 2021; Maheshwari and Perrot, 2022; Mehrabi et al., 2022, and references therein), with most of them focusing on a single sensitive axis (Lohaus et al., 2020; Agarwal et al., 2018) such as gender (e.g., Male vs. Female) or race (e.g., African-Americans However, recent stud- ies (Yang et al., 2020; Kirk al., 2021) have demonstrated that even when fairness can be en- sured the level of each individual sensitive axis, significant unfairness can still exist at the inter- section levels (e.g., Male European-Americans. Americans vs.
VS. Crropean-Americany
To address these issucs and explicitly capture the leveling down phenomena, we propose a gener-

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alization of DF, called c--Interscctonal Fairness (lF,), which takes into account both the relative performance between the groups and the absolute performance of the groups. More precisely, & is a weighted average between the relative and absolute performance of the groups, and allows the explo- ration of the whole trade-off between these two quantities by changing their relative importance via weight a C [0, 1]. Our extensive benchmarks across various datasets show that many existing fairness-inducing methods aim for different point in the aforementioned trade-off and generally show no consistent improvement over a simple uncon- strained approach.
non-binary), race: (Luropean American, African )merican), and age: {under 45, above 45). We de- fine a sensitive group g as any p-dimensional vector in the Cartesian product set G = A1 ... X Ap of these sensitive axes. A sensitive group g E g can then be written as (aI....,,y, ap) with 0j E Aj.
2.2 Problem Statement
Consider feature space X, finite discrete label space V and a ÿ representing all possible inter- sections of p sensitive axes as defined above. Let D be an unknown distribution over X y g through which we sample i.i.d a finite dataset
T = {( yi, g:)) consisting of n examples. This sample can be rewritten as T Ugeg Tz where Tg represents the subset of examples from group g. The goal of fair machine learning is then to learn an accurate model he H. with learnable parameters 0 C RDD, such that he X Y is fair with respect to a given group fairness definition like Equal Opportunity (Hardt et al., 2016), Equal Odds (Hardt et al., 2016), Accuracy Parity (Zafar et al., 2017), etc.
In summary, our primary contributions are as follows:
We showcase the shortcomings of the existing intersectional fairness definition and propose a generalization called a- e-Intersectional Fair- ness. We analyze the properties and behavior of the proposed fairness measure, and contrast them with DF.
We benchmark existing fairness approaches multiple datasets and evaluate their perfor- mance with several fairess measures, includ- ing ours. On the one hand, we find that many fairness approaches optimize for existing fair- ness measures by harming both the worst-off and best-off groups or only the best-off group. On the other hand, our measure is more care- ful in showing improvements over simple baseline than previous metrics, allowing the emphasis cases of leveling down.
Existing group faimess definitions generally sist of comparing certain performance measure, such as True Positive Rate (TPR), False Positive Rate (FPR) or accuracy, across groups. In the following. for the sake of generality, we abstract away from the particular measure and denote by m(he, Tg) C [0, 1] the group-wise performance for model ho on the group of examples Tg. with the convention that higher values of m correspond to better performance. For instance, in the case of TPR (used define Equal Opportunity) de- fine m mhe, Tg) TPR Tg). while for FPR, we define m(h,, Tg) FPR(hg, Tg).
2 Setting
In this section, we begin by introducing our nota- tions and then formally define problem statement.
Existing Intersectional Framework
While the literature on group fairness in machine learning initially considered a single sensitive axis, several works have recently proposed fairness defi- nitions for the intersectional setting (Gohar and Cheng, 2023). Kearns et al. (2018) proposed subgroup-faimmsss, which is based on the difference in performance of particular group weighted by the size of the group. Several calibration and metric fairness-based variants considered by Hébert- Johnson al. (2018) and Yona and Rothblum (2018). A shorteoming of these notions that they weight each group by its size, hence small groups may not be protected even though they are
2.1 Notations
this study, we adopt and extend the notations proposed by Morina et al. (2019). Let denote the number of distinct sensitive axes of interest, which generally correspond to socio-demographic features of population. We refer to these sensi- tive axes A AI...., Ap. each of which is set of discrete-valued sensitive atlributes. For instance, a dataset may be composed of gender, race, and age as the three sensitive axes, and each of these sensitive axes may be encoded by a set of sen- sitive attributes, such as gender: (male, female,

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often the vastdvant ones.
relative performance between groups, and the ab- solute performance of the groups.
3.1 Differential Fairness
The second factor that in intersectional fair- ness, leveling down can have a negligible effect on the overall performance of a model on the full dataset. This is because the number of groups in intersectional fairness is typically quite large (ex- ponential in the number of sensitive axes p). There- fore, the bulk of examples generally do not belong to either the worst-off or best-off group, leading situation where the performance of other groups accounts for most of the model's overall perfor- mance. This issue may be further exacerbated if the class proportions are imbalanced across groups.
To circumvent the above issue, Foulds et al. (2020) proposed Differential Fairness (DF), which puts a constraint on the relative performance between all pairs of groups. DF was originally proposed for statistical parity (Foulds al., 2020), and then extended by (Morina al., 2019) to generalize other fairness definitions such as parity in False Positive Rates and Equal Odds. Below, we provide a general definition of DF based on an arbitrary group-wise performance measure m as defined in Section 2.2.
Definition (Differential Fairness). A model hg is e-differentially fair (DF) with respect group- wise performance measure m, if
4 a-Intersectional Fairness
In order to circumvent the above issue and effec- tively capture intersectional fairnsss while taking into account the leveling down phenomena, we propose a- I-Iterrscctionll Fairness (IF Our defi- nition is essentially a convex combination of two components, namely (i) Arnt, which takes into ac- count the relative performance between the two groups, such the ratio of their performance, and (ii) Dabe. which captures the leveling down effect by accounting for the absolute performance of the worst-off group.
S. DF(hg, m) = max log m(hg, Tg) g.g' Eg mhh,, Tg')
It is important to note that DF only depends on the relative performance between the best- performing group and the worst-performing group.
3.2 Shortcomings of Differential Fairness
We now highlight what we believe to be key shortcoming of DF in the context of intersectional fairness: DF can be improved by leveling down, i.e., harming the best-off and/or worst-off group, without significantly affecting the overall perfor- mance of the model. This problem is caused by the combination of two factors.
More precisely, given a model ha and a group- wise performance measure m, let us first define a measure of fairness for pair of groups and g':
La g' ha, m) aDabs +
+11 (1 ar (1)
First, DF is a strictly egalitarian measure that only considers the relative performance between groups. This can lead to situations where model that improves the performance across all groups is deemed more unfair by DF. To illustrate this, let the group-wise performance measure to be the TPR and consider two models he and hg- Let the worst-off and best-off group-wise performance of hg be 0.50 and 0.60, respectively. For hg. let be 0.65 and 0.95. According to DF, he is more fair than h2 as the two groups are closer, while hi has better performance for both groups. In other words, he is leveling down compared to hg. but is deemed more fair. This exhibits the tension between the
where [0, 1] and
A abs = max (1 - m(hg, Tx), m(hgg Tg')): )),
And 1 - max (m(ho, Ts), m(). Tg'))
min (m(ho, (((hha,TT) Tg), mho, Tx:))
Now taking the maximum value of Ta over all pairs of groups, we get our proposed notion of a- Intersectional Fairness.
Definition 2 (o-Inntesectiinal Faimees)). A model he is (a, -iniiseerscctoonllly fair (IF,)) with respect to group-wise performance measure m, if
IFa(hg m) = max Ta(g, g', ho, m) S 7. g.g' Eg
Wee note that, in their extension to parity in False Positive Rates. Morina et al. (2019) did not account for the fact that higher FPR means lower performance. hence harming all groups always leads to better faireess. Our general formulation in Definition fixes this prohlem through the convention that higher m corresponds to hetter performance.
Note that IF (hg, m) can be equivalently ob- tained as the the value of Ta over the pair of worst performing and the best performing group. as shown by the following proposition.

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Proposition 1. If a model he is 7)- intersectionally fair with respect group-wise performance measure m, then
0 y W
W
b 1
IF (ho. m. (g th gg ho: m) y,
Figure I: Group-wise performance range comparison. The range group-wise performances of models he and model h2 are respectively [wc, b] and [ū, bj. Note that the difference x (resp. y) between the best (resp. worst) group-wise performances of he and hg can be positive or negative.
where g" = arg geg m(ha, Tg) and g" arg mäxgeg m(he. Tg)
In the following, we compare and contrast our fairness definition with DF when evaluating the fairness of the two models. We then investigate various properties of our proposed definition and discuss the impact of a. In the interest of space, we delegate our discussion around the design choices for Aret and Aobe to Appendix A.
the other model has better Are performance. The faimess in this setting depends on the relative importance of absolute and relative performance for IFa, while for DF it exclu- sively depends on absolute performance. Figure 1, this corresponds to w I w and the blue region is smaller than the red region or vice-vers versa. Here, 3a E [0, 1] for which (ha. p: m) 2 [F m) and vice versa. On the other hand, DF(hg, m)) DF(ha, m) if y m(hg, Tg") ) S x m(hg, Tgb), otherwise DF(hg m) > DF(hg,
Comparing DF and IF oF The primary differ- ence between DF and IF, when comparing two models arises when one model adversely affects the worst-off group (Auis) more than the other, despite having better relative performance (Aret). this case, DF would consistently consider one model more fair than the other, whereas IF, en- ables the exploration of this tension by varying the relative importance of both criteria through a.
To summarize, in the first two cases, one model harms the worst-off group (absolute performance), and the relative performance of that model worse than the other. Thus, a good fairness measure should assign a higher unfairness to that model, which both DF and IF,, do. In the third case, one model performs better on the worst-off group, while the other model has closer relative perfor- mance. The fairness in this setting depends on the relative importance of absolute and relative perfor- mance. Here, DF consistently assigns one model a higher fairness than the other, while IF,, enables to explore this tension and tune the relative impor- tance of both criteria through a. For instance, the previous example in Section 2 falls in the third case. On the one hand, DF will assign higher for he, in comparison to hog- On the other hand, IF, will assign higher 7 for he, for a (0.0, 0.81), while for all other a, the y would be higher for hey- We illustrate the effect of a in more details below.
We formally capture this intuition follows.
Consider two models he and hg- Let the value of the worst-off and the best-off group's performance for the model hg be w and b, respectively. Similarly, for model ha let the worst and the best group's performance be w and b, respectively. Without the loss of generality, t and b can be written as t u and i b b y. Note that and y can be either positive or negative as long as ú S h. We visualize this setup Figure 1. Based on this setup, we have following cases:
2 2 O: In this case, he harms the worst- off group (absolute performance) more, and its relative performance is worse than hig In Figure 1, this corresponds to t 2 w and the blue region smaller than the red region. Here, (L m) < (ha, m) V a E [0, 1], and DF(g m) < DF(hs, m).
S This is similar to the case above, but with hg harming the groups more than hg. In Figure 1, this corresponds to w < w and the blue region is larger than the red region. Here, (hg m) < (Fa m) V a E 10, and DF(hg, m) < DF(hg, m).
Impact of a: The parameter a allows to tune the relative importance of A and Arct- On the one end of the spectrum, = 0 corresponds to con- sidering only the relative performance Aret, while corresponds to considering only the abso- lute performance. At a = 0.0 we recover the same relative ranking of unfairness as DF, and thus DF can be seen as a special case of IF,, In other words,
All other cases: In this setting, one of the model has better Ants performance, while

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for any three models he hag and hag such that Dh m) DF(h m) 2 DF m). then IFo(ho, m) 2 1og m) 2 I m). On the other end, a - only considers the absolute performance Aats), and a = 0.5 corresponds to giving Aabs and Ayel an equal importance. In prac- tice, it is useful to visualize the complete trade-off by plotting + aF (see Section 5).
has higher impact on minority (Nora and Hor- vath, 1989; Hinojosa, 2023).
Cases where data for disadvantaged groups is unreliable due to historical underrepresentua tion and lack of opportunities. For instance, certain facial recognition systems exhibit a higher likelihood of error when analyzing im- ages of dark-skinned female individuals (Buo- lamwini and Gebru, 2018). Similarly, Sap et al. (2019) found that the hate speech detec- tion systems are biased against black people.
Intersectional Property: We have the following interscctional property.
Proposition 2.
Let the model hg be (a, 7)) intersectionally fair over the set of groups defined
by A x Ap. Let 81 5... S sy S p. and P = As₁ A., be the Cartesian product of the sensitive axes where 8j E N Then, hy is (a, 7)-iintersectionally fair over P.
In such contexts, emphasizing improvement for these disadvantaged groups more pivotal than uniform performance over all subgroups, in line with the ideas of affirmative action. These scenar- ios best align with strategies seeking Demographic Parity Equalized Odds.
In other words, the fairness value calculated over the intersectional groups also holds over in- dependent and "gerrymandering" intersectional groups (Yang et al., 2020). For instance, if model is ( inttersecctionllyy fair in space defined by gender, race, and age, then it is also (a, 7)- interscctionally fair in the space defined by gender and race, just gender. We delegate the proof to Appendix B.
Setting a 0.25 (more focus on relative perfor- mance) in:
Scenarios where no group significantly worse off, but to make sure that the algo- rithm behaves similarly for all the groups in- volved. This is related to algorithmic bias, presented by Mehrabi et al. (2021). Moreover, the misclassification costs similar in this setting.
Generalization Guarantees: a-Intersectional Fairness enjoys the same generalization guaran- tees as the ones shown for DF in (Foulds et al., 2020). Indeed, the result of Foulds et al. (2020) re- lies on generalization analysis of the group-wise performance measure m, which directly translates into generalization guarantees for iF..
Legal or regulatory requirements may man- date similar outcomes across groups, like the 4/5th employment rule?. However, one must exercise caution when extrapolating this other contexts, as can lead the "portabil- ity trap" as discussed by (Selbst al., 2019).
Guidelines for setting a: a-Intersectional Fair- ness enables exploring the tradeoff between worst- case performance and relative performance across groups. Indeed, at alpha=0.0,, only relative perfor- mance considered, aligning with strictly egalitar- ian measures. On the other extreme, at alpha=1 .0, solely the worst-off group performance is consid- ered. Based on this, we recommend:
In such a context, the emphasis is on equality among the groups. In practice, these scenarios in- dicate places where the partitioner would advocate for Accuracy Parity.
Setting a 0.75 (more focus towards worst case performance) in:
Otherwise, recommend setting a 0.50 (a neutral default) when domain context-specific insights are available. This is what we used in our experiments. Ultimately, the choice of alpha re- flects an understanding of the domain, the inherent biases the data, and the real-world consequences of misclassifications.
Situations where the cost of misclassification is not similar for each group. In these cases, leveling down would disproportionately affect those subgroups for whom the cost is higher. One example can seen education system, where the cost of denying financial assistance

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5 Experiments
Both these datasets consists of free text re- sponses and binarized scores by the medical expert. Moreover, each response associated with gender, race, age, and income. We use same pre-processing as (Lalor et al., 2022) and follow the same procedure to split the dataset as described above.
In this section, we present experiments that show- case (i) the model's performance over the worst-off group the number of sensitive axes increases, and (ii) the "leveling down" phenomenon observed in various fairness-promoting mechanisms, along with the effectiveness of a-Intersectional Fairness in uncovering it. However, before describing these experiments, we begin with an overview of the datasets, baselines, and fairness measures used.
For improved readability, we present a subset of experiments in the main article. The remaining experiments are included in the Appendix.
Methods. We evaluate the fairness performance of the following methods: (i) Unconstrained which is oblivious to any fairness measure and solely optimizes the model's accuracy; (ii) Adversarial implements standard adversarial learning approach (Li et al., 2018), where an ad- versary is added to the Unconstrained with the objective to predict the sensitive attributes; (iii) FairGrad (Maheshwari and Perrot, 2022), is an in- processing approach that iteratively learns group- specific weights based on the faimess level of the model; (iv) INLP (Ravfogel et al., 2020), a post- processing approach that iteratively trains clas- sifier to predict the sensitive attributes and then projects the representation on the classifier's 's null space. To enforce fairness across multiple sensi- tive axes in this work, we follow the extension proposed by Subramanian et al. (2021); ); (v) Fair MixUp (Chuang and Mrouch, 2021) is data aug- mentation mechanism that enforces fairness by reg- ularizing the model on the paths of interpolated samples between the sensitive groups.
Datasets: We benchmark over four datasets cov- ering both text and images, with varying numbers of examples and sensitive groups:
The dataset is derived from multilingual Twitter Hate speech cor- pus (Huang et al., 2020) consisting of twects annotated with 4 demographic factors (sensi- tive axes), namely age, race, gender, and coun- try. The primary objective is to classify indi- vidual tweets as either hate speech or non-hate speech. In this work, we focus on the English subset and binarize all the demographic fac- tors resulting in a total of 63 sensitive groups. Moreover, we only choose tweets where all the demographic factors are present. Conse- quently, our train, valid and test sets consists of 22, 818, 4, 512, and 5, 032 tweets.
Twitter Hate Speech:
CelekA (Liu et al., 2015): The dataset consists of 202, 599 images of human faces, alongside 40 binary attributes for each image. We set 'sex', 'Young', 'Attractive', and 'Plle Skin' attributes the sensitive axis for the images and 'Smiling' the class label. We split the dataset into 80% training and 20% test split, Furthermore, we set aside 20% of the training set as the validation split.
In all our experiments, we employ the same model architecture for all the approaches to have fair comparison. Specifically, we use three- hidden layer fully connected neural network with 128, 64, and 32 corresponding sizes. Furthermore, ReLU as the activation with dropout fixed to 0.5. We optimize cross-entropy loss in all cases with Adam (Kingma and Ba, 2015) as the opti- mizer using default parameters. Moreover, for Twitter Hate Speech and Numeracy datasets, we encode the text using bert-base-uncased (Devlin et al., 2019) text encoder. For CelebA, an image dataset, we employ ResNet18 (He et al., 2016) as the encoder. In all cases, we do not fine-tune the pre-trained encoders. Lastly, several previous studies have shown the effectiveness of equal sam- pling improving fairness (Kamiran and Calders, 2009; Chawla et al., 2003; Kamiran and Calders,
Psychometric dataset (Abbasi et al., 2021): The dataset is collection of 8, 502 free text responses alongside numerical scores over multiple psychometric dimensions. In this work, we focus on two dimensions:
Numeracy reflects the numerical compre- hension capability of the individual.
Anxiety reflects the anxicty level as de- scribed by the patient.
'source code is available here: https: httpg/tttub.ccogoo
2010; Zellay et al., 2021). That is, to

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3.6
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Uncztaberee ned Adveraaiial
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Fordrod INLP Poi Mieli
hardrad
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Senstive bale
(a) CelebA TPR
(b) Numeracy TPR
(c) Twitter Hate Speech TPR
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025
Hid
samsithve aca
2013 tiee aa
sensiziee sots
(d) CelebA FPR
(c) Numeracy FPR
(f) Twitter Hate Speech FPR
Figure 2: Test results over the worst-off group CelebA, Twitter Hate Speech, and (b) Numeracy by varying the number of sensitive axes, For p binary sensitive axis in the dataset, the total number of sensitive groups are pd 1. Note that in FPR, lower the value better it is, while for TPR opposite is true.
counter the imbalance in the training data, the data is resampled so that there is an equal number of examples from each group and class in the final training set. Through preliminary experiments, we determine that equal sampling improves the worst- case performance of several approaches, including Unconstrained in various settings. We thus incor- porate as a hyperparameter indicating a contin- uous scale between undersampling and oversam- pling. Note that we also incorporate a setting where no equal sampling is performed, and we take the distribution as it is.
the results over all the sampled datasets. order to evaluate the utility of various methods, we em- ploy balanced accuracy. Note that the choice of TPR Parity, and FPR Parity allows the derivation of several other faimess measures including Equal Opportunities, and Equalized Odds.
5.1 Worst-off performance and number of sensitive axis
In this experiment, we empirically evaluate the in- terplay between the number of sensitive groups and the harm towards the worst-off group. To this end, we iteratively increase the number of sensitive axes in the dataset and report the performance of the worst-off group for each approach. For instance, with CelebA we first randomly added gender (ran- domly chosen) when considering I sensitive axis. In the next iteration, we added race (randomly cho- sen) to the set with gender (previously added). Sim- ilarly, we then added age, and finally country. Note that for all the datasets, we start with random choice of sensitive axis hoping to remove any form of selection bias. To select the optimal hyperpa- rameters for this experiment, we follow the same procedure described in (Maheshwari et al., 2022) with the objective to select the hyperparameters with the best performance over the worst-off group.
Fairness performance measure. In this work we focus on True Positive Rate parity and False Positive Rate parity as the fairness measure. The corresponding group wise performance measure for these fairness measures are TPR and FPR. Formally, m in case of TPR for group g is:
m(ha, Tg) = P(ho(x)= = 1/y = Va, E Tg.
while the FPR for a group g is:
mhho, Tx)
(x) 0ly - Va, E Tk
In order to estimate the empirical probabilities, we employ the bootstrap estimation procedure as proposed by Morina et al. (2019). In total, we generate 1000 datasets by sampling from the orig- inal dataset with replacement. We then estimate the probabilities on this dataset using smoothed empirical estimation mechanism and then average
We plot the results of this experiment in Figure 2. The results over the Anxiety dataset, which fol- low similar trend, can be found in the Appendix C. Based on these results, we observe that as the num- ber of subgroups increases, the performance of the

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Unconstrained
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(a) CelebA FPR
(b) Anxiety FPR
Figure 3: Value of IF, on the test of CelehA, and Numeracy datasets for varying a E [0, 1].
Method
BA f
Best Off 4
Worst Off 4
DF +
IF.=0.5 4
Unconstrained
0.81 + 0.0
0.08 + 0.01
0.36 + 0.04
0.36 +/- 0.06
0.31 +/- 0.02
Adversarial
0.8 + 0.0
0.07 + 0.02
0.32 + 0.02
0.31 +/- 0.12
0.28 +/- 0.04
FairGrad
0.77 0.01
0.14 + 0.01
0.39 + 0.01
0.34 0.03
0,4 +/- 0.02
INLP
0.8 + 0.0
0.09 + 0.01
0.34 + 0.04 0.04
0.32 +/-
0.32 +/- 0.01
Fair MixUp
0.8 0.0
0.08 + 0.01
0.37 + 0.02
0.38 0.04
0.3 0.01
(a) Results on CelebA
Method
BA f
Best Off |
Worst Off |
DF
IFa=0.5 $
Unconstrained
0.63 + 0.01
0.27 + 0.04
0.5 + 0.03
0.38 +/- 0.05
0.55 +/- 0.06
Adversarial
0.62 + 0.01
0.28 + 0.05
0.53 + 0.09
0.43 +/- 0.04
0.55 +/- 0.06
FairGrad
0.63 + 0.01
0.33 + 0.04
0.59 + 0.06
0.49 +/- 0.05
0.61 +/- 0.03
INLP
0.63 + 0.01
0.27 + 0.04
0.49 + 0.03
0.36 +/- 0.03
0.56 +/- 0.05
Fair MixUp
0.61 0.02
0.3 + 0.03
0.61 + 0.07
0.58 +/- 0.03
0.56 +/- 0.03
(b) Results Anxiety
Table 1: Test results on (a) CelebA, and (b) Anxiery using False Positive Rate while optimizing for DF. The utility of various approaches is measured by balanced accuracy (BA), whereas fairness is measured by differential fairness DF and intersectional fairness IF.=0.5. For both faimess definition, lower is better, while for balanced accuracy, higher is better. The Best Off and Worst Off, in both cases lower is better, represents the min FPR and max FPR. Results have been averaged over different runs. We deem method to exhibit leveling down if its performance on either the worst-off or best-off group is inferior to the performance of an unconstrained model which we have highlighted using cyan
worst-off group becomes worse for all approaches in all settings. This can be attributed to the fact that the number of training examples available for each group decreases the number of sen- sitive axis in the dataset increases. In terms of the performance of other approaches in comparison to Unconstrained. we find that fairness-inducing approuches generally perform better or similar to Unconstrained when 1 or sensitive axes are con- sidered. However, when 3 or more sensitive axis are considered, the performance of all approaches tends to converge to that of Unconstrained For instance, in CelebA, the one hand, with sensi-
tive axis, all approaches significantly outperform Unconstrained with the difference between the best pest-performing method and Unconstrained be- ing 0.26. On the other hand, when 4 sensitive axes considered, the difference between the best- performing method and Unconstrained is 0.03, with only Adversarias outperforming it.
In similar fashion, when considering TPR over Numeracy dataset, Unconstras ained performs sig- nificantly worse than FairGrad and Adversarial with sensitive axis while outperforming all ap- proaches apart from INLP when sensitive axis are considered. Similar observations can be made for

---

Numeracy and Twitter Hate Speech datasets in the FPR setting, with some minor exceptions. Overall we find that most fairness approaches start barming or do not improve the worst-off group as the num- ber of sensitive axes grows the dataset. Thus is pivotal for an intersectional fairess measure to consider the harm induced by an approach while calculating its fairness.
worst-off group's performance is considered, i.e., 0.0, it rapidly becomes unfair with being one of the most unfair method at a = 1.0. Interest- ingly, in Anxiety, INLP starts as one of the worst- performing mechanisms. However, with o > 0.0, it quickly outperforms most approaches.
These findings shed light the trade-offs and complexities inherent in optimizing fairesss while maintaining worst-off group performance. It high- lights the need for comprehensive evaluation met- rics and the importance of considering the per- formance of both advantaged and disadvantaged groups in the fairesss analysis. Finally, we em- phasize that methods do not always exhibit level- ing down. In settings without leveling down, DF adequately captures unfairness, producing values similar to a- Intersectional Faimess. However, ery method displays some degree of leveling down for some combinations of datasets and metries. A robust fairness measure should expose unfairness universally, which our experiments demonstrate IF, achieves.
5.2 Benchmarking Intersectional Fairness
this experiment, we showcase the leveling down phenomena shown by various existing approaches. We also compare and contrast IF and DF. The results of this comparison over FPR parity can be found in Table la and lb for CelebA and Anxiety respectively. The results of remaining two datasets over FPR parity, and all datasets over TPR Parity can be found in Appendix C. In these experiment, we deem a method to exhibit leveling down if per- formance on either the worst-off or best-off group is inferior to the performance of an unconstrained model. In the results table, we highlight the meth- ods that show leveling down in cyan
Conclusion
We find that most of the methods have similar balanced accuracy across all the datasets, even the fairness levels are different. This observation aligns with the arguments presented in Section 3 about the relationship between group fairness mea- sure and the overall performance. In terms of fair- ness, most methods showcase leveling down. For instance, over the CelebA dataset, all methods apart from Adversarial shows leveling down. While in the case of Anxiety, all methods apart from INLP shows leveling down.
We propose new definition for measuring inter- sectional fairness of statistical models, the group classification setting. We provide various compar- ative analyses of our proposed measure, and con- trast it with existing ones. Through them, we show that our fairness defimition can uncover various no- tions of harm, including notably, the leveling down phenomenon. We further show that many fairness- inducing methods show significant improvement over simple unconstrained approach. Through this work, we provide tools the community better uncover latent vectors of harm. Further, our findings chart a path for developing new fairness- inducing approaches which optimizes for fairness without harming the groups involved.
While comparing DF and IF we find that =F5= is more conservative assigning fairness value, with most approaches performing similarly to Unconstrained. Moreover, leveling down cases may go unnoticed in DF. For instance, over the CelebA dataset, even though FairGrad and INLP showcases leveling down, the fairness value signed by DF is lower for them than the signed Unconstrained. Similar observation can be seen over Numeracy in case of INLP.
7 Acknowledgement
The authors would like to thank the Agence Na- tionale de la Recherche for funding this work under grant number ANR-19-CE2330002, as well as the reviewers for their feedback and suggestions.
A particular advantage of IF, over DF is that it cquips the practitioner with more nuanced view of the results. In Figure 3, we plot the complete trade-off between the relative and the absolute per- formance of groups by varying a. For instance, in CelebA FPR, Fair MixUp shows the lowest level unfairness at a = 0.0. However, as soon as the
8 Limitations
While appealing, a-Intersectional Fairness also has limitations. One of the primary ones that it as- sumes a minimum number of examples for cach subgroup to estimate the faimess level of the model

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correctly, Moreover, does not consider the data drift over time, as assumes a static view of the problem. Thus we recommend checking the fair- ness level over time to account for it. Further, in this definition, setting up a is left to the practitioner and thus can be abused. In the future, we aim to develop mechanisms to validate without access to the dataset or model.
7th European Conference on Principles and Prac- tice of Knowledge Discovery in Databases, Cavtat- Dubrovnik, Croatia, September 22-26, 2003, Pro- ceedings, volume 2838 of Lecture Notes in Computer Science, pages 107-119. Springer.
Ching- Yao Chuang and Youssef Mroueh. 2021. Fair mixup: Fairness via interpolation. In 9th Inter- national Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net.
Finally, we want emphasize that a hypothetical perfectly fair model might not be devoid of social harm. Firstly, vectors of harm of using statistical models are not restricted to existing definitions of group fairness. Further, if some socio-economic groups are not present in a given dataset, exist- ing fairness-inducing approaches are likely to not have any positive impact towards them when en- countered upon deployment. Such is the case with commonly used datasets in the community, which over-simplify gender and race as binary features, ignoring people of mixed heritage, or non- non-binary gender, for example. In our experiments, we too have used these datasets, owing to their prevalence, and we urge the community to create dataset with non-binary attributes. That said, our measure works with non-binary sensitive attributes, with modi- fications.
Equal Employment Opportunity Commission et al. 1990. Uniform guidelines on employec selection procedures. Fed Register, 1:216-243.
Kimberle Crenshaw. 1989. Demarginalizing the inter- section of race and sex: A black feminist critique of antidiscrimination doctrine, feminist theory and antiracist politics. The University of Chicago Legal Forum, 140:139-167.
Christophe Denis, Romuald Elie, Mohamed Hebiri, and François Hu. 2021. Faimess guarantee in multi-class classification. arXiv preprint 2rXiv:2109.1388 1364..
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Choice of Aret An alternate choice of Aret is to utilize the performance difference between the groups instead of the above mentioned ratio. How- ever, we advocate for the ratio as a superior choice for the following reasons:
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Scale-Invariant Comparison: The ratio en- ables comparing two models without the influ- ence of the scale by normalizing the relative performance of model. For instance, assume two models hg and he with the worst and the best group's 's performance for hg as 0.01 and 0.02 respectively, and 0.1 and 0.2 for hgr. this setting, the Are as the difference would always assign hg as fairer, even though both models are twice worse for the worst group compared to the best group. Note that our overall fairness measure accounts for the ef- fect of scale through the inclusion of Aabe- This is in-contrast to DFwhich does not take scale into account.
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Alignment with the 80% rule: The ratio aligns with the well-known 80% rule (Commission et al., 1990), which states that there exists le- gal evidence of discrimination if the ratio of the probabilities for favorable outcome be- tween the disadvantaged sensitive group and the advantaged sensitive group is less than 0.8. By adopting the ratio as Ayel, our metric adheres this established criterion.
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Influence of worst-case group: If Aret repre- sents the difference in performance, then at a - 0.5 the model with better worst-case per- formance will always have a lower than the one with worse worst-case performance. In other words, at 0.5, Aabs would always dominate Aret- However, this contradicts the intuitive understanding that, at a = 0.5, both Arel and Agbs should exert an equal influence.
PMLR.
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Choice of Aabs An alternate choice we explored for Aabs was the average performance of the two groups involved instead of just the worst- performing one. However, Proposition I does not hold in the average case. This implics that a pair of groups can exist for which I., is larger than the pair of groups consisting of the worst and best- performing groups. Moreover, Proposition 1 is an essential building block for intersectional property which described later.
A Design Choices
In this section, we discuss our design choices for
Aabs and Aret-

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Method
BA 1
Best Off 1
Worst Off t
DF
a.5 /
Unconstrained
0,8 + 0.01
0.84 + 0.01
0.45 + 0.04
0.62 0.03
0.43 +/- 0.01
Adversarial
0.8 + 0.01
0.84 + 0.01
0.46 + 0.04
0.6 +/- 0.04
0.44 +/ 0.01
FairGrad
0.78 0.01
0.85 + 0.02
0.44 + 0.04
0.66 +/- 0.02
0.43 +/- 0.03
INLP
0.8 + 0.0
0.85 + 0.03
0.52 + 0.05
0.49 +/- 0.04
0.41 +/- 0.02
Fair MixUp
0.79 0.01
0.85 + 0.03
0.48 + 0.05
0.57 0.05
0.43 +/- 0.04
(a) Results on CelebA
Method
BA t
Best Off |
Worst Off 1
DF +
IFa=0.5 4
Unconstrained
0.68 + 0.02
0.87 + 0.05
0.61 + 0.04
0.36 +/- 0.01
0.38 +/- 0.09
Adversarial
0.7 + 0.01
0.81 + 0.05
0.55 + 0.08
0.39 0.03
0.45 +/- 0.07
FairGrad
0.68 + 0.02
0.88 + 0.04
0.64 + 0.09
0.32 +/- 0.03
0.35 +/ 0.07
INLP
0.68 + 0.01
0.84 + 0.05
0.66 + 0.1
0.24 +/- 0.03
0.44 +/- 0.08
Fair MixUp
0.7 0.01
0.81 + 0.05
0.54 + 0.05
0.41 +/- 0.02
0.44 +/- 0.07
Results Numeracy
Method
BA f
Best Off f
Worst Off f
DF L
IF &
Unconstrained
0.79 + 0.01
0.96 + 0.01
0.77 + 0.03
0.22 +/- 0.03
0.2 +/- 0.03
Adversarial
0.76 + 0.0
0.97 + 0.01
0.81 + 0.04
0.18 +/- 0.04
0.21 +/- 0.03
FairGrad
0.76 + 0.02
0.95 + 0.01
0.78 + 0.03
0.2 +/- 0.04
0.25 +/- 0.03
INLP
0.67 + 0.01
0.73+0.03
0.38 + 0.03
0.65 0.05
0.56 +- 0.03
Fair MixUp
0.76 + 0.01
0.98 + 0.0
0.84 + 0.02
0.15 +/- 0.02
0.16 +/- 0.01
(c) Results on Twitter Hate Specch
Method
BA f
Best Off f
Worst Off f
DF
IFa=0.5 $
Unconstrained
0.63 + 0.01
0.77 + 0.02
0.47 + 0.07
0.49 +/- 0.05
0.5 0.02
Adversarial
0.63 + 0.01
0.82 + 0.05
0.51 + 0.1
0.47 +/- 0.06
0.45 +/- 0.05
FairGrad
0.63 + 0.01
0.76 + 0.01
0.47 + 0.06
0.48 +/- 0.04
0.52 +/ 0.02
INLP
0.63 + 0.01
0.76 + 0.02
0.51 + 0.04
0.4 0.01
0.51 +/- 0.03
Fair MixUp
0.62 + 0.01
0.75 + 0.07
0.45 + 0.07
0.51 +/- 0.03
0.52 +- 0.06
(d) Results on Anxiety
Table 2: Test results on (a) CelebA, (b) Numeracy, and (c) Twitter Hate Speech using True Positive Rate while optimizing for DF. The utility of various approaches is measured by balanced accuracy (BA), whereas fairness is measured by differential fairness DF and intersectional fairness F For both fairness definition, lower is better. while for balanced accuracy, higher better. The Best Off and Worst Off, in both cases higher better, represents the min TPR and max TPR. Results have been averaged over 5 different runs. We have also highlighted methods which showcase leveling down using cyan
B Intersectional Property
and P As₁ x .., be the Cartesian product of the sensitive axes where sj E N Then, he is (a, -nt-errsectionallyy fair over P.
In this section, prove the intersectional property stated in Section 4. The proof follows the same procedure as described by Foulds et (2020). The intersectional property states that:
The essential idea of the proof is show that the maximum and the minimum group wise per- formance
Proposition. Let the model hg be (a, intersectionally fair over the set of groups defined by 9 A₁ .... Ap- Let s 81 S ..... st S p.
is bounded by the maximum and the minimum group wise performance in G. After proving the above, then using Proposition 1, we

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Method
BA t
Best Off 4
Worst Off 4
DF
IF a= /
Unconstrained
0.7 0.01
0.22 + 0.03
0.04
0.44 +/ 0.1
0.51 +/- 0.04
Adversarial
0.71 0.01
0.14 + 0.03
0.38 + 0.02
0.33 0.22
0.42 +/- 0.08
FairGrad
0.02
0.19 + 0.06
0.51+
0.5 0.22
0.45 +/- 0.07
INLP
0.68 + 0.01
0.27 + 0.08
0.52 + 0.08
0.42 +/- 0.13
0.58 +/- 0.06
Fair MixUp
0.7 + 0.01
0.22 + 0.05
0.48 + 0.03
0.41 +/- 0.17
0.52 +/- 0.06
(a) Results on Numeracy
Method
BA t
Best Off 4
Worst Off 4
DF +
aF.5 +
Unconstrained
0.81 + 0.01
0.18 + 0.02
0.47 + 0.02
0.44 +/- 0.04
0.46 +/- 0.03
Adversarial
0.8 0.01
0.18 + 0.02
0.46 + 0.02
0.42 +/- 0.03
0.47 +/- 0.04
FairGrad
0.79 0.01
0.19 + 0.03
0.51 + 0.04
0.5 0.03
0.47 +/- 0.04
INLP
0.67 + 0.01
0.18 0.1
0.38 + 0.18
0.28 +/- 0.02
0.47 +/- 0.1
Fair MixUp
0.81 + 0.01
0.18 + 0.02
0.49 + 0.04
0.47 +/- 0.02
0.46 +/- 0.03
(b) Results on Twitter Hate Speech
Table 3: Test results on (a) Numeracy, and (b) Twitter Hate Speech using False Positive Rate while optimizing for DF. The utility of various approaches measured by balanced accuracy (BA), whereas fairness measured by differential fairness DF and intersectional fairness IF a.55 For both fairness definition, lower is better, while for balanced accuracy, higher is better. The Best Off and Worst Off, in both cases lower is better, represents the min FPR and max FPR. Results have been averaged over different runs. We have also highlighted methods which showcase leveling down using cyan
can show that IF& over G higher than IF, over P.
C Extended Experiments
In this section, we detail the additional results. Ta- ble 2 provides results for the True Positive Rate (TPR) faimess measure, as outlined in the Experi- ment Section 5.2. In Figure 6, vary the number of sensitive axes and plot the worst-case perfor- mance for Anxiety FPR and TPR settings. Fi- nally, Table 3 displays results related to the FPR parity fairness measure, focusing on the Twitter Hate Speech and Numeracy datasets. Notably, for TPR, each method exhibits leveling down in at least one dataset. For cxample, Adver sarial shows lev- eling down in the Numeracy dataset, whereas INLP does so both the Twitter Hate Speech and Anxi- ety datasets. Similarly, as with FPR, DF does not consistently identify leveling down. As evidence, while both FairGrad and INLP demonstrate level- ing down, they show a better fairness level than Unconstrained.
Define E = A x ... x Aa-1 x Act1... x
Ak-1 X Ak11 x... x Ap. the Cartesian product of the protected attributes included in G but not in P. Then for any model ho. y Range(ho),
max Pho (ho(x) yyp RCP:P(g(0)50
Σ (ho(l) 8/ g)
max PCP:)
oCE
Pho E = clg)
Σ
<
max 00 BEF d'EE:Phg (E=e(
max
(ho = w e', 8))
Pa(E elg)
max
max
(000)(/))))><<<<E33333 (F= 0)x00)000
Pho (ho(x) = E - = e',g)
max PMI (x()) = -yls') s'Eg:P(s(10)>0
By
similar
argument,
mingeP:P(gle)>0 Pho (ho(x) P = g >
Pho (ha(x) = ylg'). Applying
Corollary 1, we hence bound y in P by the y in g

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9.50
FarrSrud
0.50
Afsersaii
FarGred
2.5
Uncoralraree
9.45
INLF
Advessrtal
ParMsccp
0.6
Tae Nidi
Parcro
Unconeraiied sugmanted
INLP Fair Nicup
0.40
0.25
02
3.50
0.30
02
(a) CelebA TPR
(b) Numeracy TPR
Twitter Hate Speech TPR
Figure 4: Value IFa on the test set of CelebA, Numeracy, and Twitter Hate Speech datasets for varying a E [0.1].
0.575
Unconstrained
Adversarial
0.60
FairGrad
0.525
INLP
Unconstrained
0.55
Adversarial
0.500
FairGrad
0.475
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INLP
Fair MixUp
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(a) Anxicty
(b) Anxiety
Figure 5: Value of IF, on the test set of Anxiety datasets for varying E [0, 1].
0.7
0.70
Unconstrained
Adversarial
Worst
0.6
Worst TPR
0.65
FairGrad
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Unconstrained
0.60
Fair MiXUp
Adversarial
FairGrad
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sensitive axis
sensilive axis
(a) Amxicty FPR
(b) Anxicty TPR
Figure 6: Test results the worst-off group Anxiety by varying the number of sensitive For binary sensitive axis in the dataset, the total number of sensitive groups are pd - 1. Note that in FPR, lower the value better it is, while for TPR opposite true.


Reviews:
[{"review_id": "iSgWG9UeEy", "reviewer": "Reviewer_JvEV", "summary": "", "questions": "1. Could you please provide a concrete guideline for choosing $/alpha$ in the evaluation of intersectional fairness?", "limitations": "", "rating": 3, "confidence": 3, "soundness": 4, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank you for the review. Please let us know if we missed some of your points or if our comments and the other reviews raised additional questions.\n\n> Guideline for choosing alpha:\n\n$\\text{IF}_{\\alpha}$ enables exploring the tradeoff between worst-case performance and relative performance across groups. Indeed, at alpha=0.0, only relative performance is considered, aligning with strictly egalitarian measures. On the other extreme, at alpha=1.0, solely the worst-off group performance is considered. Based on this, we recommend:\n\n- Setting $\\alpha$=0.75 (more focus towards worst case performance) in:\n\t- Situations where the cost of misclassification is not similar for each group. In these cases, leveling down would disproportionately affect those subgroups for whom the cost is higher. One example can be seen in education system, where the cost of denying financial assistane has higher impact on moniroty [1][2]. \n\t- Cases where data for disadvantaged groups is unreliable due to historical underrepresentation and lack of opportunities. For instance, certain facial recognition systems exhibit a higher likelihood of error when analyzing images of dark-skinned female individuals [3]. Similarly, [4] found that the hate speech detection systems are biased against black people.  \n\nIn such contexts, emphasizing improvement for these disadvantaged groups is more pivotal than uniform performance over all subgroups, in line with the ideas of affirmative action. These scenarios best align with strategies seeking Demographic Parity or Equalized Odds.\n\n- Setting $\\alpha$=0.25 (more focus on relative performance) in:\n\t- Scenarios where no group is significantly worse off, but to make sure that the algorithm behaves similarly for all the groups involved. This is related to algorithmic bias, as presented by [5]. Moreover, the misclassification costs are similar in this setting.\n\t- Legal or regulatory requirements may mandate similar outcomes across groups, like the 4/5th employment rule [6]. However, one must exercise caution when extrapolating this to other contexts, as it can lead to the \"portability trap\" as discussed by [7].\n\nIn such a context, the emphasis is on equality among the groups. In practice, these scenarios indicate places where the partitioner would advocate for Accuracy Parity.\n\n- Otherwise, we recommend setting $\\alpha$=0.50 (a neutral default) when no domain or context-specific insights are available. This is what we used in our experiments.\n\n\nUltimately, the choice of alpha reflects an understanding of the domain, the inherent biases in the data, and the real-world consequences of misclassifications. \n\nWe agree with the reviewer that adding these discussions and guidelines will improve the quality and practical usefulness of our work. We will thus include them in the final version of the paper.\n\n\n> Result visualization needs to be improved:\n\nFor the final version of the paper, we will use a bar chart alongside Table 1 to improve the readability.\n\n> Studied topic in this paper is less solely related to NLP:\n\nWe included the CelebA dataset to demonstrate the broad applicability of our approach and show that the leveling down phenomenon generalizes beyond NLP contexts. Moreover, we believe intersectionality is pertinent to NLP, as it naturally arises in text data as multiple attributes of people are known to affect how they write [8][9].\n\nThank you for suggesting the Jigsaw dataset. However, the identities in Jigsaw are not intersectional, which is the focus of this work. Indeed, each text in Jigsaw has a toxicity score and a single demographic identity such as feminist, queer, atheist, etc. Please let us know if we have misunderstood this suggestion. Instead of Jigsaw, we ran our experiments over the Anxiety dataset, which, akin to Numeracy, is part of the Psychometrics test set developed by [10]. The test consists of a free-text response about the patient's anxiety and a numerical score assigned by the medical expert. Each text is also associated with self-reported gender, race, age, and income. For pre-processing, we followed the footsteps of [11] and binarized each sensitive attribute. Please find a snippet of the results below (along the lines of Table 1). Our preliminary results reinforce our conclusion that methods showcase leveling down, and $\\text{IF}_{\\alpha}$ helps uncover it.\n\n| Method        | BA            | Best Off      | Worst Off     | DF            | $\\text{IF}_{\\alpha=0.5}$  |\n|---------------|---------------|---------------|---------------|---------------|---------------|\n| Unconstrained | 0.63 +/- 0.01 | 0.29 +/- 0.08 | 0.58 +/- 0.11 | 1.69 +/- 0.05 | 0.55 +/- 0.08 |\n| Adversarial   | 0.62 +/- 0.01 | 0.42 +/- 0.1  | 0.73 +/- 0.12 | 2.14 +/- 0.07 | 0.66 +/- 0.08 |\n| FairGrad      | 0.63 +/- 0.01 | 0.34 +/- 0.03 | 0.58 +/- 0.01 | 1.57 +/- 0.03 | 0.61 +/- 0.03 |\n| INLP          | 0.63 +/- 0.01 | 0.29 +/- 0.04 | 0.52 +/- 0.04 | 1.47 +/- 0.04 | 0.57 +/- 0.05 |\n| Fair MixUp    | 0.62 +/- 0.01 | 0.35 +/- 0.05 | 0.68 +/- 0.09 | 2.03 +/- 0.07 | 0.61 +/- 0.05 |\n\n\n\nTest results on Anxiety dataset using False Positive Rate while optimizing for DF. The utility of various approaches is measured by balanced accuracy (BA), whereas fairness is measured by differential fairness DF and intersectional fairness IFα=0.5. For both fairness definition, lower is better, while for balanced accuracy, higher is better. The Best Off and Worst Off, in both cases lower is better, represents the min FPR and max FPR. Results have been averaged over 5 different runs.\n\n| Method        | BA            | Best Off      | Worst Off     | DF            | $\\text{IF}_{\\alpha=0.5}$  |\n|---------------|---------------|---------------|---------------|---------------|---------------|\n| Unconstrained | 0.63 +/- 0.01 | 0.79 +/- 0.02 | 0.47 +/- 0.07 | 1.68 +/- 0.10 | 0.49 +/- 0.03 |\n| Adversarial   | 0.62 +/- 0.01 | 0.83 +/- 0.04 | 0.53 +/- 0.08 | 1.56 +/- 0.09 | 0.43 +/- 0.05 |\n| FairGrad      | 0.63 +/- 0.01 | 0.8 +/- 0.02  | 0.51 +/- 0.04 | 1.56 +/- 0.12 | 0.47 +/- 0.02 |\n| INLP          | 0.63 +/- 0.01 | 0.77 +/- 0.02 | 0.51 +/- 0.04 | 1.51 +/- 0.08 | 0.51 +/- 0.02 |\n| Fair MixUp    | 0.63 +/- 0.01 | 0.79 +/- 0.02 | 0.48 +/- 0.06 | 1.64 +/- 0.15 | 0.49 +/- 0.03 |\n\nTest results on Anxiety dataset using True Positive Rate while optimizing for DF. Rest everything remains the same as above. \n\n\n[1]: Nora, Amaury, and Fran Horvath. \"Financial assistance: Minority enrollments and persistence.\" Education and Urban Society 21.3 (1989): 299-311.\n\n[2]:Hinojosa, Elisa Reyes. \"Unequal Access to Higher Education: Student Loan Debt Disproportionately Impacts Minority Students.\" Scholar 25 (2023): 63.\n\n[3]: Buolamwini, Joy, and Timnit Gebru. \"Gender shades: Intersectional accuracy disparities in commercial gender classification.\" Conference on fairness, accountability and transparency. PMLR, 2018.\n\n[4]: Sap, Maarten, et al. \"The risk of racial bias in hate speech detection.\" Proceedings of the 57th annual meeting of the association for computational linguistics. 2019.\n\n[5]: Mehrabi, Ninareh, et al. \"A survey on bias and fairness in machine learning.\" ACM computing surveys (CSUR) 54.6 (2021): 1-35.\n\n[6]: Law Cornell -  https://www.law.cornell.edu/cfr/text/29/1607.4\n\n[7]: Selbst, Andrew D., et al. \"Fairness and abstraction in sociotechnical systems.\" Proceedings of the conference on fairness, accountability, and transparency. 2019.\n\n[8]: Peersman, Claudia, Walter Daelemans, and Leona Van Vaerenbergh. \"Predicting age and gender in online social networks.\" Proceedings of the 3rd international workshop on Search and mining user-generated contents. 2011.\n\n[9]: Rangel, Francisco, et al. \"Overview of the 3rd Author Profiling Task at PAN 2015.\" CLEF2015 Working Notes. Working Notes of CLEF 2015-Conference and Labs of the Evaluation forum.. Notebook Papers, 2015.\n\n[10]: Abbasi, Ahmed, et al. \"Constructing a psychometric testbed for fair natural language processing.\" Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing. 2021.\n\n[11]: Lalor, John P., et al. \"Benchmarking intersectional biases in NLP.\" Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. 2022.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "1. The paper is technically sound, and the proposed definition could provide a better evaluation and understanding of intersectional fairness.   \n2. The comparison of differential fairness is well-written, and the experiments are comprehensive.", "reasons_to_reject": "1. The topic studied in this paper is less solely related to NLP. It is a general algorithmic fairness topic. The evaluation also uses CelebA – a vision dataset. I will suggest the author use Jigsaw dataset instead in the evaluation.\n\n2. Choosing $\\alpha$ for practitioners is hard. The author acknowledges the practitioner can abuse the choice of $/alpha$ in the limitation section. However, the author does not provide a guideline for choosing $\\alpha$ in the evaluation, which should be critical. \n\n3. Result visualization needs to be improved. For example, the results in Table 1 are hard to interpret. Maybe the bar charts could further improve the visualization by highlighting the difference between the best-offs and worst-offs.", "questions_to_authors": "1. Could you please provide a concrete guideline for choosing $/alpha$ in the evaluation of intersectional fairness?", "paper_topic_and_main_contributions": "The paper proposes a new intersectional fairness definition. The definition new definition is a linear combination of two components: (1) $\\Delta_{ref}$, which takes into account the relative performance between the two groups,  and (2) $\\Delta_{abs}$, which captures the leveling down effect by accounting for the absolute performance of the worst-off group. Through rigorous ad empirical analysis, they demonstrate the necessity of the proposed definition.", "excitement": 3, "reproducibility": 4, "ethical_concerns": "Yes", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "5EnB3kuUb1", "reviewer": "Reviewer_Msnu", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": 4, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank you for the review. Please let us know if we missed some of your points or if our comments and the other reviews raised additional questions.\n\n> Guideline for choosing alpha:\n\nPlease see our reply to the Reviewer JvEV about guidelines for choosing alpha. We will add these guidelines to the final version of the paper.\n\n> How exactly the IF improves model performance:\n\n$IF_{\\alpha}$ does not directly improve model performance since none of the evaluated methods explicitly optimize for $IF_{\\alpha}$. We hope future work develops techniques tailored to optimizing $IF_{\\alpha}$, which is out of the scope of this paper. The key takeaways of the current experiments are:\n- Experiment 1 shows existing methods can exhibit leveling down, harming involved groups. This is particularly evident in intersectional settings.\n- Experiment 2 demonstrates that current fairness measures are sometimes inadequate at uncovering the leveling down phenomena. In contrast, our $IF_{\\alpha}$ is more conservative in measuring fairness, and most methods perform similarly to Unconstrained. This showcases the need for developing better fairness-inducing mechanisms. Moreover, $IF_{\\alpha}$ helps explore tradeoffs between strictly egalitarian fairness views and improving worst-case performance (see Figure 3).\n\nImportantly, we emphasize that methods do not always exhibit leveling down. In settings without leveling down, DF adequately captures unfairness, producing values similar to $IF_{\\alpha}$. However, every method displays some degree of leveling down for some combinations of datasets and metrics. A robust fairness measure should expose unfairness universally, which our experiments demonstrate $IF_{\\alpha}$ achieves. In summary, no method displays leveling down in all cases, resulting in method's $IF_{\\alpha}$ not being consistently worse than baseline (unconstrained) in every setting. Finally, following the suggestion of Reviewer JvEV, we will also add bar plots corresponding to Table 1, making this leveling down phenomena more explicit.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "This paper effectively demonstrate the harmful impact of the 'leveling down' issue, providing both a cogent explanation and compelling motivation for its new definition. They introduce a novel definition for intersectional fairness, termed '$\\alpha$-Intersectional Fairness', whose definition is rigorous and well-articulated. ", "reasons_to_reject": "The work introduces another hyper-parameter $\\alpha$. In the paper, it does not show how to decide this parameter $\\alpha$. In the empirical experiment, there is not enough direct evidence to show how exactly the $\\alpha$IF improves model performance. ", "questions_to_authors": "", "paper_topic_and_main_contributions": "This work is about the problem of intersectional group fairness in the classification setting. The main contribution is to propose a new intersectional fairness definition called $\\alpha$-Intersectional Fairness. This new definition is mainly defined to balance the leveling down issue in old Differential Fairness. ", "excitement": 3, "reproducibility": 5, "ethical_concerns": "No", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "u4DzXvToOP", "reviewer": "Reviewer_w78s", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "soundness": 3, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank you for the review. Please let us know if we missed some of your points or if our comments and the other reviews raised additional questions.\n\n\n> The method of adding sensitive axes is not clearly explained. Did you add axes that belong to the same set, or were they added randomly?\n\n  We are not sure we understand the question exactly, but we do our best to clarify our setup below.\n  \n  Each dataset we experimented with had pre-defined sensitive axes, either from the original dataset authors or commonly used in the community. For instance, the Twitter Hate Speech dataset had 4 binary sensitive axes - age with sensitive attributes as {U45, B45}, race with sensitive attributes as {African American, European American}, gender with sensitive attributes as {Male, Female}, and country with sensitive attributes as {USA, Outside USA}. This resulted in a total of 63 sensitive groups.\n  \n- For the first experiment, we iteratively selected sensitive axes (and all the corresponding sensitive attributes) and added them to the set. For instance, with CelebA we first randomly added gender (randomly chosen) when considering 1 sensitive axis. In the next iteration, we added race (randomly chosen) to the set with gender (previously added). Similarly, we then added age, and finally country. Note that for all the datasets, we start with a random choice of sensitive axis hoping to remove any form of selection bias. \n    \n - For our second experiment we choose all the sensitive axes and their corresponding attributes. \n\n \n> The result table does not show consistent improvement from other chosen metrics when compared to the baseline. How does this support the advantages of IF-alpha?\n\nPlease see our reply to the Reviewer, Msnu, about the improvements related to choosing $IF_{\\alpha}$. In summary, $IF_{\\alpha}$ does not improve performance as none of the models explicitly optimized for $IF_{\\alpha}$. The primary objective of our experiments was to showcase leveling down phenomena and that $IF_{\\alpha}$ is more resistant to it.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "- The paper is well-written and has a clear structure. It is easy to understand the shortcomings of existing measurements and the reasons for proposing a new one.\n- The proposed measurement is explained clearly.\n- The claim about the leveling down effect is supported by experiments.\n- Multiple datasets are used for cross-validation.", "reasons_to_reject": "- The method of adding sensitive axes is not clearly explained. Did you add axes that belong to the same set, or were they added randomly?\n- The result table does not show consistent improvement from other chosen metrics when compared to the baseline. How does this support the advantages of $\\text{IF}_\\alpha$?", "questions_to_authors": "", "paper_topic_and_main_contributions": "The researchers suggest that existing fairness measurements conceal a \"leveling down\" effect. Therefore, they propose a new measurement called $\\alpha$-intersection fairness, which considers both absolute and relative performance across sensitive groups. They demonstrated their proposed measurement by benchmarking multiple popular in-processing fair machine learning approaches and observed that previous methods did not achieve improvement regarding this metric.", "excitement": 3, "reproducibility": 4, "ethical_concerns": "No", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a new intersectional fairness definition that combines both absolute and relative performance across sensitive groups. The authors demonstrate the necessity of the proposed definition. The paper is well-written and has a clear structure. This technically sound paper could provide a better evaluation and understanding of intersectional fairness.

However, this paper lacks a description of how to choose $\alpha$. And there should be better evidence to show how exactly the 
IF improves model performance. Including an NLP task is also recommended.
Decision: Accept-Main