Characterizing Intersectional Group Fairness with Worst-Case Comparisons

Introduction

The use of machine learning algorithms is ubiquitous in the developed world. It has become an integral part of society, affecting the lives of millions of people. Algorithmic decisions vary from low-stakes determinations, like product or film recommendations, to high-impact like loan or credit approval Mukerjee et al. (2002), hiring recommendations Bogen and Rieke (2018), facial recognition Vasilescu and Terzopoulos (2002) and prison recidivism Corbett-Davies and Goel (2018). With this direct impact on people’s lives, the need for fair and unbiased algorithms is paramount. It is critical that algorithms do not replicate and enhance existing societal biases, including those rooted in differences of race, gender, or sexual orientation.

To tackle these problems, both fairness and bias need to be clearly defined. Currently, there does not exist a single universally agreed upon definition of fairness. Anti-discrimination legislation exists in various jurisdictions around the world. In the US, anti-discrimination laws exist under the Civil Rights Act Berg (1964), and under specific areas like credit lending https://www.justice.gov/crt/equal-credit-opportunity-act-3 and housinghttps://www.justice.gov/crt/fair-housing-act-1. There have also been efforts to introduce legislation combating algorithmic biashttps://www.congress.gov/bill/116th-congress/house-bill/2231/all-info. In the European Union, the General Data Protection Regulation (GDPR) provides for regulations regarding digital profiling, data collection, and a \sayright to explanation Goodman and Flaxman (2017). Under Indian law, quotas for scheduled castes, scheduled tribes and other backward classes are mandated in public education and government employment.http://www.legalservicesindia.com/article/1145/Reservations-In-India.html

We begin with the broad definition of fairness as \saythe absence of prejudice or preference for an individual or group based on their characteristics. Bias can also exist in a variety of forms. Mehrabi et al. (2019) provides an excellent overview on the differing types of bias and discrimination. In general, a fair machine learning algorithm is one that does not favor or make prejudice towards an individual or a group.

While most early fairness research focused on binary fairness metrics (protected vs. unprotected groups), newer methods to address fairness have begun to incorporate intersectional frameworks. These frameworks are derived from the third wave of feminist thought, which is rooted in the understanding of the interconnected nature of social categories, like race, gender, sexual orientation, and class Crenshaw (1989). The intersection of these categories creates differing levels of privilege or disadvantage for the various possible subgroups. There exist legal precedents for discrimination under an intersectional lens : The Equal Employment Opportunity Commission (EEOC) describes some Intersectional Discrimination/Harassment examples%TT␣every␣character␣and␣hyphenate␣after␣ithttps://www.eeoc.gov/initiatives/e-race/significant-eeoc-racecolor-casescovering-private-and-federal-sectors#intersectional. Buolamwini and Gebru (2018) examined gender classification algorithms for facial image data and found that they performed substantially better on male faces than female faces. However, the largest performance drops came when both race and gender were considered, with darker skinned women disproportionately affected having a misclassification rate of $\approx$30%.

The example in Figure 1 describes the importance of intersectional fairness. In the figure, we observe equal numbers of black and white people pass. Similarly, there is an equal number of men and women passing. However, this classification is unfair because we don’t have any black women and white men that passed, and all black men and white women passed. We observe the bias only while looking at the subgroups when we take race and gender as protected attributes. This phenomenon was called \sayFairness Gerrymandering by Kearns et al. (2018).

Additionally, there are minorities that have historically faced discrimination around the world, but due to their sparse population, empirical evidence of discrimination against them is difficult to trace, for example, the indigenous population Paradies (2006); King et al. (2009), or trans people Feldman et al. (2016); Reisner et al. (2016); Bockting et al. (2016). This causes machine learning practitioners to either disinclude these groups from their training datasets due to statistical insignificance, or worse, conflate them with other minorities to create a general \sayprotected category, which leads to the same sort of neglected bias as shown in figure 1.

In this paper, we discuss the notion of intersectional group fairness. After introducing existing related work, we define a combinatorial approach giving subsets of the population. With this definition of subgroups, we introduce a measure of the worst case disparity using existing fairness metrics, to discover biases against underserved subgroups. We then show how this method can be applied to classification models, ranking models, and models with continuous output. We end the paper with a discussion about the limitations of our approach and future work.

Related Work

Fair machine learning differentiates group and individual fairness measures. While group fairness metrics focus on treating two different groups equally, individual fairness metrics focus on treating similar individuals similarly. Binns (2019) introduces those two notions and discusses the motivations behind individual and group fairness. In this paper, we focus on group fairness metrics.

2 Binary fairness metrics

A large majority of research in algorithmic fairness has covered fairness metrics for a single protected attribute Corbett-Davies and Goel (2018). Hardt et al. (2016) introduces the definitions of Equalized odds and equal opportunity, two measures for discrimination against a binary sensitive attribute. Verma and Rubin (2018) collected some known binary fairness metrics for classification models and demonstrated each metric with a unique example on the German credit dataset. In their example, the protected class is Gender, which has two values female and male.

3 Intersectional Fairness

More recently, however, some work has begun to address the issue of intersectionality in AI by providing statistical frameworks that control for bias within multiple subgroups. Hébert-Johnson et al. (2018) introduces the idea of multi-calibration which gives meaningful predictions for overlapping subgroups in a larger protected group. Kearns et al. (2018) developed an analogous method named rich subgroup fairness for false positive and negative constraints that hold over an infinitely large collection of subgroups. Kim et al. (2019) extend these methods for classifiers to be equally accurate on a combinatorially large collection of all subgroups. Mary et al. (2019) present the Renyi correlation coefficient as a fairness metric for datasets with continuous protected attributes. Finally, Foulds et al. (2020) introduce differential fairness (DF), as an intersectional fairness metric.

Intersectional group fairness metrics

In this section, we discuss our intersectional fairness metrics framework. We outline our definition of a subgroup of the population, define a worst case disparity metric that we call the min-max ratio and describe how we can operationalize the notion of min-max ratio to encompass intersectionality in existing metrics of fairness.

For the purposes of this paper, similar to Kearns et al. (2018), we define a subgroup $sg_{a_{1}....a_{n}}$ as a set containing the intersection of all members who belong to groups $g_{a_{1}}$ through $g_{a_{n}}$, where $a_{1}$, $a_{2}$…$a_{n}$ are marginal protected attributes, like race, gender, etc. Formally,

Hence, for example, if $g_{1}$(race) $\in$ {black, white} and $g_{2}$(gender) $\in$ {man, woman}, then $sg\in$ {black women, black men, white women, white men} and N = $|sg|=2\times 2=4$.

This combinatorial, or cartesian product of attributes approach gives us subsets of the original dataset, where in each subgroup, the members have all the protected attributes of the groups they were composed of.

2 Worst Case Disparity

We introduce a simple concept to measure the worst case disparity using existing fairness metrics to incorporate intersectionality - the min-max ratio. In the vein of Rawls (2001) principle for distributive justice, the idea essentially is to measure the value of the given fairness metric for every subgroup $sg_{i}$ then take the ratio of the minimum and maximum values from this given list. The further this ratio is from 1, the greater the disparity is between subgroups. This min-max ratio technique allows us to encompass the entire breadth of possible subgroups in a dataset, by considering the worst case scenario in terms of adverse impact. For fairness metrics that are already comparative ratios of two groups, we redefine it by calculating the said ratio for all possible permutations of two subgroups and then simply take the minimum, also the worst possible case.

We discuss some of the most commonly used metrics in the literature below and show how we use the worst possible case framing to incorporate intersectionality.

3 Fair Classification metrics

Several fair classification metrics exist in literature. We discuss four group fairness metrics below from Mehrabi et al. (2019) and Gartner (2020).

According to demographic parity, the proportion of each segment of a protected class should receive positive outcomes at equal rates. Mathematically, demographic parity compares the pass rate (rate of positive outcome) of two groups. Demographic parity is satisfied for a predictor $\hat{Y}$ and for a member $A$ if:

where N is the total number of subgroups. Demographic parity is also known as statistical parity Dwork et al. (2012); Kusner et al. (2017).

Using our worst case, min-max ratio definition, Demographic parity ratio (DPR) would be defined as:

Disparate impact, as defined under the guideline by the Equal Employment Opportunity Commission et al. (1979) is similar to the demographic parity metric. It is intended as a way to measure indirect and unintentional discrimination in which certain decisions disproportionately affect members of a protected group. Disparate impact compares the pass rate of one group versus another. The Four-Fifths rule states that the ratio of the pass rate of group 1 to the pass rate of group 2 has to be greater than 80% (groups 1 and 2 interchangeable). Using our worst case definition, intersectional disparate impact (DI) is defined as the minimum disparate impact between all possible pairs of subgroups $sg$.

3.2 Conditional statistical parity

Conditional statistical parity extends demographic parity by permitting a set of legitimate attributes to affect the outcome Corbett-Davies et al. (2017). Conditional statistical parity is satisfied for a predictor $\hat{Y}$, a member $A$ with a set of legitimate attributes $L$ if:

Using the worst case, min-max ratio definition, Conditional statistical parity ratio (CSPR) would be defined just like equation 3:

3.3 Equal opportunity

Equal opportunity or True Positive Rate Parity states that all members should be treated equally or similarly and not disadvantaged by prejudice or bias. Mathematically, it compares True Positive Rate (TPR) of the classifier between the protected group and the unprotected groupTPR is the probability that a ground truth positive observation is correctly classified as positive. Hardt et al. (2016). Equal opportunity for a binary predictor $\hat{Y}$ and a member $A$, is satisfied if:

Using the worst case, min-max ratio definition, Equal opportunity ratio (EOppR) would be defined as:

In a similar vein, there can exist True Negative Rate Parity, False Positive Rate Parity and False Negative Rate Parity. Hardt et al. (2016) propose Equalized Odds as a method to generalize the Equal Opportunity metric by comparing all these different parities.

3.4 Group Benefit Equality

Group Benefit Equality, introduced by Gartner (2020) aims to be useful in the domain of healthcare. Group benefit equality measures the predicted rate of passing for a subgroup compared to the actual rate of passing. Mathematically, this is defined as:

And, Group benefit ratio for a subgroup is defined as:

Using the worst case, min-max ratio definition, Group benefit ratio (GBR_INT) would be defined intersectionally as:

4 Multi-class classification models

For multiclass classification models, we present a modified version of the Equalized Odds metric, except instead of a binary positive or negative label, we measure the odds ratio for each possible discrete output, and then take the worst odds ratio among all outputs.

For instance, if a multiclass classifier has five possible output classes, we calculate the min-max ratio for each output class $y$, and then take the minimum of those five values as our final metric, since it is the worst possible scenario. Formally, Multiclass Equalized Odds Ratio (M-EOddR) is defined as:

where K is the set of all possible output classes. The closer the value of M-EOddR is to 1, the lower the disparity is of the classifier’s performance among the various subgroups for all possible output classes.

5 Models with continuous output

We can extend the worst possible case framing for models which produce a continuous output, like regression models, or recommendation models that provide relevance scores. The Kullback-Leibler (KL) divergenceHere we use KL Divergence as our base metric, although this method would work for any distribution comparison metric. between two distributions $q$ and $p$ is defined as the following:

In the context of intersectional fairness, we compute the KL divergence between the model output distributions of all possible pairs of two subgroups, and we display the maximum KL divergence value obtained, since it is the worst case scenario. If this value is close to 0, the two subgroups have similar distributions, as well as the other subgroups.

Thus, Worst case KL Divergence (W-$D_{KL}$) is formally defined as:

6 Ranking metrics

Existing fair ranking metrics in the literature can be divided broadly into two classes - representation based Yang and Stoyanovich (2017) and exposure based Singh and Joachims (2018); Sapiezynski et al. (2019). We pick one of each kind and redefine them under the light of intersectionality.

The representation-based metric we discuss is skew@k Geyik et al. (2019). For a ranked list $\tau$, the Skew for subgroup $sg_{i}$ at the top $k$ is defined as

where $p_{\tau^{k},sg_{i}}$ represents the fraction of members from subgroup $sg_{i}$ among the top $k$ items in $\tau$, and $p_{q,g_{i}}$ represents the fraction of members from to subgroup $sg_{i}$ in the overall population $q$. Ideally, $\text{Skew}_{g_{i}}@k$ should be close to one for each $sg_{i}$ and $k$, to show that people from $sg_{i}$ are represented in $\tau$ proportionally relative to the overall population.

Using our worst case method, the skew ratio at K (SR@K) is defined as:

6.2 Attention

Attention is the exposure-based metric we discuss here. Ranking problems are unique from classification problems in the sense that the position of a ranked item, even within the top K results, can draw significantly different levels of visual attention. Previous research shows that people’s attention sharply drops off after the first few items in a ranked list Mullick et al. (2019). Different papers have modeled visual attention as a function of the position K as a logarithmic distribution Singh and Joachims (2018), a geometric distribution, or other sharply falling distributions with increasing rank Sapiezynski et al. (2019). Assuming the attention distribution function of an item to be Att(k), the mean attention per subgroup is defined as:

And, using our worst case method, the attention ratio (AR) is defined as:

Case Study and LSAC Dataset

As an example application of the framework described in this paper, we present a case study on a trained tensorflow modelhttps://github.com/tensorflow/fairness-indicators/blob/master/g3doc/tutorials/Fairness_Indicators_Pandas_Case_Study.ipynb outputs on the Law School Admissions Council (LSAC) datasethttps://eric.ed.gov/?id=ED469370, where the classifier predicts whether a candidate passed the bar exam. We calculated metrics for binary values where only race (Table 1) or only gender (Table 2) is examined for the false negative rates (FNR) and also for an intersectional FNR metric, where both race and gender are used (Table 3).

The tables show clear examples of how viewing fairness metrics for only one group or protected class can obscure inequality for combined subgroups. When using only the gender lens, the FNR is lower for women, $\approx{0.006}$ when compared to men, $\approx{0.017}$ (Table 2). However the trend reverses for certain subgroups when race is added. For example with Black women having higher FNR at $\approx{0.039}$ when compared either to white men at $\approx{0.015}$ or asian men at $\approx{0.020}$ (Table 3). The min/max ratio is 0.002398/0.065327 = 0.036, which is far from the ideal value of 1. The model therefore fails to achieve intersectional fairness under FNR parity.

Discussion

In this paper, we introduce the worst-case comparison as a simple, easily comprehensible method to surface hidden biases that commonly used fairness metrics may not be able to show. We establish the importance of introducing such modifications to better serve minorities with sparse populations and show how the method can be applied to a diverse range of model metrics, thereby being easy for practitioners and researchers to adapt without significantly changing their existing fairness monitoring systems.

2 Limitations and Future Work

The idea of creating combinatorial subgroups has a couple of caveats: It does not take into account partial group membership (for instance, a person who identifies as multiracial), or continuous variables (for example, instead of treating age as an integer, we would convert the age attribute as discrete buckets). We encourage researchers to expand our method to include partial group membership and continuous attributes.

Secondly, creating a combinatorially large number of subgroups inevitably leads to subgroups which have a very small number of members, thereby demonstrating the effects of Simpson’s Paradox Blyth (1972). A possible direction of research could be to introduce statistical significance measures for such small subgroups, and suggest thumb rules for subgroup creation via empirical measurements.

The authors would like to thank Joshua Rubin, Amit Paka, Jack Reidy, Aalok Shanbhag, Christo Wilson, and the anonymous reviewers for their helpful discussions and suggestions.

References