Fair Without Leveling Down: A New Intersectional Fairness Definition (2023)

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.

## introduction the aim of fair machine learning is to develop models that are devoid of discriminatory behavior towards sensitive subgroups of the population. a broad range of approaches have been developed (see e.g., @xcite @xcite , with most of them focusing on a single sensitive axis @xcite such as gender (e.g., male vs. female) or race (e.g., african-americans vs. european-americans). however, recent studies @xcite have demonstrated that even when fairness can be ensured at the level of each individual sensitive axis, significant unfairness can still exist at the intersection levels (e.g., male european-americans vs. female african-americans). for example, @xcite showed that commercially available face recognition tools exhibit significantly higher error rates for darker-skinned females than for lighter-skinned males. similar observations have been made by several studies in nlp including contextual word representation (tan and celis, 2019), and generative models @xcite . these findings resonate with the analytical framework of intersectionality @xcite , which argues that systems of inequality based on various attributes (like gender and race) may "intersect" to create unique effects. to capture these effects in the context of machine learning, several intersectional fairness measures have been proposed @xcite @xcite . amongst them the most commonly used @xcite @xcite is differential fairness (df) @xcite , which is 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 egalitarian" view, i.e., it only considers the relative performance between the group and ignores their absolute performance. in particular, a trivial way to 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 fairness requirements in many practical use-cases @xcite . yet, we empirically observe that (i) popular fairness-promoting approaches tend to level down more in intersectional fairness, and (ii) this often goes unnoticed in the overall performance of the model due to the large number of groups induced by intersectional fairness. to address these issues and explicitly capture the leveling down phenomena, we propose a gener-alization of df, called α-intersectional fairness (if α ), which takes into account both the relative performance between the groups and the absolute performance of the groups. more precisely, if α is a weighted average between the relative and absolute performance of the groups, and allows the exploration of the whole trade-off between these two quantities by changing their relative importance via a weight α ∈ [0, 1]. our extensive benchmarks across various datasets show that many existing fairness-inducing methods aim for a different point in the aforementioned trade-off and generally show no consistent improvement over a simple unconstrained approach. in summary, our primary contributions are as follows: on the other hand, our measure is more careful in showing improvements over a simple baseline than previous metrics, allowing the emphasis on cases of leveling down. ## setting in this section, we begin by introducing our notations 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 definitions for the intersectional setting @xcite . @xcite proposed subgroup-fairness, which is based on the difference in performance of a particular group weighted by the size of the group. several calibration and metric fairness-based variants were considered by @xcite and @xcite . a shortcoming of these notions is that they weight each group by its size, hence small groups may not be protected even though they are often the disadvantaged ones. ## α-intersectional fairness in order to circumvent the above issue and effectively capture intersectional fairness while taking into account the leveling down phenomena, we propose α-intersectional fairness (if α ). our definition is essentially a convex combination of two components, namely (i) ∆ rel , which takes into account the relative performance between the two groups, such as the ratio of their performance, and (ii) ∆ abs , which captures the leveling down effect by accounting for the absolute performance of the worst-off group. more precisely, given a model h θ and a groupwise performance measure m, let us first define a measure of fairness for a pair of groups g and g ′ : now taking the maximum value of i α over all pairs of groups, we get our proposed notion of α-intersectional fairness. note that if α (h θ , m) can be equivalently obtained as the the value of i α over the pair of worst performing and the best performing group, as shown by the following proposition. proposition 1. if a model h θ is (α, γ)intersectionally fair with respect to a group-wise performance measure m, then 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 α. in the interest of space, we delegate our discussion around the design choices for ∆ rel and ∆ abs to appendix a. comparing df and if α . the primary difference between df and if α when comparing two models arises when one model adversely affects the worst-off group (∆ abs ) more than the other, despite having better relative performance (∆ rel ). in this case, df would consistently consider one model more fair than the other, whereas if α enables the exploration of this tension by varying the relative importance of both criteria through α. we formally capture this intuition as follows. consider two models h θ and h θ. let the value of the worst-off and the best-off group's performance for the model h θ be w and b, respectively. similarly, for model h θ let the worst and the best group's performance be w and b, respectively. without the loss of generality, w and b can be written as @xmath0. note that x and y can be either positive or negative as long as @xmath1. we visualize this setup in figure 1 . based on this setup, we have following cases: in figure 1 , this corresponds to @xmath2. the other model has better ∆ rel performance. the fairness in this setting depends on the relative importance of absolute and relative performance for if α , while for df it exclusively depends on absolute performance. in figure 1 , this corresponds to @xmath3. here, ∃α ∈ to summarize, in the first two cases, one model harms the worst-off group (absolute performance), and the relative performance of that model is worse than the other. thus, a good fairness measure should assign a higher unfairness to that model, which both df and if α do. in the third case, one model performs better on the worst-off group, while the other model has a closer relative performance. the fairness in this setting depends on the relative importance of absolute and relative performance. here, df consistently assigns one model a higher fairness than the other, while if α enables to explore this tension and tune the relative importance of both criteria through α. for instance, the previous example in section 2 falls in the third case. on the one hand, df will assign higher ϵ for h θ 1 in comparison to h θ 2 . on the other hand, if α will assign higher γ for h θ 1 for α ∈ (0.0, 0.81), while for all other α, the γ would be higher for h θ 2 . we illustrate the effect of α in more details below. impact of α: the parameter α allows to tune the relative importance of ∆ abs and ∆ rel . on the one end of the spectrum, α = 0 corresponds to considering only the relative performance ∆ rel , while α = 1 corresponds to considering only the absolute performance. at α = 0.0 we recover the same relative ranking of unfairness as df, and thus df can be seen as a special case of if α . in other words, for any three models h θ 1 , h θ 2 , and h θ 3 such that df on the other end, α = 1 only considers the absolute performance ∆ abs ), and α = 0.5 corresponds to giving ∆ abs and ∆ rel an equal importance. in practice, it is useful to visualize the complete trade-off by plotting α → if α (see section 5). ## experiments in this section, we present experiments foot_2 that showcase (i) the model's performance over the worst-off group as the number of sensitive axes increases, and (ii) the "leveling down" phenomenon observed in various fairness-promoting mechanisms, along with the effectiveness of α-intersectional fairness in uncovering it. however, before describing these experiments, we begin with an overview of the datasets, baselines, and fairness measures used. datasets: we benchmark over four datasets covering both text and images, with varying numbers of examples and sensitive groups: the dataset is a collection of 8, 502 free text responses alongside numerical scores over multiple psychometric dimensions. in this work, we focus on two dimensions: both these datasets consists of free text responses and binarized scores by the medical expert. moreover, each response is associated with gender, race, age, and income. we use same pre-processing as @xcite and follow the same procedure to split the dataset as described above. for improved readability, we present a subset of experiments in the main article. the remaining experiments are included in the appendix. ## conclusion we propose a new definition for measuring intersectional fairness of statistical models, in the group classification setting. we provide various comparative analyses of our proposed measure, and contrast it with existing ones. through them, we show that our fairness definition can uncover various notions of harm, including notably, the leveling down phenomenon. we further show that many fairnessinducing methods show no significant improvement over a simple unconstrained approach. through this work, we provide tools to the community to better uncover latent vectors of harm. further, our findings chart a path for developing new fairnessinducing approaches which optimizes for fairness without harming the groups involved.