Within-group fairness: A guidance for more sound between-group fairness

For a given score function $f$ and a sensitive group $Z=z$, we consider the group performance function of $f$ given as

$$q_{z}(f):=\mathbb{E}(\mathcal{E}|\mathcal{E}^{\prime},Z=z)$$

(1)

for events $\mathcal{E}$ and $\mathcal{E}^{\prime}$ that might depend on $f(\mathbf{X})$ and $Y.$ The group performance function $q_{z}$ in (1), which is considered by [7], includes various performance functions used in fairness AI. We summarize representative group performance functions having the form of (1) in Table 1.

For given group performance functions $q_{z}(\cdot),z\in\{0,1\},$ we say that $f$ satisfies the BGF constraint with respect to $q_{z}$ if $q_{0}(f)=q_{1}(f).$ A relaxed version of the BGF constraint so called the $\epsilon$-BGF constraint, is frequently considered, which requires $|q_{0}(f)-q_{1}(f)|<\epsilon$ for a given $\epsilon>0.$ Typically, AI algorithms search an optimal function $f$ among those satisfying the $\epsilon$-BGF constraint with respect to given group performance functions $q_{z}(\cdot),z\in\{0,1\}.$

Several learning algorithms have been proposed to find an accurate model $f$ satisfying a given BGF constraint, which are categorized into three groups. In this subsection, we review some methods for each group.

Pre-processing methods: Pre-processing methods remove bias in training data or find a fair representation with respect to sensitive variables before the training phase and learn AI models based on de-biased data or fair representation [11, 12, 13, 14, 15, 16, 17, 18, 19]. [11] suggested pre-processing methods to eliminate bias in training data by use of label changing, reweighing and sampling. Based on the idea that transformed data should not be able to predict the sensitive variable, [13] proposed a transformation of input variables for eliminating the disparate impact. To find a fair representation, [12, 14] proposed a data transformation mapping for preserving accuracy and alleviating discrimination simultaneously. Pre-processing methods for fair learning on text data were studied by [15, 16].

In-processing methods: In-processing methods generally train an AI model by minimizing a given cost function (e.g. the cross-entropy, the sum of squared residuals, the empirical AUC etc.) subject to a $\epsilon$-BGF constraint. Most group performance functions $q_{z}(\cdot)$ are not differentiable, and thus various surrogated group performance functions and corresponding $\epsilon$-BGF constraints have been proposed [20, 21, 22, 23, 6, 24, 25, 26, 7, 27, 28]. [20] used a fairness regularizer which is an approximation of the mutual information between the sensitive variable and the target variable. [23, 6] proposed covariance-type fairness constraints as tractable proxies targeting the disparate impact and the equality of the false positive or negative rate, and [24] used a linear surrogated group performance function for the equalized odds. On the other hand, [25, 7] derived an optimal classifier for a constrained fair classification as a form of an instance-dependent threshold. Also, for fair score functions, [27] proposed fairness constraints based on ROC curves of each sensitive group.

Post-processing methods: Post-processing methods first learn an AI model without any BGF constraint and then transform the decision boundary or score function of the trained AI model for each sensitive group to satisfy given BGF criteria [29, 30, 9, 31, 32, 33, 34, 35]. [9, 33] suggested finding sensitive group dependent thresholds to get a fair classifier with respect to equal opportunity. [34, 35] developed an algorithm to transform the original score function to achieve a BGF constraint.

Conceptually, WGF means that the classifier $C_{f}$ and $C^{\star}$ have the same ranks in each sensitive group. That is, for two individuals $\mathbf{x}_{A}$ and $\mathbf{x}_{B}$ in a same sensitive group with $C^{\star}(\mathbf{x}_{A})>C^{\star}(\mathbf{x}_{B}),$ WGF requires that $C_{f}(\mathbf{x}_{A})\geq C_{f}(\mathbf{x}_{B}).$ To materialize this concept of WGF, we define the WGF constraint as

$$\begin{split}&\Pr\left\{C^{\star}(\mathbf{X})=0,C_{f}(\mathbf{X})=1|Z=z\right\}=0\\ &\mbox{or }\Pr\left\{C^{\star}(\mathbf{X})=1,C_{f}(\mathbf{X})=0|Z=z\right\}=0\end{split}$$

(2)

for each $z\in\{0,1\}.$ Similar to the BGF, we relax the constraint (2) by requiring that either of the two probabilities is small. That is, we say that $f$ satisfies the $\delta$-WGF constraint for a given $\delta>0$ if

$$\max_{z\in\{0,1\}}\min\{a_{01|z}(f),a_{10|z}(f)\}<\delta,$$

(3)

where $a_{ij|z}(f)=\Pr\{C^{\star}(X)=i,C_{f}(X)=j|Z=z\}.$

Many BGF constraints have their own implicit directions toward which the classifier is expected to be guided in the training phase. We can design a special WGF constraint reflecting the implicit direction of a given BGF constraint which results in more desirable classifiers (better guided, more fair and frequently more accurate). Below, we present two such WGF constraints.

Disparate impact: Note that the disparate impact requires that

$$\Pr\{C_{f}(\mathbf{X})=1|Z=0\}=\Pr\{C_{f}(\mathbf{X})=1|Z=1\}.$$

Suppose that $\Pr\{C^{\star}(\mathbf{X})=1|Z=0\}<\Pr\{C^{\star}(\mathbf{X})=1|Z=1\}.$ Then, we expect that a desirable classifier $C_{f}$ achieves this BGF constraint by increasing $\Pr\{C_{f}(\mathbf{X})=1|Z=0\}$ from $\Pr\{C^{\star}(\mathbf{X})=1|Z=0\}$ and decreasing $\Pr\{C_{f}(\mathbf{X})=1|Z=1\}$ from $\Pr\{C^{\star}(\mathbf{X})=1|Z=1\}.$ To reflect this direction, we can enforce a learning algorithm to search a classifier $C_{f}$ satisfying $\Pr\{C^{\star}(\mathbf{X})=1|Z=0\}<\Pr\{C_{f}(\mathbf{X})=1|Z=0\}$ and $\Pr\{C^{\star}(\mathbf{X})=1|Z=1\}>\Pr\{C_{f}(\mathbf{X})=1|Z=1\}.$ Based on this argument, we define the directional $\delta$-WGF constraint for the disparate impact as

$$\max\{a_{10|0}(f),a_{01|1}(f)\}<\delta.$$

(4)

Equal opportunity: The equal opportunity constraint is given as

$$\Pr\{C_{f}(\mathbf{X})=1|Z=0,Y=1\}=\Pr\{C_{f}(\mathbf{X})=1|Z=1,Y=1\}.$$

Suppose that $\Pr\{C^{\star}(\mathbf{X})=1|Z=0,Y=1\}<\Pr\{C^{\star}(\mathbf{X})=1|Z=1,Y=1\}.$ A similar argument for the disparate impact leads us to define the directional $\delta$-WGF constraint for the equal opportunity as

$$\max\{a_{10|01}(f),a_{01|11}(f)\}<\delta$$

(5)

and

$$\max_{z\in\{0,1\}}\min\left\{a_{10|z0}(f),a_{01|z0}(f)\right\}<\delta,$$

(6)

where

$$a_{ij|zy}(f)=\Pr\{C^{\star}(X)=i,C_{f}(X)=j|Z=z,Y=y\}.$$

We say that $f$ satisfies the $(\epsilon,\delta)$-doubly-group fairness constraint if $B(f)<\epsilon$ and $W(f)<\delta,$ where $B$ is a given BGF constraint and $W$ is the corresponding WGF constraint proposed in the previous two subsections. In this section, we propose a relaxed version of $W(\cdot)$ for easy computation. As we review in Section 2, many relaxed versions of $B(\cdot)$ have been proposed already.

The WGF constraints considered in Sections 3.1 and 3.2 are hard to be used as themselves in the training phase since they are neither convex nor continuous. A standard approach to resolve this problem is to use a convex surrogated function. For example, a surrogated version of the WGF constraint (3) is $W_{\text{surr}}(f)<\delta,$ where

$$\begin{split}W_{\text{surr}}(f):=\max_{z\in\{0,1\}}\min\Big{\{}&\mathbb{E}\left\{\phi(-f(\mathbf{X}))|Z=z,Y^{\star}=1\right\}p_{1|z},\\ &\mathbb{E}\left\{\phi(f(\mathbf{X}))|Z=z,Y^{\star}=0\right\}p_{0|z}\Big{\}},\end{split}$$

(7)

where $Y^{\star}=C^{\star}(\mathbf{X}),p_{y|z}=\Pr(C^{\star}(\mathbf{X})=y|Z=z)$ and $\phi$ is a convex surrogated function of the indicator function $\mathds{1}(z\geq 0).$ In this paper, we use the hinge function given as $\phi_{\text{hinge}}(z)=(1+z)_{+}$ as a convex surrogated function which is popularly used for fair AI [21, 24, 36]. The surrogated versions for the other WGF constraints are derived similarly. Finally, we estimate $f$ by $\hat{f}$ that minimizes the regularized cost function

$$\mathcal{L}(f)+\lambda B_{\text{surr}}(f)+\eta W_{\text{surr}}(f),$$

(8)

where $\mathcal{L}$ is a given cost function (e.g. the cross-entropy) and $B_{\text{surr}}$ and $W_{\text{surr}}$ are the surrogated constraints of $B$ and $W,$ respectively. The nonnegative constants $\lambda$ and $\eta$ are regularization parameters which are selected so that $\hat{f}$ satisfies $B(\hat{f})<\epsilon$ and $W(\hat{f})<\delta.$

There are several fairness concepts which are somehow related to WGF. However, the existing concepts are quite different from our WGF.

1. 1.

   Unified fairness: [37] used the term ‘within-group fairness’. However, WGF of [37] is different from our WGF. [37] measured individual-level benefits of a given prediction model and they defined the model to be WGF if the individual benefits in each group are similar. They also illustrated that WGF keeps decreasing as BGF increases. Our WGF is nothing to do with individual-level benefits. Our WGF can be high even when individual-level benefits are not similar. Also, our WGF can increase even when BGF increases.

2. 2.

   Slack consistency: [38] proposed the ‘slack consistency’ which requires that the estimated scores of each individual should be monotonic with respect to slack variables used in fairness constraints. Slack consistency does not guarantee within-group fairness because the ranks of the estimated scores can change even when they move monotonically.

We consider following group performance functions for the BGF: the disparate impact (DI) [8] and the disparate mistreatment w.r.t. error rate [6], which are defined as

$$\begin{split}\text{DI}(f)&=\left|\Pr(C_{f}(\mathbf{X})=1|Z=1)-\Pr(C_{f}(\mathbf{X})=1|Z=0)\right|\\ \text{ME}(f)&=\left|\Pr(C_{f}(\mathbf{X})\neq Y|Z=0)-\Pr(C_{f}(\mathbf{X})\neq Y|Z=1)\right|.\end{split}$$

Note that the DI is directional while the ME is not. For the surrogated BGF constraints, we replace the indicator function with the hinge function in calculating the BGF constraints as is done by [21, 36]. We name the corresponding BGF constraints by Hinge-DI and Hinge-ME respectively. The results for other surrogated constraints such as the covariance type constraints proposed by [23, 6] and the linear surrogated functions considered in [42] are presented in the Supplementary material. In addition, the results for the equal opportunity constraint are summarized in the Supplementary material.

For investigating the impacts of WGF on trained classifiers, we first fix the $\epsilon$ for each BGF constraint, and we choose the regularization parameters $\lambda$ and $\eta$ to make the classifier $\hat{f}$ minimizing the regularized cost function (8) satisfy the $\epsilon$-BGF constraint. Then, we assess the prediction accuracy and the degree of WGF of $\hat{f}.$

In this section, we examine the WGF constraint for score functions. We choose the logistic loss (binary cross-entropy, BCE) and AUC (area under the ROC) as evaluation metrics for prediction accuracy. For the BGF, we consider the mean score parity (MSP, [10]):

$$\text{MSP}(f)=\left|\mathbb{E}(\sigma(f(X))|Z=1)-\mathbb{E}(\sigma(f(X))|Z=0)\right|,$$

where $\sigma:x\mapsto 1/(1+e^{-x})$ is the sigmoid function. To check how much the estimated score function $\hat{f}$ is within-group fair, we calculate the Kendall’s $\tau$ between $\hat{f}$ and the ground-truth score function $f^{\star}$ on the test data for each sensitive group, and then we average them, which is denoted by $\bar{\tau}$ in Table 5. We choose the regularization parameters $\lambda$ and $\eta$ such that $\bar{\tau}$ of $\hat{f}$ is as close to 1 as possible while maintaining the MSP value around 0.03.

Table 5 amply shows that the DF score function always improves the degree of WGF (measured by $\bar{\tau}$) and the accuracy in terms of AUC simultaneously while keeping the degree of BGF at a reasonable level. With respect to the BCE, the BGF and DF score functions are similar. The superiority of the DF score function in terms of AUC compared with the BGF score function is partly because the WGF constraint shrinks the estimated score toward the ground-truth score (Uncons. in Table 5) which is expected to be most accurate. Based on these results, we conclude that the WGF constraint is a useful guide to find a better score function with respect to AUC as well as the WGF.

Basically, pre-processing methods transform the training data in a certain way to be between-group fair and train an AI model on the transformed data. To reflect the WGF, it suffices to add a WGF constraint in the training phase. Let $\mathcal{D}_{\text{trans}}$ be the transformed training data to be between-group fair and let $\mathcal{L}_{\text{trans}}$ be the corresponding cost function. Then, we learn a model by minimizing $L_{\text{trans}}(f)+\eta W_{\text{conv}}(f)$ for $\eta>0.$

Table 6 presents the results of the models trained on the pre-processing training data and a WGF constraint for various values of $\eta,$ where the DI is used as the BGF and thus the corresponding dWGF constraint is used. In this experiment, we use the linear logistic model and the Massaging [11] for the pre-processing. Surprisingly we observed that introducing the dWGF constraint to the pre-processing method helps to improve the BGF and WGF simultaneously without sacrificing the accuracies much.

For the BGF score functions, [35] developed an algorithm to obtain two monotonically nondecreasing transformations $m_{z},z\in\{0,1\}$ such that $m_{0}\circ f^{\star}$ and $m_{1}\circ f^{\star}$ are BGF in the sense that the distributions of $m_{0}\circ f^{\star}(\mathbf{X})|Z=0$ and $m_{1}\circ f^{\star}(\mathbf{X})|Z=1$ are the same. It is easy to check that the transformed score function $m_{z}\circ f^{\star}(\mathbf{x})$ is a perfectly WGF score function even though it depends on the sensitivity group variable $z.$ Note that the algorithm in Section 4 yields score functions not depending on $z.$