A Sequentially Fair Mechanism for Multiple Sensitive Attributes

TL;DR

This work proposes a sequential framework, which allows to progressively achieve fairness across a set of sensitive features, by leveraging multi-marginal Wasserstein barycenters, which extends the standard notion of Strong Demographic Parity to the case with multiple sensitive characteristics.

Abstract

In the standard use case of Algorithmic Fairness, the goal is to eliminate the relationship between a sensitive variable and a corresponding score. Throughout recent years, the scientific community has developed a host of definitions and tools to solve this task, which work well in many practical applications. However, the applicability and effectivity of these tools and definitions becomes less straightfoward in the case of multiple sensitive attributes. To tackle this issue, we propose a sequential framework, which allows to progressively achieve fairness across a set of sensitive features. We accomplish this by leveraging multi-marginal Wasserstein barycenters, which extends the standard notion of Strong Demographic Parity to the case with multiple sensitive characteristics. This method also provides a closed-form solution for the optimal, sequentially fair predictor, permitting a clear interpretation of inter-sensitive feature correlations. Our approach seamlessly extends to approximate fairness, enveloping a framework accommodating the trade-off between risk and unfairness. This extension permits a targeted prioritization of fairness improvements for a specific attribute within a set of sensitive attributes, allowing for a case specific adaptation. A data-driven estimation procedure for the derived solution is developed, and comprehensive numerical experiments are conducted on both synthetic and real datasets. Our empirical findings decisively underscore the practical efficacy of our post-processing approach in fostering fair decision-making.

Figures (6)

• Figure 1: Synthetic data with $\boldsymbol{\tau} = (0, 0.05, 0.1)$. A sequential unfairness evaluation, $\mathcal{U}_3$, of (left pane) exact fairness, (middle) approximate $A_{1:3}$-fairness with $\boldsymbol{\varepsilon}$-RI where $\boldsymbol{\varepsilon} = \varepsilon_{1, 2, 3} = (0.2, 0.5, 0.75)$ and (right) approximate $A_{2, 1, 3}$-fairness with $\varepsilon_{2, 1, 3}$-RI.
• Figure 2: (Risk, Unfairness) phase diagrams that shows the sequential fairness approach for (left) two and (right) three sensitive features. In this study, Unfairness represents the overall unfairness $\Hat{\mathcal{U}} =\Hat{\mathcal{U}}_{1:3}$. Bottom-left corner gives the best trade-off.
• Figure 3: Applications on the folktables data set. Left pane, visualisation of the combined unfairness across two sensitive attributes and intermediate solutions rendering predictions fair on only one of them. Center pane, marginal changes to predicted income when rendering fair the predictions w.r.t. a single variable and the baseline predictions. Right pane, visualization of global metrics when correcting the score first for race, but keeping the average predicted salary of female individuals constant.
• Figure 4: Synthetic data with parameter $\tau = (0, 0.05, 0.1)$. A sequential unfairness evaluation, $\mathcal{U} = \mathcal{U}_{1:3}$, of (left pane) exact fairness, (middle pane) approximate $A_{1, 2, 3}$-fairness with $\boldsymbol{\varepsilon}$-RI where $\boldsymbol{\varepsilon} = \varepsilon_{1, 2, 3} = (0.2, 0.5, 0.75)$ and (right pane) approximate $A_{2, 1, 3}$-fairness with $\varepsilon_{2, 1, 3}$-RI. Hashed color corresponds to exact fairness.
• Figure 5: Synthetic data with parameters $\boldsymbol{\tau} = (0, 0.05, 0.1)$: sequential evaluation of exact fairness.

Theorems & Definitions (7)

• Proposition 1: Associativity of the $\mathcal{W}_2$-barycenter
• Definition 3: Fairness under Demographic Parity
• Proposition 4: Fair characterization: global approach
• Proposition 5: Sequentially fair mechanism
• Proposition 6: Characterization of approximate fairness
• Lemma 7
• Lemma 8