Group-blind optimal transport to group parity and its constrained variants

TL;DR

A single group-blind projection map that aligns the feature distributions of both groups in the source data, achieving (demographic) group parity, without requiring values of the protected attribute for individual samples in the computation of the map, as well as its use.

Abstract

Fairness holds a pivotal role in the realm of machine learning, particularly when it comes to addressing groups categorised by protected attributes, e.g., gender, race. Prevailing algorithms in fair learning predominantly hinge on accessibility or estimations of these protected attributes, at least in the training process. We design a single group-blind projection map that aligns the feature distributions of both groups in the source data, achieving (demographic) group parity, without requiring values of the protected attribute for individual samples in the computation of the map, as well as its use. Instead, our approach utilises the feature distributions of the privileged and unprivileged groups in a boarder population and the essential assumption that the source data are unbiased representation of the population. We present numerical results on synthetic data and real data.

Figures (7)

• Figure 1: Left: A cut-off point (vertical dark line) for an exam score is used to make school-admission decisions. With the hypothetical exam-score distributions of the unprivileged group (orange) and the privileged group (purple), this group-blind cut-off point would result in outcomes biased against the unprivileged group. If the score distributions of unprivileged group and the privileged group are projected closer to an target distribution (green), the same cut-off point can achieve equalised admission rates. Right: The partial repair (top two) and total repair (bottom) schemes proposed in this paper, move the score distributions of unprivileged group (orange) and the privileged group (purple) closer to the pre-defined target distribution, without access to the protected attribute of each sample in training process.
• Figure 2: Overview of the empirical distributions of $P^{X},P^{X_{s_0}},P^{X_{s_1}}$ from the generated source data and the target distribution $P^{\Tilde{X}}$ that we set arbitrarily.
• Figure 3: From left to right: the baseline coupling of Equation \ref{['equ:formulation-baseline']}, the optimal coupling $\gamma^*$ of Equations (\ref{['equ:our-formulation-2']}-\ref{['equ:convex-sets']}), when $\Theta=10^{-2}\mathbb{1},10^{-3}\mathbb{1},\mathbb{0}$. The blue and green curves represent marginal distributions $P^{X},P^{\Tilde{X}}$ that are the same across all couplings.
• Figure 4: Group-blind distributions. Solid green curves are the $S$-blind target distributions $P^{\Tilde{X}}$ used to compute couplings, which are the green curves in Figure \ref{['fig:exa_couplings']}. Dashed green curves (from left to right) are the $S$-blind empirical distributions of $P^{\Tilde{X}}$ computed from projected data from baseline, from $10^{-2}\mathbb{1}$-repair, from $10^{-3}\mathbb{1}$-repair, and from total repair. The overlap between dashed curves and solid curves shows the projected data follow the target distribution we design, and verifies that all couplings in Figure \ref{['fig:exa_couplings']} are feasible and the projection method in Definition \ref{['def:projection']} is correct.
• Figure 5: Group-wise distributions. From left to right: the $S$-wise empirical distributions of $P^{X_{s_0}},P^{X_{s_1}}$ in source data, $P^{\Tilde{X}_{s_0}},P^{\Tilde{X}_{s_1}}$ in projected data from baseline, from $10^{-2}\mathbb{1}$-repair, from $10^{-3}\mathbb{1}$-repair, and from total repair. $P^{\Tilde{X}_{s_0}}$ is plotted orange and $P^{\Tilde{X}_{s_1}}$ is plotted purple.

Theorems & Definitions (31)

• Example 1
• Example 2
• Definition 1: A protected attribute
• Definition 2: Source variables of an unprotected attribute
• Definition 3: Target variables of the unprotected attribute
• Definition 4: Projection
• Example 3
• Example 4
• Definition 5: Total repair
• Lemma 6