Fairness Through Matching

TL;DR

A new group fairness measure termed Matched Demographic Parity (MDP), which quantifies the averaged gap between predictions of two individuals matched by a given transport map, and proves that any transport map can be used in MDP to learn group-fair models.

Abstract

Group fairness requires that different protected groups, characterized by a given sensitive attribute, receive equal outcomes overall. Typically, the level of group fairness is measured by the statistical gap between predictions from different protected groups. In this study, we reveal an implicit property of existing group fairness measures, which provides an insight into how the group-fair models behave. Then, we develop a new group-fair constraint based on this implicit property to learn group-fair models. To do so, we first introduce a notable theoretical observation: every group-fair model has an implicitly corresponding transport map between the input spaces of each protected group. Based on this observation, we introduce a new group fairness measure termed Matched Demographic Parity (MDP), which quantifies the averaged gap between predictions of two individuals (from different protected groups) matched by a given transport map. Then, we prove that any transport map can be used in MDP to learn group-fair models, and develop a novel algorithm called Fairness Through Matching (FTM), which learns a group-fair model using MDP constraint with an user-specified transport map. We specifically propose two favorable types of transport maps for MDP, based on the optimal transport theory, and discuss their advantages. Experiments reveal that FTM successfully trains group-fair models with certain desirable properties by choosing the transport map accordingly.

Figures (8)

• Figure 1: Simplified illustration of MDP. Once two individuals A and B are matched ($\leftrightarrow$), a model treats the pair of matched individuals A and B similarly (as well as all the other pairs). This implicit mechanism contributes to making the model fair.
• Figure 2: An example DAG of the SCM in equation (\ref{['eq:sem']}).
• Figure 3: Fairness-prediction trade-offs: $\Delta \textup{WDP}$ vs. Acc on (Left to right) Adult, German, Dutch, Bank datasets. Similar results are observed for $\Delta \textup{DP}$ and $\Delta \overline{\textup{DP}}$ in Figure \ref{['fig:tradeoff-appen']} in Section \ref{['sec:results-appendix']} of Appendix.
• Figure 4: Fairness on random subsets: Boxplots of the levels of $\Delta \overline{\textup{DP}}$ on 1,000 randomly generated subsets $\mathcal{D}_{\textup{sub}}$ of test datasets. (Left to right) Adult, German, Dutch and Bank. The values presented under the algorithm name (e.g., 0.0151 for FTM in German) are the standard deviations.
• Figure 5: Fairness-prediction trade-offs: $\Delta \textup{DP}$ vs. Acc on (Left to right) Adult, German, Dutch, Bank datasets. FTM (joint) = FTM with the joint OT map. FTM (marginal) = FTM with the marginal OT map.

Theorems & Definitions (20)

• Proposition 3.1: Fair model $\Rightarrow$ Transport map: perfect fairness case
• Definition 3.2: Matched Demographic Parity
• Theorem 3.3: Fair model $\Rightarrow$ Transport map: relaxed fairness case
• Definition 3.4: Fair matching function of $f$
• Remark 3.6: Usage of the transport cost of the fair matching function
• Theorem 3.7: Transport map $\Rightarrow$ Group-fair model
• Definition 4.1: Subset fairness
• Theorem 4.2: Low transport cost benefits subset fairness
• Proposition 4.3: Counterfactual fairness and the marginal OT map
• proof : Proof of Proposition \ref{['prop:perfect']}