Demographic parity in regression and classification within the unawareness framework

TL;DR

This paper investigates the theoretical underpinnings of fair regression under demographic parity constraints within the unawareness framework, linking optimal solutions to barycenter problems and decision set nestedness.

Contribution

It characterizes the optimal fair regression function as a barycenter problem and establishes a connection between fair classification and regression through nested decision sets.

Findings

Optimal fair regression is characterized by a barycenter problem with transport costs.

Nested decision sets are necessary and sufficient for equivalence between classification and regression.

Thresholding the regression function yields optimal classifiers under fairness constraints.

Abstract

This paper explores the theoretical foundations of fair regression under the constraint of demographic parity within the unawareness framework, where disparate treatment is prohibited, extending existing results where such treatment is permitted. Specifically, we aim to characterize the optimal fair regression function when minimizing the quadratic loss. Our results reveal that this function is given by the solution to a barycenter problem with optimal transport costs. Additionally, we study the connection between optimal fair cost-sensitive classification, and optimal fair regression. We demonstrate that nestedness of the decision sets of the classifiers is both necessary and sufficient to establish a form of equivalence between classification and regression. Under this nestedness assumption, the optimal classifiers can be derived by applying thresholds to the optimal fair regression function; conversely, the optimal fair regression function is characterized by the family of cost-sensitive classifiers.