The price for fairness in a regression framework

We consider the problem of achieving fairness in a regression framework. Fairness is here expressed as demographic parity. We provide a control over the loss of the generalization error when fairness constraint is imposed, hence computing the cost for fairness for a regressor. Then, using optimal transport theory, we provide a way to construct a fair regressor which is optimal since it achieves the optimal generalization bound. This regressor is obtained by a post-processing methodology.