A minimax framework for quantifying risk-fairness trade-off in regression

We propose a theoretical framework for the problem of learning a real-valued function which meets fairness requirements. Leveraging the theory of optimal transport, we introduce a notion of $\alpha$-relative (fairness) improvement of the regression function. With $\alpha = 0$ we recover an optimal prediction under Demographic Parity constraint and with $\alpha = 1$ we recover the regression function. For $\alpha \in (0, 1)$ the proposed framework allows to continuously interpolate between the two. Within this framework we precisely quantify the cost in risk induced by the introduction of the $\alpha$-relative improvement constraint. We put forward a statistical minimax setup and derive a general problem-dependent lower bound on the risk of any estimator satisfying $\alpha$-relative improvement constraint. We illustrate our framework on a model of linear regression with Gaussian design and systematic group-dependent bias. Finally, we perform a simulation study of the latter setup.