  We propose a theoretical framework for the problem of learning a real-valued
function which meets fairness requirements. This framework is built upon the
notion of $\alpha$-relative (fairness) improvement of the regression function
which we introduce using the theory of optimal transport. Setting $\alpha = 0$
corresponds to the regression problem under the Demographic Parity constraint,
while $\alpha = 1$ corresponds to the classical regression problem without any
constraints. For $\alpha \in (0, 1)$ the proposed framework allows to
continuously interpolate between these two extreme cases and to study partially
fair predictors. Within this framework we precisely quantify the cost in risk
induced by the introduction of the fairness 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, deriving matching (up to absolute
constants) upper and lower bounds on the minimax risk under the introduced
constraint. We provide a general post-processing strategy which enjoys
fairness, risk guarantees and can be applied on top of any black-box algorithm.
Finally, we perform a simulation study of the linear model and numerical
experiments of benchmark data, validating our theoretical contributions.
