We study the problem of learning an optimal regression function subject to a fairness constraint. It requires that, conditionally on the sensitive feature, the distribution of the function output remains the same. This constraint naturally extends the notion of demographic parity, often used in classification, to the regression setting. We tackle this problem by leveraging on a proxy-discretized version, for which we derive an explicit expression of the optimal fair predictor. This result naturally suggests a two stage approach, in which we first estimate the (unconstrained) regression function from a set of labeled data and then we recalibrate it with another set of unlabeled data. The recalibration step can be efficiently performed via a smooth optimization. We derive rates of convergence of the proposed estimator to the optimal fair predictor both in terms of the risk and fairness constraint. Finally, we present numerical experiments illustrating that the proposed method is often superior or competitive with state-of-the-art methods.

Summary and Contributions: This paper provides a new algorithm to train a regression function subject to a demographic parity like fairness constraint. The proposed approach constructs a plug-in estimator by first training an unconstrained regression function using labeled data and calibrate the model to satisfy the fairness constraint using unlabeled data. The final model is a "regression function with discrete outputs". The authors show convergence rates to the optimal fair regression model, and demonstrate competitive empirical performance compared to previous approaches for fair regression. I'm still of the opinion that the technical gap I pointed out is an important one, and that the analysis would have been much more complete and satisfying had the guarantees for the optimization algorithm been on the gradients of the dual objective.

We study the problem of learning an optimal regression function subject to a fairness constraint. It requires that, conditionally on the sensitive feature, the distribution of the function output remains the same. This constraint naturally extends the notion of demographic parity, often used in classification, to the regression setting. We tackle this problem by leveraging on a proxy-discretized version, for which we derive an explicit expression of the optimal fair predictor. This result naturally suggests a two stage approach, in which we first estimate the (unconstrained) regression function from a set of labeled data and then we recalibrate it with another set of unlabeled data.