Demographic Parity Constrained Minimax Optimal Regression under Linear Model

We explore the minimax optimal error associated with a demographic parityconstrained regression problem within the context of a linear model. Our proposed model encompasses a broader range of discriminatory bias sources compared to the model presented by Chzhen and Schreuder [6]. Our analysis reveals that the minimax optimal error for the demographic parity-constrained regression problem under our model is characterized by Θ(dM/n), where ndenotes the sample size, d represents the dimensionality, and M signifies the number of demographic groups arising from sensitive attributes. Moreover, we demonstrate that the minimax error increases in conjunction with a larger bias present in the model.