We propose a unified framework for fair regression tasks formulated as risk minimization problems subject to a demographic parity constraint. Unlike many existing approaches that are limited to specific loss functions or rely on challenging non-convex optimization, our framework is applicable to a broad spectrum of regression tasks. Examples include linear regression with squared loss, binary classification with cross-entropy loss, quantile regression with pinball loss, and robust regression with Huber loss. We derive a novel characterization of the fair risk minimizer, which yields a computationally efficient estimation procedure for general loss functions. Theoretically, we establish the asymptotic consistency of the proposed estimator and derive its convergence rates under mild assumptions. We illustrate the method's versatility through detailed discussions of several common loss functions. Numerical results demonstrate that our approach effectively minimizes risk while satisfying fairness constraints across various regression settings.

翻译：我们提出了一个用于公平回归任务的统一框架，该框架被表述为在人口统计均等约束下的风险最小化问题。与许多局限于特定损失函数或依赖于具有挑战性的非凸优化的现有方法不同，我们的框架适用于广泛的回归任务。示例包括使用平方损失的线性回归、使用交叉熵损失的二元分类、使用分位数损失的分位数回归以及使用Huber损失的稳健回归。我们推导了公平风险最小化器的一种新颖表征，这为一般损失函数产生了一种计算高效的估计程序。理论上，我们在温和假设下建立了所提出估计量的渐近一致性，并推导了其收敛速率。我们通过对几种常见损失函数的详细讨论，阐明了该方法的普适性。数值结果表明，我们的方法在各种回归设置中能有效最小化风险，同时满足公平性约束。