论文标题
一个最小值框架，用于量化回归中风险的权衡
A minimax framework for quantifying risk-fairness trade-off in regression
论文作者
论文摘要
我们为学习满足公平要求的实现功能的问题提出了一个理论框架。该框架建立在$α$偏爱（公平性）改进回归函数的概念之上，我们使用最佳运输理论引入了回归函数。设置$α= 0 $对应于人口统计奇偶校验约束下的回归问题，而$α= 1 $对应于经典的回归问题而没有任何约束。对于$α\ in（0，1）$，所提出的框架允许在这两个极端情况下连续插入并研究部分公平的预测因子。在此框架内，我们精确地量化了通过引入公平限制引起的风险成本。我们提出了一个统计最小值设置，并根据任何满足$α$偏移的改进约束的估计器的风险获得了一般问题依赖性下限。我们以高斯设计和系统的群体依赖性偏差为线性回归模型说明了我们的框架，在引入约束下，在Minimax风险上得出了匹配（至绝对常数）上限和下限。我们提供了一般的后处理策略，该策略享有公平性，风险保证，并且可以在任何黑盒算法上应用。最后，我们对基准数据的线性模型和数值实验进行了模拟研究，从而验证了我们的理论贡献。
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 $α$-relative (fairness) improvement of the regression function which we introduce using the theory of optimal transport. Setting $α= 0$ corresponds to the regression problem under the Demographic Parity constraint, while $α= 1$ corresponds to the classical regression problem without any constraints. For $α\in (0, 1)$ 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 $α$-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.
