这篇论文用数学证明了公平和准确不可兼得，小样本会让偏见更严重，模型表达能力不够也不行。做AI公平的人都会引用它。
该论文给出No-Free-Fairness定理，指出学习系统中不公平的三个根本来源。首先，当任务在子组上存在不可约成本时，任何决策规则必须在整体性能与子组差异间权衡，形成固有的公平-成本边界。其次，即使在理想无噪声场景下存在完全公平且准确的解，有限样本学习也会导致子组间非平凡差异，且强制执行严格相对公平可能使达到低成本所需样本数呈指数增长。最后，模型类限制可独立引发差异：若模型无法表示某子组的准确解，则公平性无法通过数据或训练实现。
No-Free-Fairness: Fundamental Limits and Trade-offs in Learning Systems
In this paper, we establish a set of theoretical impossibility results, termed the No-Free-Fairness theorems, that identify three fundamental sources of disparity in learning systems. First, we show that when a task exhibits irreducible cost on a subgroup, any decision rule must trade off overall performance with disparity, yielding an inherent fairness--cost frontier. Second, we prove that even in ideal, noise-free settings where a perfectly fair and accurate solution exists, finite-sample learning alone induces nontrivial subgroup disparity, ruling out distribution-free fairness guarantees. More seriously, enforcing strict relative fairness creates a statistical bottleneck: achieving low cost may require exponentially many samples. Third, we show that limitations of the model class can independently induce disparity: if the model cannot represent accurate solutions for a subgroup, fairness remains unattainable regardless of data or training procedure. Overall, these results demonstrate that unfairness is not solely a consequence of biased data or suboptimal optimization, but arises from the intrinsic structure of decision problems, the constraints of finite data, and the expressivity of models. Our framework applies broadly beyond standard supervised learning, and suggests that achieving fairness requires explicit trade-offs and should be treated as a core design consideration.