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.

翻译：本文建立了一系列称为“无免费公平定理”的理论不可能性结果，这些定理揭示了学习系统中的三种根本性差异来源。首先，我们证明当任务在某个子群上存在不可约成本时，任何决策规则都必须在整体性能与差异之间进行权衡，从而产生固有的公平-成本边界。其次，我们证明即使在理想的无噪声环境下（存在完全公平且精确的解），仅凭有限样本学习就能诱发非平凡的群体差异，从而排除了无分布假设的公平性保证。更严重的是，强制实施严格的相对公平性会形成统计瓶颈：实现低成本可能需要指数级的样本数量。第三，我们证明模型类别的局限性可独立引发差异：若模型无法为某个子群表示精确解，则无论数据或训练过程如何，公平性均无法实现。总体而言，这些结果表明不公平性并非仅由有偏数据或次优优化导致，而是源于决策问题的内在结构、有限数据的约束以及模型的表达能力。我们的框架广泛适用于标准监督学习之外的情形，并表明实现公平性需要明确的权衡，应将其视为核心设计考量。