No-Free-Fairness: Fundamental Limits and Trade-offs in Learning Systems

2026-06-16

AI Summary

This work identifies three fundamental sources limiting fairness in learning systems: inherent task costs, finite sample sizes, and model expressivity constraints, establishing an unavoidable trade-off between fairness and performance. Leveraging statistical learning theory and minimax analysis, the study formally introduces the “No-Free-Fairness” theorem, rigorously proving that subgroup disparities persist even under unbiased data and optimal optimization, and demonstrating that the sample complexity required to achieve relative fairness grows exponentially. The paper further proposes the “fairness–cost frontier,” a universal analytical framework for fairness, underscoring that fairness must be integrated as a core design principle rather than treated as a post-hoc correction.

Abstract

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.

Problem

• fairness
• learning systems
• disparity
• impossibility results
• model expressivity

Innovation

• No-Free-Fairness
• fairness--cost frontier
• finite-sample learning
• model expressivity
• subgroup disparity