We present a theoretical framework analyzing the relationship between data distributions and fairness guarantees in equitable deep learning. We establish novel bounds that account for distribution heterogeneity across demographic groups, deriving fairness error and convergence rate bounds that characterize how distributional differences affect the fairness-accuracy trade-off. Extensive experiments across diverse modalities, including FairVision, CheXpert, HAM10000, FairFace, ACS Income, and CivilComments-WILDS, validate our theoretical findings, demonstrating that feature distribution differences across demographic groups significantly impact model fairness, with disparities particularly pronounced in racial categories. Motivated by these insights, we propose Fairness-Aware Regularization (FAR), a practical training objective that minimizes inter-group discrepancies in feature centroids and covariances. FAR consistently improves overall AUC, ES-AUC, and subgroup performance across all datasets. Our work advances the theoretical understanding of fairness in AI systems and provides a foundation for developing more equitable algorithms. The code for analysis is publicly available at https://github.com/Harvard-AI-and-Robotics-Lab/FairnessGuarantee.

翻译：本文提出一个理论框架，分析公平深度学习中的数据分布与公平性保证之间的关系。我们建立了考虑不同人口统计群体间分布异质性的新颖界，推导出公平性误差与收敛速率界，刻画了分布差异如何影响公平性与准确性的权衡。通过在多种模态数据集（包括FairVision、CheXpert、HAM10000、FairFace、ACS Income和CivilComments-WILDS）上的大量实验，验证了我们的理论发现，证明人口统计群体间的特征分布差异显著影响模型公平性，其中种族类别间的差异尤为突出。基于这些洞见，我们提出公平性感知正则化（FAR），这是一种通过最小化群体间特征质心与协方差差异的实用训练目标。FAR在所有数据集上持续提升了整体AUC、ES-AUC及子群体性能。我们的工作推进了对AI系统公平性的理论理解，并为开发更公平的算法奠定了基础。分析代码公开于https://github.com/Harvard-AI-and-Robotics-Lab/FairnessGuarantee。