用联合分布差异直接衡量公平性，提升效率且不依赖条件分布估计。
A Joint-Distribution Route to Fair Representations with Continuous Sensitive Attributes
- 通过联合分布差异替代条件分布平均，避免复杂平滑估计。
- 基于HSIC的估计器收敛速度达O(n⁻¹/²)，快于传统方法的O(n⁻²/⁵)。
- 算法FRHSIC在公平性与准确率平衡上媲美基线，训练更快。
连续敏感属性下的公平表示学习要求表示变量 $Z$ 与敏感属性 $S$ 统计独立。现有方法（如广义人口均等性、期望积分概率度量EIPM、互信息）通过在 $S$ 的分布上平均每个敏感值处条件分布 $P_{Z ext{∣} S=s}$ 与边缘分布 $P_Z$ 的差异来实现独立性，需对每个 $s$ 进行非参数条件分布估计。本文提出直接评估联合分布 $P_{Z,S}$ 与边缘乘积 $P_Z \otimes P_S$ 之间的单一联合差异 $d(P_{Z,S}, P_Z \otimes P_S)$。在可分解见证类下，该差异等于EIPM和广义人口均等性所对应的条件积分泛函。由于无需条件分布，该差异可直接通过样本依赖统计量估计。以希尔伯特-施密特独立性准则（HSIC）为例，其为闭式 $O(n^2)$ 统计量，收敛速率 $O(n^{-1/2})$，优于传统条件路线的 $O(n^{-2/5})$。我们证明该实例与条件最大均值差异（MMD）积分仅差一个显式的谱尾项。对应的算法FRHSIC在公平性-准确性权衡上达到基线水平，同时显著降低每轮训练时间。
原文摘要 · Abstract (English)
Fair representation learning with a continuous sensitive attribute $S$ requires a representation $Z$ that is statistically independent of $S$. Existing criteria, including generalized demographic parity, the expectation of integral probability metrics (EIPM), and mutual information, enforce this independence by averaging a per-value discrepancy between the conditional law $P_{Z \mid S=s}$ and the marginal $P_Z$ over the law of $S$. This approach requires a nonparametric surrogate for the conditional law at each sensitive value. We propose evaluating independence through a single joint discrepancy $d\left(P_{Z, S}, P_Z \otimes P_S\right)$ between the joint law and the product of its marginals. We establish a disintegration identity; on decomposable witness classes it equals the conditional-integral functional that EIPM and generalized demographic parity instantiate. By reaching the same target without the conditional law, this discrepancy can be estimated directly from samples via a dependence statistic rather than conditional smoothing. We take the Hilbert-Schmidt independence criterion (HSIC) as an instance of the joint discrepancy $d$ to investigate the statistical efficiency of replacing the conditional formulation. The HSIC estimator is a closed-form $O\left(n^2\right)$ statistic that converges at the $O\left(n^{-1 / 2}\right)$ rate, in contrast to the nonparametric $O\left(n^{-2 / 5}\right)$ rate of the conditional-route estimators. We prove this instance is equivalent to the conditional maximum mean discrepancy (MMD) integral up to an explicit spectral tail. The corresponding algorithmic implementation, i.e., FRHSIC, attains fairness-accuracy tradeoffs comparable to conditional-route basel es while reducing per-epoch training time.
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