This paper introduces a novel kernel-based formulation of the Equalized Odds (EO) criterion, denoted as $\operatorname{EO}_k$, for fair representation learning (FRL) in supervised settings. The central goal of FRL is to mitigate discrimination regarding a sensitive attribute $S$ while preserving prediction accuracy for the target variable $Y$. Our proposed criterion enables a rigorous and interpretable quantification of three core fairness objectives: independence ($\widehat{Y} \perp S$), separation--also known as equalized odds ($\widehat{Y} \perp S \mid Y$), and calibration ($Y \perp S \mid \widehat{Y}$). Under both unbiased ($Y \perp S$) and biased ($Y \not \perp S$) conditions, we show that $\operatorname{EO}_k$ satisfies both independence and separation in the former, and uniquely preserves predictive accuracy while lower bounding independence and calibration in the latter, thereby offering a unified analytical characterization of the tradeoffs among these fairness criteria. We further define the empirical counterpart, $\widehat{\operatorname{EO}}_k$, a kernel-based statistic that can be computed in quadratic time, with linear-time approximations also available. A concentration inequality for $\widehat{\operatorname{EO}}_k$ is derived, providing performance guarantees and error bounds, which serve as practical certificates of fairness compliance. While our focus is on theoretical development, the results lay essential groundwork for principled and provably fair algorithmic design in future empirical studies.

Unfortunately, the notion is seriously flawed on two counts [8]. First, it doesn't ensure fairness.

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Machine learning has provided a universal toolbox enhancing the decision making in many disciplines from advertising and recommender systems toeducation andcriminal justice.

The impossibility theorem of fairness is a foundational result in the algorithmic fairness literature. It states that outside of special cases, one cannot exactly and simultaneously satisfy all three common and intuitive definitions of fairness demographic parity, equalized odds, and predictive rate parity. This result has driven most works to focus on solutions for one or two of the metrics.

Figure 1 shows the change of accuracy under different cutoff value. However, for gender classification under CelebA dataset, thetrade-offbetweenλval and accuracyisnotveryclear;and wesuspect that under suchscenario, focusing on hard samples does not harm the performance of easy samples, and thus benefits the classifier. Figure 1 shows the change of fairness (equalized odds) under different cutoff value. Suppose we have a large unlabeled training set of sizeN and a small labeled validation set { xvalj,yvalj,1 j M} with M N. In each training step, we sample a small mini-batch of size n(n < N) from training set and perform random augmentation twice to obtain a subset { xi,1 i 2n} and we update the contrastive encoderf with parameterθ. During validation, we freeze the contrastive encoder and train a downstream linear classifierg with parameterω for classification task.