SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Reduction (Agarwal et al., 2018). We use AIF360 implementation of ExponentiatedGradientReduction. We vary the allowed fairness constraint violation  $\epsilon \in \{0.001, 0.01, 0.2, 0.5, 1, 2, 5, 10, 15\}$  for Adult dataset and  $\epsilon \in \{0.001, 0.01, 0.2, 0.5, 1, 2, 5, 10, 15\}$  for Adult with missing values. We vary  $\epsilon \in \{0.001, 2, 5, 10, 15, 20, 25, 30, 35, 40\}$  for COMPAS to obtain a fairness-accuracy curve, and  $\epsilon \in \{0.001, 0.1, 0.5, 1, 2, 7, 8, 10, 15, 20, 25, 30\}$  for COMPAS with  $50\%$  missing values in the minority group. We use  $\epsilon \in \{20, 50, 80, 95\}$  for German Credit dataset and  $\epsilon \in \{5, 8, 10, 20, 23\}$  when using Bayes Optimal classifier.
LevEqOpp (Chzhen et al., 2019). We use the Python implementation of LevEqopp from the Github repo in Alghamdi et al. (2022). We follow the same hyperparameters setup as in the original method.
FairProjection Alghamdi et al. (2022). We use the implementation from the Github repo in Alghamdi et al. (2022) and set use_protected = True. We use Random Forest with n_estimators = 17 as the baseline classifier to predict S from  $(\mathrm{X},\mathrm{Y})$ . We set the list of fairness violation tolerance to be  $\{0.07,0.075,0.08,0.085,0.09,0.095,0.1,0.5,0.75,1.0\}$  for Adult dataset and  $\{0.02,0.03,0.04,0.05,0.06,0.07,0.08,0.1,0.5,1.0\}$  for COMPAS dataset to obtain a fairness-accuracy curve. We set the list of fairness violation tolerance to be  $\{0.005,0.01,0.02,0.07,0.1,0.15\}$  on the German Credit dataset experiment, and  $\{0.0001,0.001,0.005,0.01,0.015,0.02,0.05\}$  when using a Bayes optimal baseline classifier. For the HSLS dataset, the list of tolerance is  $\{0.06,0.065,0.07,0.08,0.1,0.09,0.15,0.2,0.3\}$ .
Multiclass	Multigroup	Avoid disparate treatment	Multi-constraint	Curve
Hardt et al. (2016)	×	✓	×	×	×
Corbett-Davies et al. (2017)	×	✓	×	×	×
Menon and Williamson (2018)	×	×	✓	×	✓
Chzhen et al. (2019)	×	×	✓	×	×
Yang et al. (2020)	✓	✓	×	✓	✓
Zeng et al. (2022a)	×	✓	×	×	✓
Zeng et al. (2022b)	×	✓	×	×	×
Our approach	✓	✓	✓	✓	✓
Table 2: Comparison with existing work that investigate the fairness Pareto frontier. Multi-class/multigroup: can handle multiclass classification problems with multiple protected groups; Avoid disparate treatment: not require the classifier to use group attributes as an input variable; Multi-constraint: can handle multiple (group) fairness constraints simultaneously; Curve: produce fairness-accuracy trade-off curves (instead of a single point).
D More on Related Work
We provide a detailed comparison with existing work on fairness Pareto frontier and ML uncertainty in this section.
D.1 Fairness Pareto Frontier
We present in Table 2 a detailed comparison of our approach with previous studies that have investigated the fairness Pareto frontier and fair Bayes optimal classifier. In short, our approach is different from this line of research as it simultaneously combines several important aspects: it is applicable to multiclass classification problems with multiple protected groups; it avoids disparate treatment by not requiring the classifier to use group attributes as an input variable; and it can handle multiple fairness constraints simultaneously and produce fairness-accuracy trade-off curves (instead of a single point).
D.2 Aleatoric and Epistemic Uncertainty
In this paper, we divide algorithmic discrimination into aleatoric and epistemic discrimination. We borrow this notion from ML uncertainty literature (see Hüllermeyer and Waegeman, 2021, for a survey). Here we provide a detailed comparison between them.
In terms of their definitions, epistemic uncertainty arises from a lack of knowledge about the best model, such as the Bayes predictor, while epistemic discrimination results from a lack of knowledge about the optimal "fair" predictive model. On the other hand, aleatoric uncertainty is the irreducible part of uncertainty caused by the random relationship between input features and label, while aleatoric discrimination is due to inherent biases in the data-generating distribution.
In terms of their characterization, epistemic uncertainty can in principle be reduced by including additional information (e.g., more data); epistemic discrimination can be reduced in a similar approach since a data scientist can choose a more effective fairness-intervention algorithm with access to more information.
Finally, in the infinite sample regime, a consistent learner will be able to remove all epistemic uncertainty, assuming the model class is large enough and there are no computational constraints. Analogously, we demonstrate in Figure 1 that when the underlying distribution is known, SOTA fairness interventions are able to eliminate epistemic discrimination as their fairness-accuracy curves are close to the fair front.
E More on Future Work
In this paper, we present an upper bound estimate for FairFront in Algorithm 1. It is important to note that this estimate may be subjected to errors originating from various sources. These include (i)
the approximation error of the function  $g$ , (ii) estimation errors from computing the expectation in (7) with a finite dataset, and (iii) the influence of hyperparameters,  $T$  (number of running iterations of Algorithm 1) and  $k$  (number of segments in the piece-wise linear functions). Regarding the dependence on  $T$ , our Theorem 2 ensures the algorithm's asymptotic convergence as  $T \to \infty$ . However, we have not established a proof for its behavior at a finite  $T$ . Regarding the dependence on  $k$ , we conjecture that  $k = A * C$  should suffice, where  $A$  is the number of protected groups and  $C$  is the number of labels. While Blackwell proved this result for  $k = 2$  in Theorem 10 of Blackwell (1953), an extension of this proof to a general value of  $k$  appears to remain an open problem.
We define aleatoric and epistemic discrimination with respect to the entire population. Investigating their per-instance counterparts and the relationship to individual fairness would be a compelling area of future research. Additionally, a more nuanced analysis of aleatoric and epistemic discrimination is desirable, further breaking them down into fine-grained components. For instance, epistemic discrimination may be attributed to various factors including limited training data, noisy observations of labels or sensitive attributes, and limitations of learning algorithms. Finally, investigating other criteria, such as scalability, generalization, and robustness in evaluating existing fairness interventions is a significant topic for future exploration.