SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Note that  $p_{\lambda}(X)$  is a sufficient statistic for  $X$ , considering  $A$  as the parameter (it can be proved by Fisher-Neyman factorization theorem). In other words, two Markov chains hold:  $A - p_{\lambda}(X) - X$  and  $A - X - p_{\lambda}(X)$  for any distribution on  $A$ .
Consider a new set of probability distributions  $P_{\lambda}^{*}\triangleq (\mathcal{L}(p_{\lambda}(\mathrm{X}_1)),\dots ,\mathcal{L}(p_{\lambda}(\mathrm{X}_n)))$  where  $\mathcal{L}(\mathrm{X}_i) = P_i$ . Here  $P_{\lambda}^{*}$  can be viewed as a conditional distribution from  $[n]$  to  $\mathcal{P}(\Delta_n)$  since each  $\mathcal{L}(p_{\lambda}(\mathrm{X}_i))$  is a probability distribution over  $\Delta_{n}$ . The following lemma follows from the sufficiency of  $p_{\lambda}(\mathrm{X})$ .
Lemma 2 (Adaptation of Theorem 3 in Blackwell (1951)). For any  $\mathcal{A}$ ,  $\mathcal{B}(\boldsymbol{P},\mathcal{A}) = \mathcal{B}(\boldsymbol{P}_{\lambda}^{*},\mathcal{A})$ .
Proof. Suppose that  $\pmb{f}^{*}(\pmb{p}) = (a_{1}^{*}(\pmb{p}),\dots ,a_{n}^{*}(\pmb{p}))$  is a decision function for  $(P_{\lambda}^{*},\mathcal{A})$ . Accordingly, we define  $\pmb{f}(x) = (a_1^* (\pmb {p}_\lambda (x)),\dots ,a_n(\pmb {p}_\lambda (x)))$  where the function  $p_\lambda$  is defined in (10). Then it is clear that  $\pmb{f}$  is a decision function for  $(P,\mathcal{A})$ . By the law of unconscious statistician, we have
$$
\int a _ {i} ^ {*} (\boldsymbol {p}) \mathrm {d} P _ {\boldsymbol {\lambda}, i} ^ {*} (\boldsymbol {p}) = \mathbb {E} \left[ a _ {i} ^ {*} \left(\boldsymbol {p} _ {\boldsymbol {\lambda}} \left(\mathbf {X} _ {i}\right)\right) \right] = \int a _ {i} ^ {*} \left(\boldsymbol {p} _ {\boldsymbol {\lambda}} (x)\right) \mathrm {d} P _ {i} (x). \tag {11}
$$
Hence,  $\pmb{v}(\pmb{f}^{*}) = \pmb{v}(\pmb{f})$ , which implies  $\mathcal{B}(P_{\lambda}^{*}, \mathcal{A}) \subseteq \mathcal{B}(P, \mathcal{A})$ . For the other direction, suppose  $\pmb{f}(x) = (a_{1}(x), \dots, a_{n}(x))$  is a decision function for  $(P, \mathcal{A})$ . Let  $\pmb{f}^{*}(\pmb{p}) = (a_{1}^{*}(\pmb{p}), \dots, a_{n}^{*}(\pmb{p}))$  where  $a_{i}^{*}(\pmb{p}) \triangleq \mathbb{E}[a_{i}(\mathrm{X}_{i}) \mid \pmb{p}_{\lambda}(\mathrm{X}_{i}) = \pmb{p}]$ . Since  $\pmb{p}_{\lambda}(\mathrm{X})$  is a sufficient statistics, for any  $i \in [n]$
$$
\mathcal {L} \left(\mathrm {X} _ {i} \mid p _ {\lambda} \left(\mathrm {X} _ {i}\right) = \boldsymbol {p}\right) = \mathcal {L} \left(\mathrm {X} _ {1} \mid p _ {\lambda} \left(\mathrm {X} _ {1}\right) = \boldsymbol {p}\right). \tag {12}
$$
Therefore,  $\pmb{f}^{*}(\pmb{p}) = \mathbb{E}\left[\pmb{f}(\mathrm{X}_{1})|\pmb{p}_{\lambda}(\mathrm{X}_{1}) = \pmb{p}\right]$ . Since  $\mathcal{A}$  is a convex set,  $\pmb{f}^{*}$  is a decision function for  $(\pmb{P}^{*},\mathcal{A})$ . By the law of total expectation, we have
$$
\int a _ {i} ^ {*} (\boldsymbol {p}) \mathrm {d} P _ {\boldsymbol {\lambda}, i} ^ {*} (\boldsymbol {p}) = \int a _ {i} (x) \mathrm {d} P _ {i} (x). \tag {13}
$$
Hence,  $\pmb {v}(\pmb {f}) = \pmb {v}(\pmb{f}^{*})$  , which implies  $\mathcal{B}(\pmb {P},\mathcal{A})\subseteq \mathcal{B}(\pmb{P}_{\lambda}^{*},\mathcal{A})$
For a vector  $\lambda \in \Delta_{n}$  such that  $\lambda > 0$ , the condition distribution  $P_{\mathrm{X|A}}$  induces a weighted standard measure  $P_{\lambda}^{*} \triangleq \mathcal{L}(p_{\lambda}(\bar{\mathrm{X}}))$  where  $\mathcal{L}(\bar{\mathrm{X}}) = \lambda_1 P_1 + \dots + \lambda_n P_n$ .
Theorem 3 (Adaptation of Theorem 4 in Blackwell (1951)). For any two conditional distributions  $P_{\mathrm{X|A}}$  and  $Q_{\mathrm{Y|A}}$ , let  $P_{\lambda}^{*}$  and  $Q_{\lambda}^{*}$  be their weighted standard measures, respectively. Then  $\mathcal{B}(P_{\mathrm{X|A}}, \mathcal{A}) \supseteq \mathcal{B}(Q_{\mathrm{Y|A}}, \mathcal{A})$  for any closed, bounded, convex set  $\mathcal{A}$  if and only if for any continuous convex  $\phi: \Delta_n \to \mathbb{R}$ ,  $\int \phi(\pmb{p}) dP_{\lambda}^{*}(\pmb{p}) \geq \int \phi(\pmb{p}) dQ_{\lambda}^{*}(\pmb{p})$ .
Proof. First, by Lemma 2, we know  $\mathcal{B}(P_{\mathrm{X|A}},\mathcal{A}) = \mathcal{B}(P_{\lambda}^{*},\mathcal{A})$  and  $\mathcal{B}(Q_{\mathrm{Y|A}},\mathcal{A}) = \mathcal{B}(Q_{\lambda}^{*},\mathcal{A})$ . We denote  $\Lambda = \operatorname {diag}(\lambda_1,\dots ,\lambda_n)$ . Consider any  $\mathcal{A} = \operatorname {conv}(a_1,\dots ,a_k)$ . Let
$$
\boldsymbol {f} ^ {*} (\boldsymbol {p}) = \underset {\boldsymbol {a} \in \mathcal {A}} {\operatorname {a r g m i n}} \boldsymbol {p} ^ {T} \boldsymbol {\Lambda} ^ {- 1} \boldsymbol {a}. \tag {14}
$$
Note that  $f^{*}(\pmb{p}) \in \{\pmb{a}_{1}, \dots, \pmb{a}_{k}\}$  since this set contains all the extreme points of  $\mathcal{A}$ .<sup>3</sup> By definition, for any decision function w.r.t.  $(P_{\lambda}^{*}, \mathcal{A})$ , we have
$$
\boldsymbol {p} ^ {T} \boldsymbol {\Lambda} ^ {- 1} \boldsymbol {f} (\boldsymbol {p}) \geq \boldsymbol {p} ^ {T} \boldsymbol {\Lambda} ^ {- 1} \boldsymbol {f} ^ {*} (\boldsymbol {p}), \quad \forall \boldsymbol {p}. \tag {15}
$$
Let  $\pmb{v} = \pmb{v}(\pmb{f})$ . By the same reason with (11), we have
$$
\begin{array}{l} v _ {j} = \int a _ {j} \left(\boldsymbol {p} _ {\boldsymbol {\lambda}} (x)\right) \mathrm {d} P _ {j} (x) (16) \\ = \frac {1}{\lambda_ {j}} \int a _ {j} \left(\boldsymbol {p} _ {\boldsymbol {\lambda}} (x)\right) \frac {\lambda_ {j} \mathrm {d} P _ {j}}{\lambda_ {1} \mathrm {d} P _ {1} + \cdots + \lambda_ {n} \mathrm {d} P _ {n}} (x) \left(\lambda_ {1} \mathrm {d} P _ {1} + \dots + \lambda_ {n} \mathrm {d} P _ {n}\right) (x) (17) \\ = \frac {1}{\lambda_ {j}} \int a _ {j} \left(\boldsymbol {p} _ {\boldsymbol {\lambda}} (x)\right) \left[ \boldsymbol {p} _ {\boldsymbol {\lambda}} (x) \right] _ {j} \left(\lambda_ {1} \mathrm {d} P _ {1} + \dots + \lambda_ {n} \mathrm {d} P _ {n}\right) (x) (18) \\ = \frac {1}{\lambda_ {j}} \mathbb {E} \left[ a _ {j} \left(\boldsymbol {p} _ {\boldsymbol {\lambda}} (\bar {\mathrm {X}})\right) \left[ \boldsymbol {p} _ {\boldsymbol {\lambda}} (\bar {\mathrm {X}}) \right] _ {j} \right] (19) \\ = \frac {1}{\lambda_ {j}} \int a _ {j} (\boldsymbol {p}) p _ {j} \mathrm {d} P _ {\boldsymbol {\lambda}} ^ {*} (\boldsymbol {p}), (20) \\ \end{array}
$$
where the last step is due to the law of unconscious statistician. Therefore,
$$
\begin{array}{l} \sum_ {j = 1} ^ {n} v _ {j} = \int \boldsymbol {p} ^ {T} \boldsymbol {\Lambda} ^ {- 1} \boldsymbol {f} (\boldsymbol {p}) \mathrm {d} P _ {\boldsymbol {\lambda}} ^ {*} (\boldsymbol {p}) (21) \\ \geq \int \boldsymbol {p} ^ {T} \boldsymbol {\Lambda} ^ {- 1} \boldsymbol {f} ^ {*} (\boldsymbol {p}) \mathrm {d} P _ {\lambda} ^ {*} (\boldsymbol {p}) (22) \\ = \int \min  _ {i} \left\{\boldsymbol {p} ^ {T} \boldsymbol {\Lambda} ^ {- 1} \boldsymbol {a} _ {i} \right\} \mathrm {d} P _ {\lambda} ^ {*} (\boldsymbol {p}). (23) \\ \end{array}
$$
The equality is achieved by  $\pmb{v}(\pmb{f}^{*})$ . Hence, for any  $\mathcal{A} = \mathrm{conv}(a_1, \dots, a_k)$
$$
\min  _ {\boldsymbol {v} \in \mathcal {B} \left(P _ {\mathrm {X} | \mathrm {A}}, \mathcal {A}\right)} \sum_ {j = 1} ^ {n} v _ {j} = \int \min  _ {i} \left\{\boldsymbol {a} _ {i} ^ {T} \boldsymbol {\Lambda} ^ {- 1} \boldsymbol {p} \right\} \mathrm {d} P _ {\boldsymbol {\lambda}} ^ {*} (\boldsymbol {p}). \tag {24}
$$
Recall that Theorem 2.(3) in Blackwell (1951) states
$\mathcal{B}(P_{\mathrm{X|A}},\mathcal{A})\supseteq \mathcal{B}(P_{\mathrm{Y|A}},\mathcal{A})$  for every closed, bounded, convex  $\mathcal{A}$
$\Leftrightarrow \min_{\pmb{v}\in \mathcal{B}(P_{\mathrm{X|A}},\mathcal{A})}\sum_{j = 1}^{n}v_{j}\leq \min_{\pmb{v}\in \mathcal{B}(P_{\mathrm{Y|A}},\mathcal{A})}\sum_{j = 1}^{n}v_{j}\quad \text{for every closed, bounded, convex}\mathcal{A}.$
By approximation theory, the second condition can be relaxed to any  $\mathcal{A}$  that is a convex hull of a finite set. By (24), this relaxed condition is equivalent to
$$
\int \phi (\boldsymbol {p}) \mathrm {d} P _ {\boldsymbol {\lambda}} ^ {*} (\boldsymbol {p}) \geq \int \phi (\boldsymbol {p}) \mathrm {d} Q _ {\boldsymbol {\lambda}} ^ {*} (\boldsymbol {p}) \tag {25}
$$
for all  $\phi (\pmb {p})$  that are the maximum of finitely many linear functions. By approximation theory again, the above condition is equivalent to the one holding for any continuous convex function  $\phi$
B Omitted Proofs
B.1 Proof of Lemma 3
Proof. Clearly,  $\mathcal{C}$  is a subset of  $\mathcal{T}(C|AC)$ . Let  $\lambda \in (0,1)$  and  $P_{\hat{\mathrm{Y}}_0|\mathrm{S},\mathrm{Y}}, P_{\hat{\mathrm{Y}}_1|\mathrm{S},\mathrm{Y}} \in \mathcal{C}$ . Now we introduce a Bernoulli random variable  $\mathbf{B}$  such that  $\operatorname*{Pr}(\mathbf{B} = 0) = \lambda$ . Finally, we define  $\hat{\mathrm{Y}}_\lambda = \mathrm{B}\hat{\mathrm{Y}}_1 + (1 - \mathrm{B})\hat{\mathrm{Y}}_0$ . By definition, we have  $(\mathbf{S},\mathbf{Y}) - \mathbf{X} - \hat{\mathrm{Y}}_\lambda$  so  $P_{\hat{\mathrm{Y}}_\lambda |\mathrm{S},\mathrm{Y}} \in \mathcal{C}$ . Moreover,
$$
P _ {\hat {\mathrm {Y}} _ {\lambda} | \mathrm {S}, \mathrm {Y}} = \lambda P _ {\hat {\mathrm {Y}} _ {0} | \mathrm {S}, \mathrm {Y}} + (1 - \lambda) P _ {\hat {\mathrm {Y}} _ {1} | \mathrm {S}, \mathrm {Y}}.
$$
3If (14) has multiple optimal solutions, we always choose the one from  $\{\pmb {a}_1,\dots ,\pmb {a}_k\}$
Hence,  $\mathcal{C}$  is convex.
Let  $\lambda \in (0,1)$ . Assume  $\pmb{P}$  and  $\bar{\pmb{P}}$  achieve the maximal values of Proposition 1 under  $(\alpha_{\mathrm{SP}},\alpha_{\mathrm{EO}},\alpha_{\mathrm{OAE}})$  and  $(\bar{\alpha}_{\mathrm{SP}},\bar{\alpha}_{\mathrm{EO}},\bar{\alpha}_{\mathrm{OAE}})$ , respectively. We define  $P_{\lambda} = \lambda P + (1 - \lambda)\bar{P}$ , which satisfies the constraints of Proposition 1 with thresholds  $(\lambda \alpha_{\mathrm{SP}} + (1 - \lambda)\bar{\alpha}_{\mathrm{SP}},\lambda \alpha_{\mathrm{EO}} + (1 - \lambda)\bar{\alpha}_{\mathrm{EO}},\lambda \alpha_{\mathrm{OAE}} + (1 - \lambda)\bar{\alpha}_{\mathrm{OAE}})$ . Finally, since the objective function of Proposition 1 is a linear function, it is equal to  $\lambda$  FairFront  $(\alpha_{\mathrm{SP}},\alpha_{\mathrm{EO}},\alpha_{\mathrm{OAE}}) + (1 - \lambda)$  FairFront  $(\bar{\alpha}_{\mathrm{SP}},\bar{\alpha}_{\mathrm{EO}},\bar{\alpha}_{\mathrm{OAE}})$  under  $P_{\lambda}$ .
B.2 Proof of Theorem 1
Proof. The proof relies on Theorem 3 and Lemma 1. For simplicity, we write the conditional  $P_{\hat{\mathrm{Y}}|\mathrm{S},\mathrm{Y}}$  as its corresponding transition matrix  $\pmb{P}$ . Let  $\pmb{\mu} = (\operatorname*{Pr}(\mathbf{S} = 1,\mathbf{Y} = 1),\dots ,\operatorname*{Pr}(\mathbf{S} = A,\mathbf{Y} = C))$ . The function (10) in our setting can be written as:
$$
\boldsymbol {p} _ {\boldsymbol {\mu}} (\hat {y}) = \left(\frac {\mu_ {1 , 1} P _ {(1 , 1) , \hat {y}}}{\sum_ {s , y} \mu_ {s , y} P _ {(s , y) , \hat {y}}}, \dots , \frac {\mu_ {A , C} P _ {(A , C) , \hat {y}}}{\sum_ {s , y} \mu_ {s , y} P _ {(s , y) , \hat {y}}}\right). \tag {26}
$$
$$
\boldsymbol {p} _ {\boldsymbol {\mu}} (x) = \left(\frac {\mu_ {1 , 1} \mathrm {d} P _ {\mathrm {X} | \mathrm {S} = 1 , \mathrm {Y} = 1}}{\sum_ {s , y} \mu_ {s , y} \mathrm {d} P _ {\mathrm {X} | \mathrm {S} = s , \mathrm {Y} = y}} (x), \dots , \frac {\mu_ {A , C} \mathrm {d} P _ {\mathrm {X} | \mathrm {S} = A , \mathrm {Y} = C}}{\sum_ {s , y} \mu_ {s , y} \mathrm {d} P _ {\mathrm {X} | \mathrm {S} = s , \mathrm {Y} = y}} (x)\right). \tag {27}
$$
Note that  $p_{\mu}(x) = g(x)$  due to Bayes' rule. By Lemma 1, we can rewrite  $\mathcal{C}$  in Definition 2 as
$$
\mathcal {C} = \left\{\boldsymbol {P} \mid P _ {\hat {X} | S, Y} \text {i s m o r e i n f o r m a t i v e t h a n} \boldsymbol {P} \right\}. \tag {28}
$$
By Lemma 1 and Theorem 3, the above set is further equivalent to all transition matrices  $P \in \mathcal{T}(C|AC)$  satisfying
$$
\sum_ {\hat {y} = 1} ^ {C} \phi \left(\frac {\mu_ {1 , 1} P _ {(1 , 1) , \hat {y}}}{\sum_ {s , y} \mu_ {s , y} P _ {(s , y) , \hat {y}}}, \dots , \frac {\mu_ {A , C} P _ {(A , C) , \hat {y}}}{\sum_ {s , y} \mu_ {s , y} P _ {(s , y) , \hat {y}}}\right) \sum_ {s, y} \mu_ {s, y} P _ {(s, y), \hat {y}} \leq \mathbb {E} [ \phi (\boldsymbol {g} (\mathbf {X})) ] \tag {29}
$$
for any function  $\phi : \Delta_{AC} \to \mathbb{R}$  which is the maximum of finitely many linear functions. Now we can write  $\phi(\pmb{p}) = \max_{i \in [k]} \{\pmb{a}_i^T \pmb{p}\}$  —we ignore the bias term because  $\pmb{a}_i^T \pmb{p} + b_i = (\pmb{a}_i + b_i \pmb{1})^T \pmb{p}$ . Then the inequality in (29) can be simplified as
$$
\sum_ {\hat {y} = 1} ^ {C} \max  _ {i \in [ k ]} \left\{\boldsymbol {a} _ {i} ^ {T} \boldsymbol {\Lambda} _ {\mu} \boldsymbol {p} _ {\hat {y}} \right\} \leq \mathbb {E} \left[ \max  _ {i \in [ k ]} \left\{\boldsymbol {a} _ {i} ^ {T} \boldsymbol {g} (\mathrm {X}) \right\} \right], \tag {30}
$$
where  $\pmb{p}_{\hat{y}}$  is the  $\hat{y}$ -th column of  $\pmb{P}$  and  $\Lambda_{\mu} = \mathrm{diag}(\mu_{1,1},\dots ,\mu_{A,C})$ . Finally, we can always normalize the above inequality so that each  $\pmb{a}_i\in [-1,1]^{AC}$ .
B.3 Proof of Theorem 2
Proof. We denote
$$
f (\boldsymbol {P}) \triangleq \sum_ {s = 1} ^ {A} \sum_ {y = 1} ^ {C} \mu_ {s, y} P _ {(s, y), y},
$$
$$
g (\pmb {P}; \pmb {a} _ {1}, \dots , \pmb {a} _ {k}) \triangleq \sum_ {\hat {y} = 1} ^ {C} \max _ {i \in [ k ]} \left\{\pmb {a} _ {i} ^ {T} \pmb {\Lambda} _ {\mu} \pmb {p} _ {\hat {y}} \right\} - \mathbb {E} \left[ \max _ {i \in [ k ]} \{\pmb {a} _ {i} ^ {T} \pmb {g} (\mathbf {X}) \} \right],
$$
$$
\mathcal {F} \triangleq \mathcal {C} _ {k} \cap \left\{\boldsymbol {P} \in \mathcal {T} (C | A C) \mid \mathrm {S P} \leq \alpha_ {\mathrm {s p}}, \mathrm {E O} \leq \alpha_ {\mathrm {e o}}, \mathrm {O A E} \leq \alpha_ {\mathrm {o a e}} \right\}.
$$
Let  $\mathcal{F}^t$  be the constraint set of  $\pmb{P}$  at the  $t$ -th iteration of our algorithm. Note that  $\mathcal{F} \subseteq \mathcal{F}^t$  by definition. If the algorithm stops at the  $t$ -th iteration, then for any  $\{\pmb{a}_i \mid \pmb{a}_i \in [-1, 1]^{AC}, i \in [k]\}$ ,  $\pmb{P}^t$  satisfies
$$
g \left(\boldsymbol {P} ^ {t}; \boldsymbol {a} _ {1}, \dots , \boldsymbol {a} _ {k}\right) \leq 0,
$$
which implies  $P^t\in \mathcal{F}$  . Consequently,
$$
f (\boldsymbol {P} ^ {t}) = \max _ {\boldsymbol {P} \in \mathcal {F} ^ {t}} f (\boldsymbol {P}) \geq \max _ {\boldsymbol {P} \in \mathcal {F}} f (\boldsymbol {P}) \geq f (\boldsymbol {P} ^ {t}).
$$
As a result,  $f(\pmb{P}^t) = \max_{\pmb{P} \in \mathcal{F}} f(\pmb{P})$  so  $\pmb{P}^t$  is an optimal solution of  $\text{FairFront}_k(\alpha_{\text{SP}}, \alpha_{\text{EO}}, \alpha_{\text{OAE}})$ .
If the algorithm never stops, consider any convergent sub-sequence of  $P^t$  that converges to a limit point  $P^* \in \mathcal{T}(C|AC)$ . To simplify our notation, we assume  $P^t \to P^*$  as  $t \to \infty$ . Since  $\{\mathcal{F}^t\}_{t \geq 1}$  is non-increasing and they all contain  $\mathcal{F}$ , there exists a set  $\mathcal{F}^*$  such that
$$
\lim  _ {t \to \infty} \mathcal {F} ^ {t} = \mathcal {F} ^ {*}, \quad \mathcal {F} \subseteq \mathcal {F} ^ {*}.
$$
$$
f (\boldsymbol {P} ^ {*}) = \lim  _ {t \rightarrow \infty} f (\boldsymbol {P} ^ {t}) = \lim  _ {t \rightarrow \infty} \max  _ {\boldsymbol {P} \in \mathcal {F} ^ {t}} f (\boldsymbol {P}) = \max  _ {\boldsymbol {P} \in \mathcal {F} ^ {*}} f (\boldsymbol {P}).
$$
Since  $\mathcal{F} \subseteq \mathcal{F}^*$ , we have
$$
f (\boldsymbol {P} ^ {*}) = \max  _ {\boldsymbol {P} \in \mathcal {F} ^ {*}} f (\boldsymbol {P}) \geq \max  _ {\boldsymbol {P} \in \mathcal {F}} f (\boldsymbol {P}).
$$
If  $P^* \notin \mathcal{F}$ , then there exists a  $(\bar{a}_1, \dots, \bar{a}_k)$ , such that  $g(P^*; \bar{a}_1, \dots, \bar{a}_k) > 0$ . Let  $(a_{1,t}, \dots, a_{k,t})$  be the output of Step 2 at  $t$ -th iteration. Since  $P^* \in \mathcal{F}^t$  for all  $t$ , we have
$$
g \left(\boldsymbol {P} ^ {*}; \boldsymbol {a} _ {1, t}, \dots , \boldsymbol {a} _ {k, t}\right) \leq 0. \tag {31}
$$
By the optimality of  $(\pmb{a}_{1,t},\dots ,\pmb{a}_{k,t})$  , we have
$$
g \left(\boldsymbol {P} ^ {t}; \boldsymbol {a} _ {1, t}, \dots , \boldsymbol {a} _ {k, t}\right) \geq g \left(\boldsymbol {P} ^ {t}; \bar {\boldsymbol {a}} _ {1}, \dots , \bar {\boldsymbol {a}} _ {k}\right). \tag {32}
$$
Suppose that some sub-sequence of  $(\pmb{a}_{1,t},\dots ,\pmb{a}_{k,t})$  converges to a vector  $(\pmb{a}_1^*,\dots ,\pmb{a}_k^*)$ . For the sake of simplicity, we assume  $(\pmb{a}_{1,t},\dots ,\pmb{a}_{k,t})\to (\pmb{a}_1^*,\dots ,\pmb{a}_k^*)$  as  $t\rightarrow \infty$ . On the one hand, taking limit of  $t\to \infty$  on both sides of (32) leads to
$$
g \left(\boldsymbol {P} ^ {*}; \boldsymbol {a} _ {1} ^ {*}, \dots , \boldsymbol {a} _ {k} ^ {*}\right) \geq g \left(\boldsymbol {P} ^ {*}; \bar {\boldsymbol {a}} _ {1}, \dots , \bar {\boldsymbol {a}} _ {k}\right).
$$
On the other hand, taking limit of  $t\to \infty$  on both sides of (31) leads to
$$
g \left(\boldsymbol {P} ^ {*}; \boldsymbol {a} _ {1} ^ {*}, \dots , \boldsymbol {a} _ {k} ^ {*}\right) \leq 0.
$$
$$
0 \geq g \left(\boldsymbol {P} ^ {*}; \boldsymbol {a} _ {1} ^ {*}, \dots , \boldsymbol {a} _ {k} ^ {*}\right) \geq g \left(\boldsymbol {P} ^ {*}; \bar {\boldsymbol {a}} _ {1}, \dots , \bar {\boldsymbol {a}} _ {k}\right) > 0,
$$
which is impossible. Therefore,  $P^{*} \in \mathcal{F}$  and, as a result, we have
$$
f \left(\boldsymbol {P} ^ {*}\right) = \max  _ {\boldsymbol {P} \in \mathcal {F} ^ {*}} f (\boldsymbol {P}) \geq \max  _ {\boldsymbol {P} \in \mathcal {F}} f (\boldsymbol {P}) \geq f \left(\boldsymbol {P} ^ {*}\right) \Rightarrow \max  _ {\boldsymbol {P} \in \mathcal {F}} f (\boldsymbol {P}) = f \left(\boldsymbol {P} ^ {*}\right).
$$
B.4 Additional Results
We establish basic properties of  $\mathcal{C}$  and  $\mathrm{FairFront}(\alpha_{\mathrm{SP}},\alpha_{\mathrm{EO}},\alpha_{\mathrm{OAE}})$  in the following lemma.
Lemma 3.  $\mathcal{C}$  is a convex subset of  $\mathcal{T}(C|AC)$  and FairFront  $(\alpha_{SP},\alpha_{EO},\alpha_{OAE})$  is a concave function w.r.t.  $\alpha_{SP},\alpha_{EO},\alpha_{OAE}$ . Here the constants  $A$  and  $C$  denote the number of protected groups and the number of classes.
Next, we discuss a special case— $\mathbf{X}$  is discrete—under which  $\mathcal{C}$  has a simple characterization.
Remark 3. If  $\mathrm{X}$  is a discrete variable with a finite support  $[D]$ , we can write  $P_{\mathrm{X|S,Y}}$  as a transition matrix  $\Phi \in \mathcal{T}(D|AC)$ . By introducing an auxiliary variable  $M \in \mathcal{T}(C|D)$ , we can write  $P \in \mathcal{C}$  equivalently as linear constraints:  $P = \Phi M$  by using the last condition of Lemma 1. Consequently, Proposition 1 boils down to a linear program. However, this characterization fails to generalize to continuous data because  $\Phi$  and  $M$  will have an infinite dimension; for categorical data, this characterization suffers from the curse of dimensionality since the support size of  $X$  grows exponentially fast w.r.t. the number of features.
C Details on the Experimental Results
C.1 Additional Experiments
In this section, we present additional experimental results to further support our findings. We reproduce our experimental results on the German Credit dataset (Bache and Lichman, 2013) and HSLS (High School Longitudinal Study) dataset (Ingels et al., 2011; Jeong et al., 2022) in Figure 4 and Figure 5. In particular, the HSLS dataset experiment is a multi-group, multi-label experiment. Our observation is consistent with those on the previous two datasets—the fairness-accuracy curves given by SOTA fairness interventions, such as Reduction and FairProjection, are close to the information-theoretic limit.
Figure 4: We reproduce our experiments on the German Credit dataset. Our observation is consistent with those on the previous two datasets—the fairness-accuracy curves given by SOTA fairness interventions, such as Reduction and FairProjection, are close to the information-theoretic limit.
Figure 5: We reproduce our experiments on the HSLS dataset with multi-group and multi-label pre-processing. On the right, we also demonstrate that FairFront can take into account multiple fairness considerations at once. We show how the fairness-accuracy curve changes as we add new types of group fairness constraints (i.e., adding OAE and SP constraints in addition to EO).
C.2 Dataset
Adult. We use sex (female or male) as the group attribute and income ( $>50\mathrm{K}$  or  $<= 50\mathrm{K}$ ) as the target for prediction. We use sex, hours-per-week, education-num, age, marital status, relationship status (husband or wife) as the input features—we include the group attribute as an input feature. We group age into a total of 12 disjoint intervals:  $[0,20), [20,25), \dots, [65,70), [70,\infty)$ ; we group hours-per-week into a total of 14 disjoint intervals:  $[0,10), [10,15), \dots, [65,70), [70,\infty)$ .
COMPAS. We use race (African-American or Caucasian) as the group attribute and is_recid (recid. or no recid.) as the target for prediction. We use race, age, c_fee的程度, sex, priors_count, c Jail_in, c Jail_out as the input features—we include the group attribute as an input feature. We use the last two features by taking their difference to be their length_of Stay. We remove entries where COMPAS case could not be found (is_recid = -1) and entries with inconsistent arrest information. We also binarize sex and remove traffic offenses. We quantize age the same way we do in the Adult dataset and quantize length_of Stay by every 30 days and let 0 be a separate category.
German Credit. We use age (below or above 25 years old) as the group attribute and the credit column, which represents whether the loan was a good decision, as the target for prediction. We use
loan duration in month, credit amount, age, number of existing credits at this bank, sex, credit history, savings, and length of present employment as input features. We include the group attribute age as an input feature. We group credit amount into three disjoint intervals: [0, 5000), [5000, 10000),  $[10000,\infty)$ . We group duration of loan into two categories: under 36 months and over 36 months.
HSLS. We use race as the group attribute and mathematics test score (number of questions answered correctly out of 72) as the target for prediction. This is a multi-group and multi-label dataset. The entire population is grouped by 4 categories: White, Asian, African American, and Others. We seek to predict the mathematics test performance from a set of attributes, including the scale of student's mathematical identity, scale of student's mathematics utility, scale of one's mathematics self-efficacy, parent's education, parent's income, scale of student's sense of school belonging, race, and sex. Note that we include the group attribute as an input feature. We group the target column (estimated number of questions answered correctly) into a total of 5 disjoint intervals:  $[0, 30)$ ,  $[30, 40)$ ,  $[40, 50)$ ,  $[50, 60)$ ,  $[60, \infty)$ ; we group the scale of student's mathematical identity, mathematics utility, mathematics self-efficacy, and sense of school belonging into a total of 4 disjoint intervals, characterized by standard deviations away from the mean:  $(- \infty, -1)$ ,  $[-1, 0)$ ,  $[0, 1)$ ,  $[1, \infty)$ .
C.3 Benchmark
Each benchmark method's hyper-parameter values are provided below. Each point in Figure 2 for Baseline, Eq0dds, CalEq0dds, Reduction, LevEq0pp, and FairProjection is obtained by applying the obtained classifier to 10 different test sets. For the Adult dataset, we use Random Forest with n_estimators=15, min_samples_leaf=3, criterion = log_loss, bootstrap = False as our baseline classifier; for the COMPAS dataset, we use Random Forest with n_estimators = 17 as our baseline classifier. For the German Credit dataset, we use Random Forest with n_estimators=100,min_samples_split =2,min_samples_leaf=1 as our baseline classifier. They are all implemented by Scikit-learn (Pedregosa et al., 2011).
EqOdds (Hardt et al., 2016). We use AIF360 implementation of EqOddsPostprocessing and the default hyper-parameter setup.
CalEqOdds (Pleiss et al., 2017). We use AIF360 implementation of CalibratedEqOddsPostprocessing and the default hyper-parameter setup.