\begin{algorithmic}[1] % The number [1] indicates that lines are numbered

\Statex \textbf{Input:} Classifiers $h_0^*$ and $h_1^*$
\Statex \textbf{Output:} A randomized classifier $h^*_{\text{Fair}}:\mathcal{X}\times\{0,1\}\rightarrow\{0,1\}$

\State Compute $\alpha = \mathbb{P}_{\mu^X_0} (h^*_0(X) = 1 )$ and $\beta = \mathbb{P}_{\mu^X_1}(h^*_1(X) = 1 )$. W.L.O.G. assume $\alpha \geq \beta$
\State For $(x,a)$, randomly sample $s$ from the uniform distribution $U(0,1)$
\State Construct $h^*_{\text{Fair}}$ as follows:

\Statex $h_{\text {Fair }}^*(x, a):= \begin{cases}
a=0: & \begin{cases}0 & \text{if } h_0^*(x)=0 \text{ or } h_0^*(x)=1 \text{ and } s>\frac{\alpha+\beta}{2 \alpha} \\
1 & \text{if } h_0^*(x)=1 \text{ and } s \leq \frac{\alpha+\beta}{2 \alpha}\end{cases} \\
a=1: & \begin{cases}0 & \text{if } h_1^*(x)=0 \text{ and } s>\frac{\alpha-\beta}{2(1-\beta)} \\
1 & \text{if } h_1^*(x)=1 \text{ or } h_1^*(x)=0 \text{ and } s \leq \frac{\alpha-\beta}{2(1-\beta)}\end{cases}
\end{cases}$

\Statex \textbf{return} $h^*_{\text{Fair}}$
\end{algorithmic}

\begin{algorithmic}[1]
% The number [1] indicates that lines are numbered
\Statex \textbf{Input:} Classifiers $h_0^*$ and $h_1^*$
\Statex \textbf{Output:} A randomized classifier $h^*_{\text{Fair}}:\mathcal{X}\times\{0,1\}\rightarrow\{0,1\}$
\State Compute $\alpha = \mathbb{P}_{\mu^X_0} (h^*_0(X) = 1 )$ and $\beta = \mathbb{P}_{\mu^X_1}(h^*_1(X) = 1 )$. W.L.O.G. assume $\alpha \geq \beta$
\State For $(x,a)$, randomly sample $s$ from the uniform distribution $U(0,1)$
\State Construct $h^*_{\text{Fair}}$ as follows:
\Statex $h_{\text {Fair }}^*(x, a):= \begin{cases}
a=0: & \begin{cases}
0 & \text{if } h_0^*(x)=0 \text{ or } h_0^*(x)=1 \text{ and } s>\frac{\alpha+\beta}{2 \alpha} \\
1 & \text{if } h_0^*(x)=1 \text{ and } s \leq \frac{\alpha+\beta}{2 \alpha}\end{cases} \\
a=1: & \begin{cases}
0 & \text{if } h_1^*(x)=0 \text{ and } s>\frac{\alpha-\beta}{2(1-\beta)} \\
1 & \text{if } h_1^*(x)=1 \text{ or } h_1^*(x)=0 \text{ and } s \leq \frac{\alpha-\beta}{2(1-\beta)}\end{cases}
\end{cases}$
\Statex \textbf{return} $h^*_{\text{Fair}}$
\end{algorithmic}
{
    "title": "differentially private fair binary classifications",
    "id": "2402.15603",
    "abstract": "in this work, we investigate binary classification under the constraints of both differential privacy and fairness. we first propose an algorithm based on the decoupling technique for learning a classifier with only fairness guarantee. this algorithm takes in classifiers trained on different demographic groups and generates a single classifier satisfying statistical parity. we then refine this algorithm to incorporate differential privacy. the performance of the final algorithm is rigorously examined in terms of privacy, fairness, and utility guarantees. empirical evaluations conducted on the adult and credit card datasets illustrate that our algorithm outperforms the state-of-the-art in terms of fairness guarantees, while maintaining the same level of privacy and utility.",
    "categories": "cs.lg cs.cr cs.it math.it stat.ml",
    "doi": "",
    "created": "2024-02-23",
    "updated": "",
    "authors": [
        "hrad ghoukasian",
        "shahab asoodeh"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2402.15603"
}
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