\begin{algorithm}[h!]
\caption{zCDP-NFL}
\label{alg_zCDP_NFL}
\begin{algorithmic}[1]
    \item[] \textbf{Initialization:} ${\bf w}_k^{(0)}=\mathbf{0}$, $\boldsymbol{\gamma}_k^{(0)}=\mathbf{0}$, $\forall k \in \mathcal{K}$
    \item[] \textit{-- Procedure at client $k$ --}
    \item[] \textbf{For} iteration $n = 1, 2, \hdots$:
    \begin{align}
        \label{PrimalUpdate}
        {\bf w}_k^{(n)} &= \arg\min_{{\bf w}_k}   \hat{f_k}({\bf w}_k;\widetilde{\mathcal{V}}^{(n-1)}) + {\bf w}_k^\mathsf{T}\boldsymbol{\gamma}_k^{(n-1)} + \rho\sum_{l\in\mathcal{N}_k}\bigg\|{\bf w}_k-\frac{\widetilde{{\bf w}}_k^{(n-1)}+\widetilde{{\bf w}}_l^{(n-1)}}{2}\bigg\|^2  \\
        \label{NoisePerturbation}
        \widetilde{{\bf w}}_k^{(n)} &=   {\bf w}_k^{(n)} {+} \boldsymbol{\xi}_k^{(n)} \\
        \label{DualUpdate}
        \boldsymbol{\gamma}_k^{(n)} & = \boldsymbol{\gamma}_k^{(n-1)} {+} \rho\sum_{l\in\mathcal{N}_k}\left(\widetilde{{\bf w}}_k^{(n)} {-} \widetilde{{\bf w}}_l^{(n)}\right)
    \end{align}
    \item[] \textbf{End For}
\end{algorithmic} 
\end{algorithm}

\begin{algorithm}
[h!]
\caption{zCDP-NFL}
\begin{algorithmic}
[1]
\item[] \textbf{Initialization:} ${\bf w}_k^{(0)}=\mathbf{0}$, $\boldsymbol{\gamma}_k^{(0)}=\mathbf{0}$, $\forall k \in \mathcal{K}$
\item[] \textit{-- Procedure at client $k$ --}
\item[] \textbf{For} iteration $n = 1, 2, \hdots$:
\begin{align*}
{\bf w}_k^{(n)} &= \arg\min_{{\bf w}_k} \hat{f_k}({\bf w}_k;\widetilde{\mathcal{V}}^{(n-1)}) + {\bf w}_k^\mathsf{T}\boldsymbol{\gamma}_k^{(n-1)} + \rho\sum_{l\in\mathcal{N}_k}\bigg\|{\bf w}_k-\frac{\widetilde{{\bf w}}_k^{(n-1)}+\widetilde{{\bf w}}_l^{(n-1)}}{2}\bigg\|^2 \\
\widetilde{{\bf w}}_k^{(n)} &= {\bf w}_k^{(n)} {+} \boldsymbol{\xi}_k^{(n)} \\
\boldsymbol{\gamma}_k^{(n)} & = \boldsymbol{\gamma}_k^{(n-1)} {+} \rho\sum_{l\in\mathcal{N}_k}\left(\widetilde{{\bf w}}_k^{(n)} {-} \widetilde{{\bf w}}_l^{(n)}\right)
\end{align*}
\item[] \textbf{End For}
\end{algorithmic}
\end{algorithm}
{
    "title": "private networked federated learning for nonsmooth objectives",
    "id": "2306.14012",
    "abstract": "this paper develops a networked federated learning algorithm to solve nonsmooth objective functions. to guarantee the confidentiality of the participants with respect to each other and potential eavesdroppers, we use the zero-concentrated differential privacy notion (zcdp). privacy is achieved by perturbing the outcome of the computation at each client with a variance-decreasing gaussian noise. zcdp allows for better accuracy than the conventional $(\\epsilon, \\delta)$-dp and stronger guarantees than the more recent r\\'enyi-dp by assuming adversaries aggregate all the exchanged messages. the proposed algorithm relies on the distributed alternating direction method of multipliers (admm) and uses the approximation of the augmented lagrangian to handle nonsmooth objective functions. the developed private networked federated learning algorithm has a competitive privacy accuracy trade-off and handles nonsmooth and non-strongly convex problems. we provide complete theoretical proof for the privacy guarantees and the algorithm's convergence to the exact solution. we also prove under additional assumptions that the algorithm converges in $o(1/n)$ admm iterations. finally, we observe the performance of the algorithm in a series of numerical simulations.",
    "categories": "math.oc stat.ml",
    "doi": "",
    "created": "2023-06-24",
    "updated": "2024-02-21",
    "authors": [
        "fran\u00e7ois gauthier",
        "cristiano gratton",
        "naveen k. d. venkategowda",
        "stefan werner"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2306.14012"
}
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