	\begin{algorithmic}[1]
		\For{$k=0,1,2,\ldots$}
		\Statex \textcolor{blue}{\textbf{Local updates}} for all clients: \State  $v^k_{x_m} = \alpha x_m^k + (1 - \alpha) u_{x_m}^k$, \ \ \ \ $v^k_{y_m} = \alpha y_m^k + (1 - \alpha) u_{y_m}^k$ \label{alg1:line2}
		\Statex 
		All clients \textcolor{red}{\textbf{communicate}} to locally compute 
        \State  $\bar v_{x_m}^{k+1} = \sum_{i=1}^M w_{m,i} v^k_{x_i}$, \ \ \ \ $\bar v_{y_m}^{k+1} = \sum_{i=1}^M w_{m,i} v^k_{y_i}$ \label{alg1:line:comm}
        \State and use \textcolor{blue}{\textbf{local}} method $\mathcal{M}$ to find a solution $(\hat x_m^{k+1}, \hat y_m^{k+1})$ of: \label{alg1_line:subproblem}
		\begin{equation}\label{alg1:subproblem}
			 \min_{x_m}  \max_{y_m} \left\{A_m^k(x_m, y_m) = \lambda\langle  \bar v_{x_m}^{k+1},x_m \rangle + \tfrac{1}{2\eta} \|x_m- x_m^k \|^2 + f_m(x_m,y_m) - \lambda\langle \bar v_{y_m}^{k+1},y_m \rangle
			  - \tfrac{1}{2\eta} \|y_m - y_m^k \|^2\right\},
		\end{equation} 
        \Statex such that $\| \nabla A_m^k (\hat x_m^{k+1}, \hat y_m^{k+1})\|^2  \leq \frac{1}{6\eta^2}\left(\|\hat x_m^{k+1} - x^k\|^2 + \|\hat y_m^{k+1} - y^k\|^2\right)$.
        \Statex \textcolor{blue}{\textbf{Local updates}} for all clients: \State \label{alg1_line:update}$x_m^{k+1} = x_m^k - \eta(\lambda \bar v_m^{k+1} + \nabla_{x} f_m(\hat x_m^{k+1}, \hat y_m^{k+1}))$ \State $y_m^{k+1} = y_m^k - \eta(\lambda \bar v_m^{k+1} - \nabla_{y} f_m(\hat x_m^{k+1}, \hat y_m^{k+1}))$        
		\State\label{alg1_line:acc} $u_{x_m}^{k+1} =  v^k_{x_m} + \alpha(\hat x_m^{k+1} - x_m^k), \ \ u_{y_m}^{k+1} =  v^k_{y_m} + \alpha(\hat y_m^{k+1} - y_m^k)$ 
		\EndFor
	\end{algorithmic}

\begin{algorithmic}
[1]
\For{$k=0,1,2,\ldots$}
\Statex \textcolor{blue}{\textbf{Local updates}} for all clients: \State $v^k_{x_m} = \alpha x_m^k + (1 - \alpha) u_{x_m}^k$, \ \ \ \ $v^k_{y_m} = \alpha y_m^k + (1 - \alpha) u_{y_m}^k$   \Statex
All clients \textcolor{red}{\textbf{communicate}} to locally compute
\State $\bar v_{x_m}^{k+1} = \sum_{i=1}^M w_{m,i} v^k_{x_i}$, \ \ \ \ $\bar v_{y_m}^{k+1} = \sum_{i=1}^M w_{m,i} v^k_{y_i}$ \State and use \textcolor{blue}{\textbf{local}} method $\mathcal{M}$ to find a solution $(\hat x_m^{k+1}, \hat y_m^{k+1})$ of:   \begin{equation*}
\min_{x_m} \max_{y_m} \left\{A_m^k(x_m, y_m) = \lambda\langle \bar v_{x_m}^{k+1},x_m \rangle + \tfrac{1}{2\eta} \|x_m- x_m^k \|^2 + f_m(x_m,y_m) - \lambda\langle \bar v_{y_m}^{k+1},y_m \rangle
- \tfrac{1}{2\eta} \|y_m - y_m^k \|^2\right\},
\end{equation*}
\Statex such that $\| \nabla A_m^k (\hat x_m^{k+1}, \hat y_m^{k+1})\|^2 \leq \frac{1}{6\eta^2}\left(\|\hat x_m^{k+1} - x^k\|^2 + \|\hat y_m^{k+1} - y^k\|^2\right)$.
\Statex \textcolor{blue}{\textbf{Local updates}} for all clients: \State $x_m^{k+1} = x_m^k - \eta(\lambda \bar v_m^{k+1} + \nabla_{x} f_m(\hat x_m^{k+1}, \hat y_m^{k+1}))$ \State $y_m^{k+1} = y_m^k - \eta(\lambda \bar v_m^{k+1} - \nabla_{y} f_m(\hat x_m^{k+1}, \hat y_m^{k+1}))$
\State$u_{x_m}^{k+1} = v^k_{x_m} + \alpha(\hat x_m^{k+1} - x_m^k), \ \ u_{y_m}^{k+1} = v^k_{y_m} + \alpha(\hat y_m^{k+1} - y_m^k)$
\EndFor
\end{algorithmic}
{
    "title": "decentralized personalized federated learning for min-max problems",
    "id": "2106.07289",
    "abstract": "personalized federated learning (pfl) has witnessed remarkable advancements, enabling the development of innovative machine learning applications that preserve the privacy of training data. however, existing theoretical research in this field has primarily focused on distributed optimization for minimization problems. this paper is the first to study pfl for saddle point problems encompassing a broader range of optimization problems, that require more than just solving minimization problems. in this work, we consider a recently proposed pfl setting with the mixing objective function, an approach combining the learning of a global model together with locally distributed learners. unlike most previous work, which considered only the centralized setting, we work in a more general and decentralized setup that allows us to design and analyze more practical and federated ways to connect devices to the network. we proposed new algorithms to address this problem and provide a theoretical analysis of the smooth (strongly) convex-(strongly) concave saddle point problems in stochastic and deterministic cases. numerical experiments for bilinear problems and neural networks with adversarial noise demonstrate the effectiveness of the proposed methods.",
    "categories": "cs.lg cs.dc math.oc",
    "doi": "",
    "created": "2021-06-14",
    "updated": "2024-01-24",
    "authors": [
        "ekaterina borodich",
        "aleksandr beznosikov",
        "abdurakhmon sadiev",
        "vadim sushko",
        "nikolay savelyev",
        "martin tak\u00e1\u010d",
        "alexander gasnikov"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2106.07289"
}
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