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A Distributed First-Order Optimization Method for Strongly Concave-Convex Saddle-Point Problems | IEEE Conference Publication | IEEE Xplore

A Distributed First-Order Optimization Method for Strongly Concave-Convex Saddle-Point Problems


Abstract:

In this paper, we propose a first-order optimization method for solving saddle-point problems when the data is distributed over a strongly connected weight-balanced netwo...Show More

Abstract:

In this paper, we propose a first-order optimization method for solving saddle-point problems when the data is distributed over a strongly connected weight-balanced network of nodes. Our solution is based on the gradient descent-ascent where each node iteratively computes partial gradients of its local cost function to implement the corresponding steps. The proposed method further uses gradient tracking for both de-scent and ascent updates to tackle the local versus global cost gaps. We show that the proposed method converges linearly to the unique saddle-point when the global problem is strongly concave-convex. The numerical experiments il-lustrate the performance comparison of the proposed method with related work for different classes of problems.
Date of Conference: 10-13 December 2023
Date Added to IEEE Xplore: 31 January 2024
ISBN Information:
Conference Location: Herradura, Costa Rica

1. Introduction and Related Work

Many applications related to signal processing, machine learning, and robust optimization naturally take the form of min-max problems [1]–[8]. The objective is to simultaneously minimize and maximize the cost function with respect to specific variables such that we find the point of equilibrium (or the saddle-point). In this paper, we consider the cost function : ; and assume that is convex in x and concave in y, where and . Thus, the objective is to find the saddle-point by minimizing with respect to x and maximizing with respect to y, i.e., \begin{equation*}\min_{\mathbf{x}\in \mathbb{R}^{p_{x}}}\max_{\mathbf{y}\in\mathbb{R}^{p_{y}}}F(\mathbf{x,y}).\end{equation*}

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