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.

翻译：个性化联邦学习（PFL）已取得显著进展，推动了保护训练数据隐私的创新机器学习应用的发展。然而，该领域现有的理论研究主要集中于最小化问题的分布式优化。本文首次研究了针对鞍点问题的PFL，这类问题涵盖更广泛的优化问题，而不仅仅需要求解最小化问题。在本工作中，我们考虑了一种最近提出的、采用混合目标函数的PFL设置，该方法结合了全局模型的学习与本地分布式学习器。与大多数仅考虑集中式设置的先前工作不同，我们工作在更一般化的去中心化设置中，这使我们能够设计并分析更实用且更符合联邦方式的设备网络连接方法。我们提出了解决该问题的新算法，并针对随机和确定性情况下光滑（强）凸-（强）凹鞍点问题提供了理论分析。针对双线性问题和含对抗噪声的神经网络的数值实验证明了所提出方法的有效性。