  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.
