This work introduces a novel decentralized framework to interpret federated learning (FL) and, consequently, correct the biases introduced by arbitrary client participation and data heterogeneity, which are two typical traits in practical FL. Specifically, we first reformulate the core processes of FedAvg - client participation, local updating, and model aggregation - as stochastic matrix multiplications. This reformulation allows us to interpret FedAvg as a decentralized algorithm. Leveraging the decentralized optimization framework, we are able to provide a concise analysis to quantify the impact of arbitrary client participation and data heterogeneity on FedAvg's convergence point. This insight motivates the development of Federated Optimization with Exact Convergence via Push-pull Strategy (FOCUS), a novel algorithm inspired by the decentralized algorithm that eliminates these biases and achieves exact convergence without requiring the bounded heterogeneity assumption. Furthermore, we theoretically prove that FOCUS exhibits linear convergence (exponential decay) for both strongly convex and non-convex functions satisfying the Polyak-Lojasiewicz condition, regardless of the arbitrary nature of client participation.

翻译：本研究提出了一种新颖的去中心化框架，用于解释联邦学习（FL），并据此修正由任意客户端参与和数据异质性引入的偏差——这两者是实际FL中的典型特征。具体而言，我们首先将FedAvg的核心流程——客户端参与、本地更新和模型聚合——重新表述为随机矩阵乘法。这种重新表述使我们能够将FedAvg解释为一种去中心化算法。借助去中心化优化框架，我们能够提供简洁的分析，以量化任意客户端参与和数据异质性对FedAvg收敛点的影响。这一见解推动了FOCUS（通过推拉策略实现精确收敛的联邦优化）的开发，这是一种受去中心化算法启发的新算法，它消除了这些偏差，并在不需要有界异质性假设的情况下实现了精确收敛。此外，我们从理论上证明，对于满足Polyak-Lojasiewicz条件的强凸和非凸函数，无论客户端参与的任意性如何，FOCUS都表现出线性收敛（指数衰减）特性。