Data similarity assumptions have traditionally been relied upon to understand the convergence behaviors of federated learning methods. Unfortunately, this approach often demands fine-tuning step sizes based on the level of data similarity. When data similarity is low, these small step sizes result in an unacceptably slow convergence speed for federated methods. In this paper, we present a novel and unified framework for analyzing the convergence of federated learning algorithms without the need for data similarity conditions. Our analysis centers on an inequality that captures the influence of step sizes on algorithmic convergence performance. By applying our theorems to well-known federated algorithms, we derive precise expressions for three widely used step size schedules: fixed, diminishing, and step-decay step sizes, which are independent of data similarity conditions. Finally, we conduct comprehensive evaluations of the performance of these federated learning algorithms, employing the proposed step size strategies to train deep neural network models on benchmark datasets under varying data similarity conditions. Our findings demonstrate significant improvements in convergence speed and overall performance, marking a substantial advancement in federated learning research.

翻译：传统上，理解联邦学习方法的收敛行为依赖于数据相似性假设。遗憾的是，这种方法通常要求根据数据相似性程度来微调步长。当数据相似性较低时，这些较小的步长会导致联邦学习方法收敛速度过慢，令人难以接受。本文提出了一种新颖且统一的分析框架，用于分析联邦学习算法的收敛性，而无需数据相似性条件。我们的分析核心在于一个捕捉步长对算法收敛性能影响的不等式。通过将我们的定理应用于知名的联邦算法，我们推导出了三种广泛使用的步长调度策略的精确表达式：固定步长、递减步长和阶梯衰减步长，这些表达式均独立于数据相似性条件。最后，我们在不同的数据相似性条件下，使用所提出的步长策略在基准数据集上训练深度神经网络模型，对这些联邦学习算法的性能进行了全面评估。我们的研究结果表明，收敛速度和整体性能均得到了显著提升，这标志着联邦学习研究取得了实质性进展。