In this paper, we present a novel analysis of \FedAvg with constant step size, relying on the Markov property of the underlying process. We demonstrate that the global iterates of the algorithm converge to a stationary distribution and analyze its resulting bias and variance relative to the problem's solution. We provide a first-order bias expansion in both homogeneous and heterogeneous settings. Interestingly, this bias decomposes into two distinct components: one that depends solely on stochastic gradient noise and another on client heterogeneity. Finally, we introduce a new algorithm based on the Richardson-Romberg extrapolation technique to mitigate this bias.

翻译：本文针对恒定步长的联邦平均算法提出了一种基于马尔可夫过程特性的全新分析框架。我们证明了该算法的全局迭代序列会收敛至一个平稳分布，并分析了该分布相对于问题解所产生的偏差与方差。我们在同质与异质两种设定下给出了偏差的一阶展开式。值得注意的是，该偏差可分解为两个独立成分：一个仅依赖于随机梯度噪声，另一个则源于客户端异质性。最后，我们提出了一种基于理查森-龙贝格外推技术的新算法以有效抑制此类偏差。