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 expansion of the bias 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：本文提出了一种基于底层过程马尔可夫性的、针对恒定步长FedAvg的新颖分析。我们证明了该算法的全局迭代收敛于一个平稳分布，并分析了其相对于问题解的偏差与方差。我们在同质与异质两种设定下给出了偏差的一阶展开。有趣的是，该偏差可分解为两个独立部分：一部分仅依赖于随机梯度噪声，另一部分则源于客户端异质性。最后，我们引入了一种基于Richardson-Romberg外推技术的新算法以减轻此偏差。