Title: On the Convergence of FedProx with Extrapolation and Inexact Prox Show Chinese title

Title: 关于带外推和不精确邻近的FedProx的收敛性

Abstract: Enhancing the FedProx federated learning algorithm (Li et al., 2020) with server-side extrapolation, Li et al. (2024a) recently introduced the FedExProx method. Their theoretical analysis, however, relies on the assumption that each client computes a certain proximal operator exactly, which is impractical since this is virtually never possible to do in real settings. In this paper, we investigate the behavior of FedExProx without this exactness assumption in the smooth and globally strongly convex setting. We establish a general convergence result, showing that inexactness leads to convergence to a neighborhood of the solution. Additionally, we demonstrate that, with careful control, the adverse effects of this inexactness can be mitigated. By linking inexactness to biased compression (Beznosikov et al., 2023), we refine our analysis, highlighting robustness of extrapolation to inexact proximal updates. We also examine the local iteration complexity required by each client to achieved the required level of inexactness using various local optimizers. Our theoretical insights are validated through comprehensive numerical experiments. Show Chinese abstract

Abstract: 增强FedProx联邦学习算法（Li等，2020），Li等（2024a）最近引入了FedExProx方法，该方法采用服务器端外推。然而，他们的理论分析依赖于每个客户端精确计算某个近似算子的假设，这在实际情况下几乎不可能实现。在本文中，我们研究了在平滑且全局强凸设置下，不考虑此精确性假设的FedExProx的行为。我们建立了一个一般的收敛结果，表明不精确性会导致收敛到解的邻域。此外，我们证明了通过仔细控制，可以减轻这种不精确性的不利影响。通过将不精确性与有偏压缩（Beznosikov等，2023）联系起来，我们改进了分析，突出了外推对不精确近似更新的鲁棒性。我们还考察了每个客户端使用各种本地优化器达到所需不精确水平所需的局部迭代复杂度。我们的理论见解通过全面的数值实验得到了验证。