Clustering clients with similar objectives and learning a model per cluster is an intuitive and interpretable approach to personalization in federated learning. However, doing so with provable and optimal guarantees has remained an open challenge. In this work, we formalize personalized federated learning as a stochastic optimization problem where the stochastic gradients on a client may correspond to one of $K$ distributions. In such a setting, we show that using i) a simple thresholding-based clustering algorithm, and ii) local client gradients obtains optimal convergence guarantees. In fact, our rates asymptotically match those obtained if we knew the true underlying clustering of the clients. Furthermore, our algorithms are provably robust in the Byzantine setting where some fraction of the gradients are corrupted.

翻译：将目标相似的用户聚类并为每个聚类学习一个模型，是联邦学习中一种直观且可解释的个性化方法。然而，以可证明且最优的保证实现这一目标仍是一个开放性挑战。在本工作中，我们将个性化联邦学习形式化为一个随机优化问题，其中用户上的随机梯度可能对应于K种分布之一。在此设定下，我们证明：使用(i)一种基于简单阈值聚类的算法，以及(ii)本地用户梯度，即可获得最优收敛性保证。事实上，我们的收敛速率渐近地匹配了已知用户真实底层聚类情况下的结果。此外，我们的算法在存在部分梯度被破坏的拜占庭设定下具有可证明的鲁棒性。