FedProx is a simple yet effective federated learning method that enables model personalization via regularization. Despite remarkable success in practice, a rigorous analysis of how such a regularization provably improves the statistical accuracy of each client's local model hasn't been fully established. Setting the regularization strength heuristically presents a risk, as an inappropriate choice may even degrade accuracy. This work fills in the gap by analyzing the effect of regularization on statistical accuracy, thereby providing a theoretical guideline for setting the regularization strength for achieving personalization. We prove that by adaptively choosing the regularization strength under different statistical heterogeneity, FedProx can consistently outperform pure local training and achieve a \textit{minimax-optimal} statistical rate. In addition, to shed light on resource allocation, we design an algorithm, provably showing that stronger personalization reduces communication complexity without increasing the computation cost overhead. Finally, our theory is validated on both synthetic and real-world datasets and its generalizability is verified in a non-convex setting.

翻译：FedProx 是一种简单而有效的联邦学习方法，它通过正则化实现模型个性化。尽管在实践中取得了显著成功，但关于这种正则化如何可证明地提升每个客户端本地模型的统计精度的严格分析尚未完全建立。启发式地设置正则化强度存在风险，因为不恰当的选择甚至可能降低精度。本文通过分析正则化对统计精度的影响填补了这一空白，从而为设置实现个性化的正则化强度提供了理论指导。我们证明，通过在不同统计异质性下自适应地选择正则化强度，FedProx 能够持续优于纯本地训练，并达到 \textit{极小极大最优} 的统计速率。此外，为阐明资源分配问题，我们设计了一种算法，可证明更强的个性化能在不增加计算成本开销的情况下降低通信复杂度。最后，我们的理论在合成数据集和真实数据集上均得到了验证，并在非凸设定中验证了其泛化性。