Federated bilevel optimization has received increasing attention in various emerging machine learning and communication applications. Recently, several Hessian-vector-based algorithms have been proposed to solve the federated bilevel optimization problem. However, several important properties in federated learning such as the partial client participation and the linear speedup for convergence (i.e., the convergence rate and complexity are improved linearly with respect to the number of sampled clients) in the presence of non-i.i.d.~datasets, still remain open. In this paper, we fill these gaps by proposing a new federated bilevel algorithm named FedMBO with a novel client sampling scheme in the federated hypergradient estimation. We show that FedMBO achieves a convergence rate of $\mathcal{O}\big(\frac{1}{\sqrt{nK}}+\frac{1}{K}+\frac{\sqrt{n}}{K^{3/2}}\big)$ on non-i.i.d.~datasets, where $n$ is the number of participating clients in each round, and $K$ is the total number of iteration. This is the first theoretical linear speedup result for non-i.i.d.~federated bilevel optimization. Extensive experiments validate our theoretical results and demonstrate the effectiveness of our proposed method.

翻译：摘要：联邦双层优化在各类新兴机器学习和通信应用中日益受到关注。近期，多种基于海森-向量的算法被提出以解决联邦双层优化问题。然而，联邦学习中的若干重要特性——如部分客户端参与、非独立同分布（non-i.i.d.）数据集下的收敛线性加速（即收敛速率与复杂度随采样客户端数量线性提升）——仍属未解问题。本文通过提出一种名为FedMBO的新型联邦双层算法，并引入联邦超梯度估计中独创的客户端采样方案，填补了上述空白。我们证明，在非独立同分布数据集上，FedMBO的收敛速率可达$\mathcal{O}\big(\frac{1}{\sqrt{nK}}+\frac{1}{K}+\frac{\sqrt{n}}{K^{3/2}}\big)$，其中$n$为每轮参与客户端数量，$K$为总迭代次数。这是非独立同分布联邦双层优化领域首个理论线性加速结果。大量实验验证了我们的理论推导，并证实了所提方法的有效性。