This paper deals with federated learning (FL) in the presence of malicious Byzantine attacks and data heterogeneity. A novel Robust Average Gradient Algorithm (RAGA) is proposed, which leverages the geometric median for aggregation and can freely select the round number for local updating. Different from most existing resilient approaches, which perform convergence analysis based on strongly-convex loss function or homogeneously distributed dataset, we conduct convergence analysis for not only strongly-convex but also non-convex loss function over heterogeneous dataset. According to our theoretical analysis, as long as the fraction of dataset from malicious users is less than half, RAGA can achieve convergence at rate $\mathcal{O}({1}/{T^{2/3- \delta}})$ where $T$ is the iteration number and $\delta \in (0, 2/3)$ for non-convex loss function, and at linear rate for strongly-convex loss function. Moreover, stationary point or global optimal solution is proved to obtainable as data heterogeneity vanishes. Experimental results corroborate the robustness of RAGA to Byzantine attacks and verifies the advantage of RAGA over baselines on convergence performance under various intensity of Byzantine attacks, for heterogeneous dataset.

该研究论文研究了在存在恶意拜占庭攻击和数据异质性的情况下的联邦学习。提出了一种新颖的鲁棒平均梯度算法 (RAGA)，其利用几何中位数进行聚合，并可以自由选择局部更新的轮次。通过对不仅是强凸而且非凸损失函数进行收敛性分析，以及考虑到恶意用户的数据集比例小于一半的情况下，本文论证了RAGA可以实现收敛。此外，证明了随着数据异质性消失，可以得到稳定点或全局最优解。实验结果验证了RAGA在不同强度的拜占庭攻击和异质数据集下的收敛性能优于基准方法。

具有适应性数据异构性的拜占庭弹性联邦学习