Hierarchical federated learning (HFL) is well suited for large-scale wireless and Internet of Things systems, where devices communicate with nearby edge servers before reaching the cloud. In these environments, uplink bandwidth and latency impose strict communication constraints, making aggressive gradient compression essential. One-bit sign-based stochastic gradient descent methods provide an attractive solution in flat federated settings, but their behavior in hierarchical edge--cloud architectures remains insufficiently understood, especially under inter-cluster data heterogeneity. To address this gap, we develop a sign-based HFL framework in which devices transmit binary stochastic-gradient signs to edge servers, edge servers apply majority voting, and the cloud periodically aggregates edge models. Our analysis reveals that inter-cluster heterogeneity induces a persistent bias term in the convergence bound, reflecting the drift of edge models toward local objectives. This term cannot be removed by increasing the number of training rounds or by tuning standard hyperparameters alone. We therefore propose \(\mathtt{DC\text{-}HierSignSGD}\), a drift-corrected sign-based HFL algorithm in which devices apply a cloud-assisted gradient correction before taking the sign. We show that this pre-sign correction mitigates the non-vanishing heterogeneity-induced bias while preserving binary device--edge communication during the repeated local sign-update steps. Experiments under severe inter-cluster heterogeneity demonstrate that \(\mathtt{DC\text{-}HierSignSGD}\) improves the stability and accuracy of sign-based HFL and achieves performance comparable to full-precision hierarchical SGD with substantially lower device--edge communication.

翻译：分层联邦学习（HFL）适用于大规模无线和物联网系统，其中设备在与云端交互前先与附近的边缘服务器通信。在这些环境中，上行带宽和时延对通信施加严格约束，使得激进的梯度压缩成为必需。基于单比特符号的随机梯度下降法在平面联邦设置中提供了有吸引力的解决方案，但其在分层边缘-云架构中的行为，特别是在集群间数据异质性下，仍未被充分理解。为填补这一空白，我们提出了一种基于符号的HFL框架，其中设备向边缘服务器传输二值化的随机梯度符号，边缘服务器采用多数投票机制，云端则周期性地聚合边缘模型。我们的分析表明，集群间异质性会在收敛界中引入一个持续性偏差项，反映边缘模型向局部目标漂移的现象。该偏差无法通过增加训练轮数或单独调整标准超参数来消除。因此，我们提出了\(\mathtt{DC\text{-}HierSignSGD}\)，一种带有漂移校正的基于符号的HFL算法，其中设备在取符号之前先应用云端辅助的梯度校正。我们证明，这种取符号前的校正能减轻非消失的异质性诱导偏差，同时在重复的本地符号更新步骤中保持设备与边缘之间的二值化通信。在严重集群间异质性下的实验表明，\(\mathtt{DC\text{-}HierSignSGD}\)提升了基于符号的HFL的稳定性和准确性，并在显著降低设备-边缘通信量的情况下实现了与全精度分层SGD相当的性能。