Local Updates Meet Error Feedback: Communication-Efficient Vertical Minimax Federated Learning

Vertical federated learning (VFL) couples model blocks held by different clients, making block-coordinate methods a natural approach to optimization. In minimax learning, local updates and compressed exchanges introduce errors that interact with the evolving maximization variables. We develop two local block methods: FedBCD-Max combines approximate server-side maximization with client-local descent, whereas FedBCDA performs client-local descent-ascent when the maximization variables are partitioned across clients. We further equip both methods with bidirectional error feedback to correct uplink and downlink compression errors. We establish the linear convergence guarantee for strongly convex-strongly concave problems, and an $O(T^{-1})$ convergence rate for nonconvex-strongly concave problems. The analysis jointly controls local trajectories, maximization error, and two coupled compression memories. It characterizes admissible effective step sizes and preserves the uncompressed accuracy dependence with an explicit ${\omega }^{-2}$ factor in the leading communication-round bounds, where $\omega$ is the compressor contraction parameter. Experiments on fair vertical classification and vertical adversarial learning illustrate the outperformance of our proposed algorithms, and further show that error feedback mitigates the degradation caused by direct compression.