A Multi-Token Coordinate Descent Method for Semi-Decentralized Vertical Federated Learning

Communication efficiency is a major challenge in federated learning (FL). In client-server schemes, the server constitutes a bottleneck, and while decentralized setups spread communications, they do not necessarily reduce them due to slower convergence. We propose Multi-Token Coordinate Descent (MTCD), a communication-efficient algorithm for semi-decentralized vertical federated learning, exploiting both client-server and client-client communications when each client holds a small subset of features. Our multi-token method can be seen as a parallel Markov chain (block) coordinate descent algorithm and it subsumes the client-server and decentralized setups as special cases. We obtain a convergence rate of $\mathcal{O}(1/T)$ for nonconvex objectives when tokens roam over disjoint subsets of clients and for convex objectives when they roam over possibly overlapping subsets. Numerical results show that MTCD improves the state-of-the-art communication efficiency and allows for a tunable amount of parallel communications.