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We study a federated linear bandits model, where $M$ clients communicate witha central server to solve a linear contextual bandits problem with finiteadversarial action sets that may be different across clients. To address theunique challenges of adversarial finite action sets, we propose theFedSupLinUCB algorithm, which extends the principles of SupLinUCB and OFULalgorithms in linear contextual bandits. We prove that FedSupLinUCB achieves atotal regret of $\tilde{O}(\sqrt{d T})$, where $T$ is the total number of armpulls from all clients, and $d$ is the ambient dimension of the linear model.This matches the minimax lower bound and thus is order-optimal (up to polylogterms). We study both asynchronous and synchronous cases and show that thecommunication cost can be controlled as $O(d M^2 \log(d)\log(T))$ and$O(\sqrt{d^3 M^3} \log(d))$, respectively. The FedSupLinUCB design is furtherextended to two scenarios: (1) variance-adaptive, where a total regret of$\tilde{O} (\sqrt{d \sum \nolimits_{t=1}^{T} \sigma_t^2})$ can be achieved with$\sigma_t^2$ being the noise variance of round $t$; and (2) adversarialcorruption, where a total regret of $\tilde{O}(\sqrt{dT} + d C_p)$ can beachieved with $C_p$ being the total corruption budget. Experiment resultscorroborate the theoretical analysis and demonstrate the effectiveness ofFedSupLinUCB on both synthetic and real-world datasets.

Federated Linear Bandits with Finite Adversarial Actions