  This paper introduces a federated learning framework tailored for online
combinatorial optimization with bandit feedback. In this setting, agents select
subsets of arms, observe noisy rewards for these subsets without accessing
individual arm information, and can cooperate and share information at specific
intervals. Our framework transforms any offline resilient single-agent
$(\alpha-\epsilon)$-approximation algorithm, having a complexity of
$\tilde{\mathcal{O}}(\frac{\psi}{\epsilon^\beta})$, where the logarithm is
omitted, for some function $\psi$ and constant $\beta$, into an online
multi-agent algorithm with $m$ communicating agents and an $\alpha$-regret of
no more than $\tilde{\mathcal{O}}(m^{-\frac{1}{3+\beta}} \psi^\frac{1}{3+\beta}
T^\frac{2+\beta}{3+\beta})$. This approach not only eliminates the $\epsilon$
approximation error but also ensures sublinear growth with respect to the time
horizon $T$ and demonstrates a linear speedup with an increasing number of
communicating agents. Additionally, the algorithm is notably
communication-efficient, requiring only a sublinear number of communication
rounds, quantified as $\tilde{\mathcal{O}}\left(\psi
T^\frac{\beta}{\beta+1}\right)$. Furthermore, the framework has been
successfully applied to online stochastic submodular maximization using various
offline algorithms, yielding the first results for both single-agent and
multi-agent settings and recovering specialized single-agent theoretical
guarantees. We empirically validate our approach to a stochastic data
summarization problem, illustrating the effectiveness of the proposed
framework, even in single-agent scenarios.
