Multi-Agent Multi-Armed Bandits with Limited Communication

We consider the problem where $N$ agents collaboratively interact with an instance of a stochastic $K$ arm bandit problem for $K \gg N$. The agents aim to simultaneously minimize the cumulative regret over all the agents for a total of $T$ time steps, the number of communication rounds, and the number of bits in each communication round. We present Limited Communication Collaboration - Upper Confidence Bound (LCC-UCB), a doubling-epoch based algorithm where each agent communicates only after the end of the epoch and shares the index of the best arm it knows. With our algorithm, LCC-UCB, each agent enjoys a regret of $\tilde{O}\left(\sqrt{({K/N}+ N)T}\right)$, communicates for $O(\log T)$ steps and broadcasts $O(\log K)$ bits in each communication step. We extend the work to sparse graphs with maximum degree $KG$, and diameter $D$ and propose LCC-UCB-GRAPH which enjoys a regret bound of $\tilde{O}\left(D\sqrt{(K/N+ KG)DT}\right)$. Finally, we empirically show that the LCC-UCB and the LCC-UCB-GRAPH algorithm perform well and outperform strategies that communicate through a central node