A Tight Lower Bound for Non-stochastic Multi-armed Bandits with Expert Advice

We determine the minimax optimal expected regret in the classic non-stochastic multi-armed bandit with expert advice problem, by proving a lower bound that matches the upper bound of Kale (2014). The two bounds determine the minimax optimal expected regret to be $\mathrm{\Theta }\left(\sqrt{TK\log (N\mathrm{/}K)}\right)$, where $K$ is the number of arms, $N$ is the number of experts, and $T$ is the time horizon.