Abstract

We consider the best arm identification problem in a fixed budget stochastic multi-armed bandit in which arm means exhibit unimodal structure, i.e., there is only one local maximum. We establish that the probability of misidentifying the optimal arm within a budget of $T$ is lower bounded as $\mathcal{O}\left(\exp\left\{-T/\bar{H}\right\}\right)$, where $\bar{H}$ depends on the sub-optimality gaps of arms in the neighborhood of the optimal arm. % where $\bar{H}\leq 2\Delta^{-2}$. In contrast to the lower bound for the unstructured case, the error exponent in this bound does not depend on the number of arms $K$ and is smaller by a factor $\log K$, which captures the gain achievable by exploiting the unimodal structure. We then develop an algorithm named {\it Fixed Budget Best Arm Unimodal Bandits ( FB-BAUB)} that exploits unimodality to achieve the gain. Specifically, we show that the error probability of \algo{} is upper bounded as $\mathcal{O}\left(\log_2 K\exp\left\{-T\Delta^2\right\}\right)$, where $\Delta$ is the gap between the neighboring arms and $\bar{H}\leq 2\Delta^{-2}$. We demonstrate that \algo{} outperforms the state-of-the-art algorithms through extensive simulations. Moreover, \algo{} is parameter-free and simple to implement.