We consider a multi-armed bandit problem specified by a set of Gaussian or Bernoulli distributions endowed with a unimodal structure. Although this problem has been addressed in the literature (Combes and Proutiere, 2014), the state-of-the-art algorithms for such structure make appear a forced-exploration mechanism. We introduce IMED-UB, the first forced-exploration free strategy that exploits the unimodal-structure, by adapting to this setting the Indexed Minimum Empirical Divergence (IMED) strategy introduced by Honda and Takemura (2015). This strategy is proven optimal. We then derive KLUCB-UB, a KLUCB version of IMED-UB, which is also proven optimal. Owing to our proof technique, we are further able to provide a concise finite-time analysis of both strategies in an unified way. Numerical experiments show that both IMED-UB and KLUCB-UB perform similarly in practice and outperform the state-of-the-art algorithms.
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