Unimodal Bandits without Smoothness

We consider stochastic bandit problems with a continuum set of arms and where the expected re-ward is a continuous and unimodal function of the arm. No further assumption is made regarding thesmoothness and the structure of the expected reward function. We propose Stochastic Pentachotomy(SP), an algorithm for which we derive finite-time regret upper bounds. In particular, we show that, forany expected reward function μ that behaves as μ(x) = μ(x)− C|x− x| locally around its maxi-mizer x for some ξ, C > 0, the SP algorithm is order-optimal, i.e., its regret scales asO(√T log(T ))when the time horizon T grows large. This regret scaling is achieved without the knowledge of ξ andC. Our algorithm is based on asymptotically optimal sequential statistical tests used to successivelyprune an interval that contains the best arm with high probability. To our knowledge, the SP algo-rithm constitutes the first sequential arm selection rule that achieves a regret scaling as O(√T ) up toa logarithmic factor for non-smooth expected reward functions, as well as for smooth functions withunknown smoothness.