An algorithm with nearly optimal pseudo-regret for both stochastic and adversarial bandits

We present an algorithm that achieves almost optimal pseudo-regret bounds against adversarial and stochastic bandits. Against adversarial bandits the pseudo-regret is O(K\sqrtn \log n) and against stochastic bandits the pseudo-regret is O(\sum_i (\log n)/\Delta_i). We also show that no algorithm with O(\log n) pseudo-regret against stochastic bandits can achieve \tildeO(\sqrtn) expected regret against adaptive adversarial bandits. This complements previous results of Bubeck and Slivkins (2012) that show \tildeO(\sqrtn) expected adversarial regret with O((\log n)^2) stochastic pseudo-regret.