  This paper considers two fundamental sequential decision-making problems: the
problem of prediction with expert advice and the multi-armed bandit problem. We
focus on stochastic regimes in which an adversary may corrupt losses, and we
investigate what level of robustness can be achieved against adversarial
corruptions. The main contribution of this paper is to show that optimal
robustness can be expressed by a square-root dependency on the amount of
corruption. More precisely, we show that two classes of algorithms, anytime
Hedge with decreasing learning rate and algorithms with second-order regret
bounds, achieve $O( \frac{\log N}{\Delta} + \sqrt{ \frac{C \log N }{\Delta} }
)$-regret, where $N, \Delta$, and $C$ represent the number of experts, the gap
parameter, and the corruption level, respectively. We further provide a
matching lower bound, which means that this regret bound is tight up to a
constant factor. For the multi-armed bandit problem, we also provide a nearly
tight lower bound up to a logarithmic factor.
