  The multi-armed bandit formalism has been extensively studied under various
attack models, in which an adversary can modify the reward revealed to the
player. Previous studies focused on scenarios where the attack value either is
bounded at each round or has a vanishing probability of occurrence. These
models do not capture powerful adversaries that can catastrophically perturb
the revealed reward. This paper investigates the attack model where an
adversary attacks with a certain probability at each round, and its attack
value can be arbitrary and unbounded if it attacks. Furthermore, the attack
value does not necessarily follow a statistical distribution. We propose a
novel sample median-based and exploration-aided UCB algorithm (called
med-E-UCB) and a median-based $\epsilon$-greedy algorithm (called
med-$\epsilon$-greedy). Both of these algorithms are provably robust to the
aforementioned attack model. More specifically we show that both algorithms
achieve $\mathcal{O}(\log T)$ pseudo-regret (i.e., the optimal regret without
attacks). We also provide a high probability guarantee of $\mathcal{O}(\log T)$
regret with respect to random rewards and random occurrence of attacks. These
bounds are achieved under arbitrary and unbounded reward perturbation as long
as the attack probability does not exceed a certain constant threshold. We
provide multiple synthetic simulations of the proposed algorithms to verify
these claims and showcase the inability of existing techniques to achieve
sublinear regret. We also provide experimental results of the algorithm
operating in a cognitive radio setting using multiple software-defined radios.
