  We study bandit algorithms under data poisoning attacks in a bounded reward
setting. We consider a strong attacker model in which the attacker can observe
both the selected actions and their corresponding rewards and can contaminate
the rewards with additive noise. We show that any bandit algorithm with regret
$O(\log T)$ can be forced to suffer a regret $\Omega(T)$ with an expected
amount of contamination $O(\log T)$. This amount of contamination is also
necessary, as we prove that there exists an $O(\log T)$ regret bandit
algorithm, specifically the classical UCB, that requires $\Omega(\log T)$
amount of contamination to suffer regret $\Omega(T)$. To combat such attacks,
our second main contribution is to propose verification based mechanisms, which
use limited verification to access a limited number of uncontaminated rewards.
In particular, for the case of unlimited verifications, we show that with
$O(\log T)$ expected number of verifications, a simple modified version of the
ETC type bandit algorithm can restore the order optimal $O(\log T)$ regret
irrespective of the amount of contamination used by the attacker. We also
provide a UCB-like verification scheme, called Secure-UCB, that also enjoys
full recovery from any attacks, also with $O(\log T)$ expected number of
verifications. To derive a matching lower bound on the number of verifications,
we prove that for any order-optimal bandit algorithm, this number of
verifications $\Omega(\log T)$ is necessary to recover the order-optimal
regret. On the other hand, when the number of verifications is bounded above by
a budget $B$, we propose a novel algorithm, Secure-BARBAR, which provably
achieves $O(\min\{C,T/\sqrt{B} \})$ regret with high probability against weak
attackers where $C$ is the total amount of contamination by the attacker, which
breaks the known $\Omega(C)$ lower bound of the non-verified setting if $C$ is
large.
