  We study adversarial attacks on linear stochastic bandits: by manipulating
the rewards, an adversary aims to control the behaviour of the bandit
algorithm. Perhaps surprisingly, we first show that some attack goals can never
be achieved. This is in sharp contrast to context-free stochastic bandits, and
is intrinsically due to the correlation among arms in linear stochastic
bandits. Motivated by this finding, this paper studies the attackability of a
$k$-armed linear bandit environment. We first provide a complete necessity and
sufficiency characterization of attackability based on the geometry of the
arms' context vectors. We then propose a two-stage attack method against LinUCB
and Robust Phase Elimination. The method first asserts whether the given
environment is attackable; and if yes, it poisons the rewards to force the
algorithm to pull a target arm linear times using only a sublinear cost.
Numerical experiments further validate the effectiveness and cost-efficiency of
the proposed attack method.
