  In this work, we develop linear bandit algorithms that automatically adapt to
different environments. By plugging a novel loss estimator into the
optimization problem that characterizes the instance-optimal strategy, our
first algorithm not only achieves nearly instance-optimal regret in stochastic
environments, but also works in corrupted environments with additional regret
being the amount of corruption, while the state-of-the-art (Li et al., 2019)
achieves neither instance-optimality nor the optimal dependence on the
corruption amount. Moreover, by equipping this algorithm with an adversarial
component and carefully-designed testings, our second algorithm additionally
enjoys minimax-optimal regret in completely adversarial environments, which is
the first of this kind to our knowledge. Finally, all our guarantees hold with
high probability, while existing instance-optimal guarantees only hold in
expectation.
