  We study regret minimization in a stochastic multi-armed bandit setting and
establish a fundamental trade-off between the regret suffered under an
algorithm, and its statistical robustness. Considering broad classes of
underlying arms' distributions, we show that bandit learning algorithms with
logarithmic regret are always inconsistent and that consistent learning
algorithms always suffer a super-logarithmic regret. This result highlights the
inevitable statistical fragility of all `logarithmic regret' bandit algorithms
available in the literature---for instance, if a UCB algorithm designed for
$\sigma$-subGaussian distributions is used in a subGaussian setting with a
mismatched variance parameter, the learning performance could be inconsistent.
Next, we show a positive result: statistically robust and consistent learning
performance is attainable if we allow the regret to be slightly worse than
logarithmic. Specifically, we propose three classes of distribution oblivious
algorithms that achieve an asymptotic regret that is arbitrarily close to
logarithmic.
