  We study stochastic structured bandits for minimizing regret. The fact that
the popular optimistic algorithms do not achieve the asymptotic
instance-dependent regret optimality (asymptotic optimality for short) has
recently alluded researchers. On the other hand, it is known that one can
achieve bounded regret (i.e., does not grow indefinitely with $n$) in certain
instances. Unfortunately, existing asymptotically optimal algorithms rely on
forced sampling that introduces an $\omega(1)$ term w.r.t. the time horizon $n$
in their regret, failing to adapt to the "easiness" of the instance. In this
paper, we focus on the finite hypothesis case and ask if one can achieve the
asymptotic optimality while enjoying bounded regret whenever possible. We
provide a positive answer by introducing a new algorithm called CRush Optimism
with Pessimism (CROP) that eliminates optimistic hypotheses by pulling the
informative arms indicated by a pessimistic hypothesis. Our finite-time
analysis shows that CROP $(i)$ achieves a constant-factor asymptotic optimality
and, thanks to the forced-exploration-free design, $(ii)$ adapts to bounded
regret, and $(iii)$ its regret bound scales not with $K$ but with an effective
number of arms $K_\psi$ that we introduce. We also discuss a problem class
where CROP can be exponentially better than existing algorithms in
\textit{nonasymptotic} regimes. This problem class also reveals a surprising
fact that even a clairvoyant oracle who plays according to the asymptotically
optimal arm pull scheme may suffer a linear worst-case regret.
