  The best arm identification problem (BEST-1-ARM) is the most basic pure
exploration problem in stochastic multi-armed bandits. The problem has a long
history and attracted significant attention for the last decade. However, we do
not yet have a complete understanding of the optimal sample complexity of the
problem: The state-of-the-art algorithms achieve a sample complexity of
$O(\sum_{i=2}^{n} \Delta_{i}^{-2}(\ln\delta^{-1} + \ln\ln\Delta_i^{-1}))$
($\Delta_{i}$ is the difference between the largest mean and the $i^{th}$
mean), while the best known lower bound is $\Omega(\sum_{i=2}^{n}
\Delta_{i}^{-2}\ln\delta^{-1})$ for general instances and $\Omega(\Delta^{-2}
\ln\ln \Delta^{-1})$ for the two-arm instances. We propose to study the
instance-wise optimality for the BEST-1-ARM problem. Previous work has proved
that it is impossible to have an instance optimal algorithm for the 2-arm
problem. However, we conjecture that modulo the additive term
$\Omega(\Delta_2^{-2} \ln\ln \Delta_2^{-1})$ (which is an upper bound and worst
case lower bound for the 2-arm problem), there is an instance optimal algorithm
for BEST-1-ARM. Moreover, we introduce a new quantity, called the gap entropy
for a best-arm problem instance, and conjecture that it is the instance-wise
lower bound. Hence, resolving this conjecture would provide a final answer to
the old and basic problem.
