Title: PAC Identification of Many Good Arms in Stochastic Multi-Armed Bandits,
Abstract: We consider the problem of identifying any $k$ out of the best $m$ arms in an
$n$-armed stochastic multi-armed bandit. Framed in the PAC setting, this
particular problem generalises both the problem of `best subset selection' and
that of selecting `one out of the best m' arms [arcsk 2017]. In applications
such as crowd-sourcing and drug-designing, identifying a single good solution
is often not sufficient. Moreover, finding the best subset might be hard due to
the presence of many indistinguishably close solutions. Our generalisation of
identifying exactly $k$ arms out of the best $m$, where $1 \leq k \leq m$,
serves as a more effective alternative. We present a lower bound on the
worst-case sample complexity for general $k$, and a fully sequential PAC
algorithm, \GLUCB, which is more sample-efficient on easy instances. Also,
extending our analysis to infinite-armed bandits, we present a PAC algorithm
that is independent of $n$, which identifies an arm from the best $\rho$
fraction of arms using at most an additive poly-log number of samples than
compared to the lower bound, thereby improving over [arcsk 2017] and
[Aziz+AKA:2018]. The problem of identifying $k > 1$ distinct arms from the best
$\rho$ fraction is not always well-defined; for a special class of this
problem, we present lower and upper bounds. Finally, through a reduction, we
establish a relation between upper bounds for the `one out of the best $\rho$'
problem for infinite instances and the `one out of the best $m$' problem for
finite instances. We conjecture that it is more efficient to solve `small'
finite instances using the latter formulation, rather than going through the
former.