  We introduce the probably approximately correct (PAC) \emph{Battling-Bandit}
problem with the Plackett-Luce (PL) subset choice model--an online learning
framework where at each trial the learner chooses a subset of $k$ arms from a
fixed set of $n$ arms, and subsequently observes a stochastic feedback
indicating preference information of the items in the chosen subset, e.g., the
most preferred item or ranking of the top $m$ most preferred items etc. The
objective is to identify a near-best item in the underlying PL model with high
confidence. This generalizes the well-studied PAC \emph{Dueling-Bandit} problem
over $n$ arms, which aims to recover the \emph{best-arm} from pairwise
preference information, and is known to require $O(\frac{n}{\epsilon^2} \ln
\frac{1}{\delta})$ sample complexity \citep{Busa_pl,Busa_top}. We study the
sample complexity of this problem under various feedback models: (1) Winner of
the subset (WI), and (2) Ranking of top-$m$ items (TR) for $2\le m \le k$. We
show, surprisingly, that with winner information (WI) feedback over subsets of
size $2 \leq k \leq n$, the best achievable sample complexity is still $O\left(
\frac{n}{\epsilon^2} \ln \frac{1}{\delta}\right)$, independent of $k$, and the
same as that in the Dueling Bandit setting ($k=2$). For the more general
top-$m$ ranking (TR) feedback model, we show a significantly smaller lower
bound on sample complexity of $\Omega\bigg( \frac{n}{m\epsilon^2} \ln
\frac{1}{\delta}\bigg)$, which suggests a multiplicative reduction by a factor
${m}$ owing to the additional information revealed from preferences among $m$
items instead of just $1$. We also propose two algorithms for the PAC problem
with the TR feedback model with optimal (upto logarithmic factors) sample
complexity guarantees, establishing the increase in statistical efficiency from
exploiting rank-ordered feedback.
