  We study the problem of identifying the best arm in a stochastic multi-armed
bandit game. Given a set of $n$ arms indexed from $1$ to $n$, each arm $i$ is
associated with an unknown reward distribution supported on $[0,1]$ with mean
$\theta_i$ and variance $\sigma_i^2$. Assume $\theta_1 > \theta_2 \geq \cdots
\geq\theta_n$. We propose an adaptive algorithm which explores the gaps and
variances of the rewards of the arms and makes future decisions based on the
gathered information using a novel approach called \textit{grouped median
elimination}. The proposed algorithm guarantees to output the best arm with
probability $(1-\delta)$ and uses at most $O \left(\sum_{i = 1}^n
\left(\frac{\sigma_i^2}{\Delta_i^2} + \frac{1}{\Delta_i}\right)(\ln \delta^{-1}
+ \ln \ln \Delta_i^{-1})\right)$ samples, where $\Delta_i$ ($i \geq 2$) denotes
the reward gap between arm $i$ and the best arm and we define $\Delta_1 =
\Delta_2$. This achieves a significant advantage over the variance-independent
algorithms in some favorable scenarios and is the first result that removes the
extra $\ln n$ factor on the best arm compared with the state-of-the-art. We
further show that $\Omega \left( \sum_{i = 1}^n \left(
\frac{\sigma_i^2}{\Delta_i^2} + \frac{1}{\Delta_i} \right) \ln \delta^{-1}
\right)$ samples are necessary for an algorithm to achieve the same goal,
thereby illustrating that our algorithm is optimal up to doubly logarithmic
terms.
