  In this paper, we consider the multi-armed bandit problem with
high-dimensional features. First, we prove a minimax lower bound,
$\mathcal{O}\big((\log d)^{\frac{\alpha+1}{2}}T^{\frac{1-\alpha}{2}}+\log
T\big)$, for the cumulative regret, in terms of horizon $T$, dimension $d$ and
a margin parameter $\alpha\in[0,1]$, which controls the separation between the
optimal and the sub-optimal arms. This new lower bound unifies existing regret
bound results that have different dependencies on T due to the use of different
values of margin parameter $\alpha$ explicitly implied by their assumptions.
Second, we propose a simple and computationally efficient algorithm inspired by
the general Upper Confidence Bound (UCB) strategy that achieves a regret upper
bound matching the lower bound. The proposed algorithm uses a properly centered
$\ell_1$-ball as the confidence set in contrast to the commonly used ellipsoid
confidence set. In addition, the algorithm does not require any forced sampling
step and is thereby adaptive to the practically unknown margin parameter.
Simulations and a real data analysis are conducted to compare the proposed
method with existing ones in the literature.
