  This paper considers contextual bandits with a finite number of arms, where
the contexts are independent and identically distributed $d$-dimensional random
vectors, and the expected rewards are linear in both the arm parameters and
contexts. The LinUCB algorithm, which is near minimax optimal for related
linear bandits, is shown to have a cumulative regret that is suboptimal in both
the dimension $d$ and time horizon $T$, due to its over-exploration. A
truncated version of LinUCB is proposed and termed "Tr-LinUCB", which follows
LinUCB up to a truncation time $S$ and performs pure exploitation afterwards.
The Tr-LinUCB algorithm is shown to achieve $O(d\log(T))$ regret if $S =
Cd\log(T)$ for a sufficiently large constant $C$, and a matching lower bound is
established, which shows the rate optimality of Tr-LinUCB in both $d$ and $T$
under a low dimensional regime. Further, if $S = d\log^{\kappa}(T)$ for some
$\kappa>1$, the loss compared to the optimal is a multiplicative $\log\log(T)$
factor, which does not depend on $d$. This insensitivity to overshooting in
choosing the truncation time of Tr-LinUCB is of practical importance.
